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Fixed point theorems and applications in pvector spaces
Fixed Point Theory and Algorithms for Sciences and Engineering volume 2023, Article number: 10 (2023)
Abstract
The goal of this paper is to develop new fixed points for quasi upper semicontinuous setvalued mappings and compact continuous (singlevalued) mappings, and related applications for useful tools in nonlinear analysis by applying the best approximation approach for classes of semiclosed 1set contractive setvalued mappings in locally pconvex and pvector spaces for \(p \in (0, 1]\). In particular, we first develop general fixed point theorems for quasi upper semicontinuous setvalued and singlevalued condensing mappings, which provide answers to the Schauder conjecture in the affirmative way under the setting of locally pconvex spaces and topological vector spaces for \(p \in (0, 1]\); then the best approximation results for quasi upper semicontinuous and 1set contractive setvalued mappings are established, which are used as tools to establish some new fixed points for nonself quasi upper semicontinuous setvalued mappings with either inward or outward set conditions under various boundary situations. The results established in this paper unify or improve corresponding results in the existing literature for nonlinear analysis, and they would be regarded as the continuation of the related work by Yuan (Fixed Point Theory Algorithms Sci. Eng. 2022:20, 2022)–(Fixed Point Theory Algorithms Sci. Eng. 2022:26, 2022) recently.
1 Introduction
It is known that the class of pseminorm spaces (\(0 < p \leq 1\)) is an important generalization of the usual normed spaces with rich topological and geometrical structures, and related studies have received a lot of attention (e.g., see Alghamdi et al. [5], Balachandran [7], Bayoumi [8], Bayoumi et al. [9], Bernuées and Pena [13], Chang et al. [26], Ding [34], Ennassik and Taoudi [38], Ennassik et al. [37], Gal and Goldstein [45], Gholizadeh et al. [46], Jarchow [61], Kalton [62, 63], Kalton et al. [64], Machrafi and Oubbi [82], Park [100], Qiu and Rolewicz [109], Rolewicz [114], Sezer et al. [119], Silva et al. [123], Simons [124], Tabor et al. [127], Tan [128], Wang [131], Xiao and Lu [134], Xiao and Zhu [135], Yuan [142–145], and many others). However, to the best of our knowledge, the corresponding basic tools and associated results in the category of nonlinear functional analysis have not been well developed, thus the goal of this paper is to develop some important tools in nonlinear analysis for semiclosed 1set contractive mappings under the framework of pvector spaces, in particular, in locally pconvex spaces by including nonexpansive setvalued mappings as a special class under uniformly convex Banach spaces or locally convex spaces with Opial condition.
In particular, we first develop the general fixed point theorems for upper semicontinuous (USC) setvalued 1set contractive mappings, which provide answer to Schauder conjecture since 1930s in the affirmative under the general framework of locally pconvex spaces (when \(p=1\) being locally convex spaces), then the best approximation results for upper semicontinuous and 1set contractive mappings are given with various boundary condition, which are used as tools to establish fixed points for nonself setvalued mappings with either inward or outward set conditions; and finally, we give existence results for solutions of Birkhoff–Kellogg problems, the general principle of nonlinear alternative by including Leray–Schauder alternative, and related results as special classes. The results given in this paper do not only include the corresponding results in the existing literature as special cases, but also are expected to be useful for the study of nonlinear problems arising from social science, engineering, applied mathematics, and related topics and areas.
Before discussing the study of best approximations and related nonlinear analysis tools under the framework of pvector spaces, we would like first to share with readers that though most of results in nonlinear analysis are normally highly associated with the convexity hypotheses under the locally convex spaces by including normed spaces, Banach spaces, and metric spaces special classes, it seems that pvector spaces provide some nice properties for pconvex subsets, which would play very important roles for us to describe Birkhoff and Kellogg problems, and related nonlinear problems such as fixed point problem comparing with convexity in topological vector spaces (TVS) for p in \((0, 1)\) (see the properties given by Remark 2.1(1), and Lemma 2.1(ii) in Sect. 2 in detail).
Here, we would also like to recall that the first Birkhoff–Kellogg theorem was proved by Birkhoff and Kellogg [14] in 1922 in discussing the existence of solutions for the equation \(x = \lambda F(x)\), where λ is a real parameter and F is a general nonlinear nonself mapping defined on an open convex subset U of a topological vector space E. Thus the general form of the Birkhoff–Kellogg problem is to find an invariant direction for the nonlinear setvalued mappings F, i.e., to find \(x_{0} \in \overline{U}\) and \(\lambda > 0\) such that \(\lambda x_{0} \in F(x_{0})\).
Since the Birkhoff and Kellogg theorem given by Birkhoff and Kellogg in 1920s, the study on Birkhoff–Kellogg problem has received a lot of attention from scholars. For example, in 1934, one of the fundamental results in nonlinear functional analysis, famously called the Leray–Schauder alternative, by Leray and Schauder [76] was established via topological degree theory, and thereafter, certain other types of Leray–Schauder alternatives were proved using different techniques other than by using the topological degree approach (see the works by Granas and Dugundji [53], Furi and Pera [44] in the Banach space setting and applications to the boundary value problems for ordinary differential equations in noncompact cases, a general class of mappings for nonlinear alternative of Leray–Schauder type in normal spaces, and Birkhoff–Kellogg type theorems for general class mappings in topological vector spaces by Agarwal et al. [1], Agarwal and O’Regan [2, 3], Park [98], and O’Regan [91] (see the related references therein).
In this paper, based on the application of our best approximation as a tool for quasi upper semicontinuous 1set contractive setvalued mappings, we first establish general principles for the existence of solutions for Birkhoff–Kellogg problems and related nonlinear alternatives, which then also allows us to give general existence of Leray–Schauder type and related fixed point theorems for nonself mappings in general vector pspaces, in particular, locally pconvex spaces for \(p \in (0, 1]\). The results established in this paper not only include the corresponding results in the existing literature as special cases, but are also expected to be useful tools for the study of nonlinear problems arising from theory to practice under the framework of pvector spaces. In particular, the work in this paper can be regarded as the continuation of related work established by Yuan [144, 145] recently.
Now we give a brief discussion and background on the best approximation method related to the study of nonlinear analysis.
We all know that the best approximation method is related to fixed points for nonself mappings, which tightly links with the classical Leray–Schauder alternative based on the Leray–Schauder continuation theorem by Leray and Schauder [76], which is a remarkable result in nonlinear analysis; in addition, there exist several continuation theorems, which have many applications in the study of nonlinear functional equations (see O’Regan and Precup [93]). Historically, it seems that the continuation theorem is based on the idea of obtaining a solution of a given equation, starting from one solution for a simpler equation, the essential part of this theorem is the “Leray–Schauder boundary condition”. But indeed, it seems that “continuation method” was initiated by Poincare [107], Bernstein [12]. Certainly, Leray and Schauder [76] in 1934 gave the first abstract formulation of “continuation principle” using the topological degree theory (see also Granas and Dugundji [53], Isac [60], Rothe [115, 116], Zeidler [146]). But in this paper, we will see how the best approximation method could be used for the study of fixed point theorems in pvector space (\(0 < p \leq 1\)), which as a basic tool, will help us to develop the principle of nonlinear alterative, Leray–Schauder alternative, fixed point theorems of Rothe, Petryshyn, Atlman type for setvalued nonself mappings, and nonlinear alternative with different boundary conditions. Moreover, the new results given in this paper are highly expected to become useful tools for the study on optimization, nonlinear programming, variational inequality, complementarity, game theory, mathematical economics, and related other social science area.
It is well known that Fan’s best approximation theorem given by Fan [42] in 1969 acts as a very powerful tool in nonlinear analysis, as discussed by the book of Singh et al. [125] for the study on the fixed point theory and best approximation with the KKMmap principle, among them, the related tools are Rothe type and the principle of Leray–Schauder alterative in topological vector spaces (in short, TVS) and local convex spaces (in short, LCS), which are also comprehensively studied by Chang et al. [27–30], Carbone and Conti [21], Ennassik and Taoudi [38], Ennassik et al. [37], Guo [54], Guo et al. [55], Granas and Dugundji [53], Isac [60], Kirk and Shahzad [68], Liu [81], Park [101], Rothe [115, 116], Shahzad [120–122], Xu [136], Yuan [142–145], Zeidler [146], and the references therein.
Moreover, since the celebrated socalled KKM principle established in 1929 in [70] (see also Mauldin [84]) was based on the celebrated Sperner combinatorial lemma and first applied to a simple proof of the Brouwer fixed point theorem, later it became clear that these three theorems are mutually equivalent and they were regarded as a sort of mathematical trinity (Park [101]). In particular, since Fan extended the classical KKM theorem to infinitedimensional spaces in 1961 (see Fan [41–43]), there have been a number of generalizations and applications in numerous areas of nonlinear analysis and fixed points in TVS and LCS as developed by Browder [15–20] and the related references therein. Among them, Schauder’s fixed point theorem [118] in normed spaces is one of the powerful tools in dealing with nonlinear problems in analysis. Most notably, it has played a major role in the development of fixed point theory and related nonlinear analysis and mathematical theory of partial and differential equations and others.
A generalization of Schauder’s theorem from a normed space to general topological vector spaces is an old conjecture in fixed point theory, which is explained by Problem 54 of the book “The Scottish Book” by Mauldin [84] and stated as Schauder’s conjecture: “Every nonempty compact convex set in a topological vector space has the fixed point property, or in its analytic statement, does a continuous function defined on a compact convex subset of a topological vector space to itself have a fixed point?”
Based on the discussion by Ennassik and Taoudi [38], Cauty [22, 23] tried to solve the Schauder conjecture, and Ennassik and Taoudi [38] gave the positive answer to the Schauder conjecture for singlevalued continuous mappings under the framework of pvector spaces, where \(p \in (0, 1]\). Indeed, from the respective of development on the study of fixed point theory and related topics in nonlinear analysis, a number of works have been contributed by Górniewicz [51], Górniewicz et al. [52], Ennassik et al. [37] by using the pseminorm method under pvector spaces; plus corresponding contributions by Askoura and GodetThobie [6], Chang [25], Chang et al. [27], Chen [32], Dobrowolski [35], Gholizadeh et al. [46], Huang et al. [57], Isac [60], Li [79], Li et al. [78], Liu [81], Mańka [83], Nhu [87], Okon [89], Park [100–102], Reich [110], Smart [126], Weber [132, 133], Xiao and Lu [134], Xiao and Zhu [135], Xu [139], Xu et al. [140], Yuan [142–145], and the related references therein under the general framework of pvector spaces for even nonself setvalued mappings (\(0 < p \leq 1\)).
The goal of this paper is to develop new fixed points for quasi upper semicontinuous setvalued mappings, and related some useful tools for nonlinear analysis by applying the best approximation approach for classes of semiclosed 1set contractive setvalued mappings in locally pconvex or pvector spaces for \(p \in (0, 1]\). In particular, we first develop general fixed point theorems for quasi upper semicontinuous setvalued and singlevalued condensing mappings, which provide answers to the Schauder conjecture in the affirmative way under the setting of locally pconvex (and pvector spaces). Then the best approximation results for quasi upper semicontinuous and 1set contractive setvalued are established, which are used as tools to establish some new fixed points for nonself quasi upper semicontinuous setvalued mappings with either inward or outward set conditions under various situations. These results unify or improve corresponding results in the existing literature for nonlinear analysis. We do wish that these new results such as the best approximation, Birkhoff–Kellogg type, nonlinear alternative, fixed point theorems for nonself setvalued mappings with boundary conditions, Rothe, Petryshyn type, Altman type, Leray–Schedule type, other related nonlinear problems would play important roles for the development of nonlinear analysis of pseminorm spaces for \(0 < p \leq 1\). The results discussed in this paper do not only unify or improve corresponding results in the existing literature for nonlinear analysis, but they can also be regarded as the continuation of (or) related work established by Yuan [144, 145] recently.
The paper consists of eight sections. Section 1 is the introduction. Section 2 describes general concepts for pvector spaces, locally pconvex spaces, and pconvexity for \(p \in (0, 1]\). In Sect. 3, some basic results of the KKM principle related to abstract convex spaces are given. In Sect. 4, as an application of the KKM principle in abstract convex spaces, which include pconvex vector spaces as a special class for \(p \in (0, 1]\), by combining the graph approximation lemma for quasi upper semicontinuous setvalued mappings in locally pconvex spaces, we provide general fixed point theorems for upper semicontinuous selfmappings defined on locally pconvex compact and 1set contractive upper semicontinuous setvalued mappings defined on noncompact pconvex subsets in locally pconvex spaces. In Sect. 5, the general best approximation result for 1set contractive upper semicontinuous mappings is first given under the framework of locally pconvex spaces, which is used as a tool to establish the general existence theorems for fixed points and the principle of nonlinear alternative and solutions for Birkhoff–Kellogg problem, including Leray–Schauder alternative, Rothe type, Altman type associated with various boundary conditions. In Sect. 7, we focus on the study of the general principle for nonlinear alternative for semiclosed contractive setvalued mappings under various boundary conditions. In Sect. 8, we develop fixed points and a related principle of nonlinear alterative for the classes of semiclosed 1set mappings including nonexpansive setvalued mappings as a special class under uniformly convex Banach spaces or locally convex spaces with the Opial condition.
For convenience of our discussion, throughout this paper, all pvector spaces, locally pconvex spaces are assumed to be Hausdorff and p satisfies the condition for \(p \in (0, 1]\) unless specified otherwise. We also denote by \(\mathbb{N}\) the set of all positive integers, i.e., \(\mathbb{N}:=\{1,2, \ldots , \}\). For a set X, the \(2^{X}\) denotes the family of all subsets of X.
2 The basic results of pvector spaces
We now recall some notion and definitions of pconvexities, pvector spaces for Hausdorff topological vector spaces, and locally pconvex spaces, which will be used in what follows (see Jarchow [61], Kalton [62], Rolewicz [114], Bayoumi [8], Gholizadeh et al. [46], or Ennassik and Taoudi [37]).
Definition 2.1
Let \(p \in (0, 1]\). A set A in a vector space X is said to be pconvex if for any \(x, y\in A\) we have \(s x+t y \in A\), whenever \(0\leq s, t\leq 1\) with \(s^{p} + t^{p}=1\); the set A is said to be absolutely pconvex if for any \(x, y\in A\) we have \(s x+ t y \in A\), whenever \(s^{p} + t^{p} \leq 1\). In the case \(p=1\), the concept of the (absolutely) 1convexity is simply the usually (absolutely) convex defined in vector spaces.
Definition 2.2
Let \(p \in (0, 1]\). If A is a subset of a topological vector space X, the closure of A is denoted by A̅, then the pconvex hull of A and its closed pconvex hull are denoted by \(C_{p}(A)\) and \(\overline{C}_{p}(A)\), respectively, which is the smallest pconvex set containing A and the smallest closed pconvex set containing A, respectively.
Definition 2.3
Let \(p \in (0, 1]\), A be pconvex and \(x_{1}, \ldots , x_{n}\in A\), and \(t_{i}\geq 0\), \(\sum_{1}^{n}t_{i}^{\mathrm{p}}=1\). Then \(\sum_{1}^{n}t_{i}x_{i}\) is called a pconvex combination of \(\{x_{i}\}\) for \(i=1, 2, \ldots , n\). If \(\sum_{1}^{n}t_{i}^{\mathrm{p}}\leq 1\), then \(\sum_{1}^{n}t_{i}x_{i}\) is called an absolutely pconvex combination. It is easy to see that \(\sum_{1}^{n}t_{i}x_{i}\in A\) for a pconvex set A.
Definition 2.4
A subset A of a vector space X is called balanced (or circled) if \(\lambda A \subset A\) holds for all scalars λ satisfying \(\lambda  \leq 1\). We say that A is absorbing if for each \(x \in X\) there is a real number \(\rho _{x} >0\) such that \(\lambda x \in A\) for all \(\lambda > 0\) with \(\lambda \leq \rho _{x}\).
By Definition 2.4, it is easy to see that the system of all balanced (circled) subsets of X is easily seen to be closed under the formation of linear combinations, arbitrary unions, and arbitrary intersections. A balanced set A is symmetric, and thus \(A =A\). In particular, every set \(A \subset X\) determines the smallest circled subset Â of X in which it is contained: Â is called the circled hull of A. It is clear that \(\hat{A} =\bigcup_{\lambda \leq 1} \lambda A\) holds so that A is circled if and only if (in short, iff) \(\hat{A} =A\). We use \(\overline{\hat{A}}\) to denote the closed circled hull of \(A\subset X\). In addition, if X is a topological vector space, then we use the \(\operatorname{int}(A)\) to denote the interior of set \(A \subset X\) and if \(0 \in \operatorname{int}(A)\), then \(\operatorname{int}(A)\) is also circled; and we use ∂A to denote the boundary of A in X.
Definition 2.5
Let X be a vector space and \(\mathbb{R}^{+}\) be a nonnegative part of a real line \(\mathbb{R}\). Then a mapping \(P: X\longrightarrow \mathbb{R}^{+}\) is said to be a pseminorm if it satisfies the requirements for (\(0 < p \leq 1\)):

(i)
\(P(x) \geq 0\) for all \(x \in X\);

(ii)
\(P(\lambda x) = \lambda ^{p} P(x)\) for all \(x\in X\) and \(\lambda \in R\);

(iii)
\(P(x + y) \leq P(x) + P(y)\) for all \(x, y \in X\).
An pseminorm P is called a pnorm if \(x=0\) whenever \(P(x)=0\). A topological vector space with a specific pnorm is called a pnormed space. Of course if \(p=1\), then X is the usual normed space. By Lemma 3.2.5 of Balachandra [7], the following proposition gives a necessary and sufficient condition for a pseminorm to be continuous.
Proposition 2.1
Let X be a topological vector space, P be a pseminorm on X and \(V: =\{ x\in X: P(x) < 1\}\). Then P is continuous if and only if \(0 \in \operatorname{int}(V)\), where \(\operatorname{int}(V)\) is the interior of V.
Now, given an pseminorm P, the pseminorm topology determined by P (in short, the ptopology) is the class of unions of open balls \(B(x, \epsilon ): = \{ y \in X: P(yx) < \epsilon \}\) for \(x \in X\) and \(\epsilon > 0\).
We also need the following notion for the socalled pgauge (see Balachandra [7]).
Definition 2.6
Let A be an absorbing subset of a vector space X. For \(x \in X\) and \(0 < p \leq 1\), set \(P_{A}=\inf \{\alpha >0: x \in \alpha ^{\frac{1}{p}}A\}\), then the nonnegative realvalued function \(P_{A}\) is called pgauge (gauge if \(p=1\)). The pgauge of A is also known as the Minkowski pfunctional.
By Proposition 4.1.10 of Balachandra [7], we have the following proposition.
Proposition 2.2
Let A be an absorbing subset of X. Then a pgauge \(P_{A}\) has the following properties:

(i)
\(P_{A}(0)=0\);

(ii)
\(P_{A}(\lambda x) = \lambda ^{p} P_{A}(x)\) if \(\lambda \geq 0\);

(iii)
\(P_{A}(\lambda x) = \lambda ^{p} P_{A}(x)\) for all \(\lambda \in R\) provided A is circled;

(iv)
\(P_{A}(x + y) \leq P_{A}(x) + P_{A}(y)\) for all \(x, y \in A\) provided A is pconvex.
In particular, \(P_{A}\) is a pseminorm if A is absolutely pconvex (and also absorbing).
Remark 2.1
It is worthwhile to note that a 0neighborhood in a topological vector space is absolutely 0neighborhoods, which are also absorbing (see Lemma 2.1.16 of Balachandran [7] or Proposition 2.2.3 of Jarchow [61]), thus it makes sense for us to define a topological vector space E to be a topological pvector space (in short, pvector space) for \(p \in (0, 1]\) by using the concept of the Minkowski pfunctional, as given below.
Definition 2.7
A topological vector space X is said to be a topological pvector space (in short, pvector space) if the base of the origin in X is generated by a family of Minkowski pfunctionals (pgauges) (defined by Definition 2.6), where \(p \in (0, 1]\).
By incorporating Proposition 2.2, it seems that the following is a natural way to lead us to have the definition for a pvector space being locally pconvex, where \(p \in (0, 1]\).
Definition 2.8
A topological vector space X is said to be locally pconvex if the origin in X has a fundamental set of absolutely pconvex 0neighborhoods. This topology can be determined by pseminorms which are defined in the obvious way (see p. 52 of Bayoumi [8], Jarchow [61], or Rolewicz [114]). When \(p=1\), a locally pconvex space X is reduced to being a usual locally convex space.
By Proposition 4.1.12 of Balachandra [7], we also have the following proposition.
Proposition 2.3
Let A be a subset of a vector space X, which is absolutely pconvex (\(0 < p \leq 1\)) and absorbing. Then, we have that

(i)
The pgauge \(P_{A}\) is a pseminorm such that if \(B_{1}: =\{x \in X: P_{A}(x) < 1\}\) and \(\overline{B_{1}}=\{ x \in X: P_{A}(x) \leq 1\}\), then \(B_{1}\subset A \subset \overline{B_{1}}\); in particular, \(\ker P_{A} \subset A\), where \(\ker P_{A}: =\{ x \in X: P_{A}(x) = 0 \}\).

(ii)
\(A = B_{1}\) or \(\overline{B_{1}}\), according to whether A is open or closed in the \(P_{A}\)topology.
Remark 2.2
Let X be a topological vector space, and let U be an open absolutely pconvex neighborhood of the origin, and let ϵ be given. If \(y \in \epsilon ^{\frac{1}{p}} U\), then \(y=\epsilon ^{\frac{1}{p}} u\) for some \(u \in U\) and \(P_{U}(y)= P_{U}(\epsilon ^{\frac{1}{p}} u)= \epsilon P_{U}(u) \leq \epsilon \) (as \(u \in U\) implies that \(P_{U}(u) \leq 1\)). Thus, \(P_{U}\) is continuous at zero, and therefore \(P_{U}\) is continuous everywhere. Moreover, we have \(U=\{ x \in X: P_{U}(x) < 1\}\).
Indeed, since U is open and the scalar multiplication is continuous, we have that for any \(x \in U\) there exists \(0 < t < 1\) such that \(x \in t^{\frac{1}{p}} U\), and so \(P_{U}(x) \leq t < 1\). This shows that \(U \subset \{ x\in X: P_{U}(x) < 1\}\). The conclusion follows by Proposition 2.3.
The following result is a very important and useful result which allows us to make the approximation for convex subsets in topological vector spaces by pconvex subsets in pconvex vector spaces (see Lemma 2.1 of Ennassik and Taoudi [37], Remark 2.1 of Qiu and Rolewicz [109], or Lemma 2.1 of Yuan [144, 145]), thus we omit their proof.
Lemma 2.1
Let A be a subset of a vector space X, then we have:

(i)
If A is rconvex with \(0 < r < 1\), then \(\alpha x \in A\) for any \(x \in A\) and any \(0 < \alpha \leq 1\).

(ii)
If A is convex and \(0 \in A\), then A is sconvex for any \(s \in (0, 1]\).

(iii)
If A is rconvex for some \(r \in (0, 1)\), then A is sconvex for any \(s \in (0, r]\).
Remark 2.3
We would like to point out that results (i) and (iii) of Lemma 2.1 do not hold for \(p = 1\). Indeed, any singleton \(\{x\} \subset X\) is convex in topological vector spaces; but if \(x \neq 0\), then it is not pconvex for any \(p \in (0, 1)\).
We also need the following proposition, which is Proposition 6.7.2 of Jarchow [61].
Proposition 2.4
Let K be compact in a topological vector X and (\(1< p \leq 1\)). Then the closure \(\overline{C}_{p}(K)\) of the pconvex hull and the closure \(\overline{AC}_{p}(K)\) of absolutely pconvex hull of K are compact if and only if \(\overline{C}_{p}(K)\) and \(\overline{AC}_{p}(K)\) are complete, respectively.
We also need the following fact, which is a special case of Lemma 2.4 of Xiao and Zhu [135].
Lemma 2.2
Let C be a (bounded) closed pconvex subset of a topological vector space X and \(0 \in \operatorname{int} C\), where (\(0< p\leq 1\)). For each \(x\in X\), we define an operator by \(r(x):=\frac{x}{\max \{1, (P_{C}(x))^{\frac{1}{p}}\}}\), where \(P_{C}\) is the Minkowski pfunctional of C. Then C is a retract of X and \(r: X \rightarrow C\) is continuous such that:

(1)
if \(x \in C\), then \(r(x)=x\);

(2)
if \(x \notin C\), then \(r(x) \in \partial C\);

(3)
if \(x \notin C\), then the Minkowski pfunctional \(P_{C}(x) >1 \).
Proof
Taking \(s = p\) in Lemma 2.4 of Xiao and Zhu [135], Proposition 2.3, and Remark 2.2, the proof is complete. □
Remark 2.4
As discussed in Remark 2.2, Lemma 2.2 still holds if “the bounded closed pconvex subset C of the pnormed space \((X, \\cdot \_{p})\)” is replaced by “X is a pseminorm vector space and C is a bounded closed absorbing pconvex subset with \(0 \in \operatorname{int} C\) of X”.
For a given pconvex subset C in a given pvector space E with the origin (zero element) \(0 \in \operatorname{int}(C)\) with the pseminorm p (for example, thinking of the pseminorm \(P_{U}\), which is the Minkowski pfunctional of U), we also denote by \(d_{P}(x, C): =\inf \{P_{U}(xy): y \in C\}\) the distance of \(\{x\}\) with the set C in space E for \(p \in (0, 1]\).
For the convenience of our discussion, throughout this paper, we also assume all topological vector spaces and locally pconvex spaces are Hausdorff unless specified for \(p \in (0, 1]\).
3 The KKM principle in abstract convex spaces
As mentioned above, Knaster, Kuratowski, and Mazurkiewicz (in short, KKM) [70] in 1929 obtained the socalled KKM principle (theorem) to give a new proof for the Brouwer fixed point theorem in finite dimensional spaces; and later in 1961, Fan [41] (see also Fan [43]) extended the KKM principle (theorem) to any topological vector spaces and applied it to various results including the Schauder fixed point theorem. Since then there have appeared a large number of works devoted to applications of the KKM principle (theorem). In 1992, such a research field was called the KKM theory for the first time by Park [95]. Then the KKM theory was extended to general abstract convex spaces by Park [99] (see also Park [100] and [101], Mauldin [84], Granas and Dugundji [53], Yuan [143], and the related references therein), which actually include locally pconvex spaces (\(0 < p \leq 1\)) as a special class.
Here we first give some notion and a brief introduction on the abstract convex spaces, which play an important role in the development of the KKM principle and related applications. Once again, for the corresponding comprehensive discussion on the KKM theory and its various applications to nonlinear analysis and related topics, we refer to Agarwal et al. [1], Granas and Dugundji [53], Mauldin [84], Park [101] and [102], Yuan [143], and the related comprehensive references therein.
Let \(\langle D\rangle \) denote the set of all nonempty finite subsets of a given nonempty set D, and let \(2^{D}\) denote the family of all subsets of D. We have the following definition for abstract convex spaces essentially introduced by Park [99].
Definition 3.1
An abstract convex space \((E, D; \Gamma )\) consists of a topological space E, a nonempty set D, and a setvalued mapping \(\Gamma : \langle D\rangle \rightarrow 2^{E}\) with nonempty values \(\Gamma _{A}: = \Gamma (A)\) for each \(A \in \langle D\rangle \), we have Γconvex hull of any \(D' \subset D\) is denoted and defined by \(\mathrm{c}\mathrm{o}_{\Gamma}D': = \cup \{\Gamma _{A} A \in \langle D'\rangle \}\subset E\).
A subset X of E is said to be a Γconvex subset of \((E, D; \Gamma )\) relative to \(D' \) if, for any \(N \in \langle D' \rangle \), we have \(\Gamma _{N} \subseteq X\), that is, \(\mathrm{c}\mathrm{o}_{\Gamma}D'\subset X\). For the convenience of our discussion, in the case \(E=D\), the space \((E, E; \Gamma )\) is simply denoted by \((E; \Gamma )\) unless specified otherwise.
Definition 3.2
Let \((E, D; \Gamma )\) be an abstract convex space and Z be a topological space. For a setvalued mapping (or, say, multivalued mapping) \(F: E \rightarrow 2^{Z}\) with nonempty values, if a setvalue mapping \(G: D\rightarrow 2^{Z}\) satisfies \(F(\Gamma _{A}) \subset G(A):=\bigcup_{y\in A}G(y)\) for all \(A\in \langle D \rangle \), then G is called a KKM mapping with respect to F. Clearly, a classical KKM mapping (see Mauldin [84]) \(G: D\rightarrow 2^{E}\) is just a KKM mapping with respect to the identity map \(1_{E}\) defined above.
Definition 3.3
The partial KKM principle for an abstract convex space \((E, D; \Gamma )\) is that, for any closedvalued KKM mapping \(G: D\rightarrow 2^{E}\), the family \(\{G(y)\}_{y\in D}\) has the finite intersection property. The KKM principle is that the same property also holds for any openvalued KKM mapping.
An abstract convex space is called a (partial) KKM space if it satisfies the (partial) KKM principle (resp.). We now give some known examples of (partial) KKM spaces (see Park [99] and also [100]) as follows.
Definition 3.4
A \(\phi _{A}\)space \((X, D;\{\phi _{A}\}_{A\in \langle D\rangle})\) consists of a topological space X, a nonempty set D, and a family of continuous functions \(\phi _{A}: \Delta _{n}\rightarrow X\) (that is, singular nsimplices) for \(A \in \{D\}\) with \(A=n+1\). By putting \(\Gamma _{A}: = \phi _{A}(\Delta _{n})\) for each \(A\in \langle D \rangle \), the triple \((X, D; \Gamma )\) becomes an abstract convex space.
Remark 3.1
For a \(\phi _{A}\)space \((X, D;\{\phi _{A}\})\), we see that any setvalued mapping \(G: D\rightarrow X\) satisfying \(\phi _{A}(\Delta _{J})\subset G(J)\) for each \(A \in \langle D \rangle \) and \(J \in \langle A \rangle \) is a KKM mapping.
By the definition given above, it is clear that every \(\phi _{A}\)space is a KKM space, thus we have the following fact (see Lemma 1 of Park [100]).
Lemma 3.1
Let \((X, D; \Gamma )\) be a \(\phi _{A}\)space and \(G: D \rightarrow 2^{X}\) be a setvalued (multimap) with nonempty closed [resp. open] values. Suppose that G is a KKM mapping, then \(\{G(a)\}_{a\in D}\) has the finite intersection property.
By Definition 2.7, we recall that a topological vector space is said to be locally pconvex if the origin has a fundamental set of absolutely pconvex 0neighborhoods. This topology can be determined by pseminorms, which are defined in the obvious way (see Jarchow [61] or p. 52 of Bayoumi [8]).
Now we have a new KKM space as follows inducted by the concept of pconvexity (see Lemma 2 of Park [100]).
Lemma 3.2
Suppose that X is a subset of the topological vector space E and \(p \in (0,1]\), and D is a nonempty subset of X such that \(C_{p}(D)\subset X\). Let \(\Gamma _{N}: =C_{p}(N)\) for each \(N\in \langle D\rangle \) for each \(p in (0, 1]\). Then \((X, D; \Gamma )\) is clearly a \(\phi _{A}\)space.
Proof
Since \(C_{p}(D)\subset X\), \(\Gamma _{N}\) is well defined. For each \(N=\{x_{0}, x_{1}, \ldots , x_{n}\}\subset D\), we define \(\phi _{N}: \Delta _{n}\rightarrow \Gamma _{N}\) by \(\sum_{i=0}^{n}t_{i}e_{i}\mapsto \sum_{i=0}^{n}(t_{i})^{ \frac{1}{\mathrm{p}}}x_{i}\) for \(p \in (0, 1]\). Then, clearly, \((X, D; \Gamma )\) is a \(\phi _{A}\)space. This completes the proof. □
4 Fixed point theorems for setvalued and singlevalued mappings in locally pconvex and pvector spaces
In this section, we mainly give fixed point theorems for quasi upper semicontinuous setvalued mappings in locally pconvex spaces and compact continuous singlevalued mappings in pvector spaces. These fixed points will allow us to establish Rothe’s principle, Leray–Schauder alternative in the next section, which would be useful tools in nonlinear analysis for the study of nonlinear problems arising from theory to practice. Here, we first gather together necessary definitions, notations, and known facts needed in this section.
Definition 4.1
Let X and Y be two topological spaces. A setvalued mapping (also called multifunction) \(T: X \longrightarrow 2^{Y}\) is a point to set function such that for each \(x \in X\), \(T(x)\) is a subset of Y. The mapping T is said to be upper semicontinuous (USC) if the subset \(T^{1}(B): = \{ x\in X: T(x) \cap B \neq \emptyset \}\) (equivalently, the set \(\{x \in X: T(x) \subset B\}\)) is closed (equivalently, open) for any closed (resp., open) subset B in Y. The function \(T: X \rightarrow 2^{Y}\) is said to be lower semicontinuous (LSC) if the set \(T^{1}(A)\) is open for any open subset A in Y.
As an application of the KKM principle for general abstract convex spaces, we have the following general existence result for the “approximation” of fixed points for upper and lower semicontinuous setvalued mappings in locally pconvex spaces for \(0 < p \leq 1\) (see also the corresponding results given by Theorem 2.7 of Gholizadeh et al. [46], Theorem 5 of Park [100], and related discussion therein).
Theorem 4.1
Let A be a pconvex compact subset of a locally pconvex space X, where \(0 < p \leq 1\). Suppose that \(T: A \rightarrow 2^{A}\) is lower (resp. upper) semicontinuous with nonempty pconvex values. Then, for any given U, which is a pconvex neighborhood of zero in X, there exists \(x_{U} \in A\) such that \(T(x_{U}) \cap (x_{U} + U) \neq \emptyset \).
Proof
Suppose that U is any given pconvex element of \(\mathfrak{U}\), there is a symmetric open pconvex neighborhood V of zero for which \(\overline{V} + \overline{V} \subset U\) in pconvex neighborhood of zero, we prove the results by two cases: T is lower semicontinuous (LSC) and upper semicontinuous (USC).
Case 1, by assuming that T is lower semicontinuous: As X is a locally pconvex vector space, suppose that \(\mathfrak{U}\) is a family of neighborhoods of 0 in X. For any element U of \(\mathfrak{U}\), there is a symmetric open pconvex neighborhood V of zero for which \(\overline{V} + \overline{V} \subset U\). Since A is compact, there exist \(x_{0}, x_{1}, \ldots , x_{n}\) in A such that \(A \subset \bigcup_{i=0}^{n} (x_{i} + V)\). By using the fact that A is pconvex, we find \(D: =\{b_{0}, b_{2}, \ldots , b_{n}\} \subset A\) for which \(b_{i}  x_{i} \in V\) for all \(i \in \{0, 1, \ldots , n\}\), and we define C by \(C: = C_{p}(D) \subset A\). By the fact that T is LSC, it follows that the subset \(F(b_{i}): = \{c \in C: T(c) \cap (x_{i} +V) = \emptyset \}\) is closed in C (as the set \(x_{i} +V\) is open) for each \(i \in \{0, 1, \ldots , n\}\). For any \(c \in C\), we have \(\emptyset \neq T(c)\cap A \subset T(c)\cap \bigcup_{i=0}^{n}(x_{i}+ V)\), it follows that \(\bigcap_{i=0}^{n} F(b_{i})=\emptyset \). Now, we apply Lemma 3.1 and Lemma 3.2, which implies that there is \(N:= \{b_{i_{0}}, b_{i_{1}}, \ldots , b_{i_{k}}\} \in \langle D \rangle \) and \(x_{U} \in C_{p}(N) \subset A\) for which \(x_{U} \notin F(N)\), and so \(T(x_{u}) \cap (x_{i_{j}} + V) \neq \emptyset \) for all \(j \in \{0, 1, \ldots , k\}\). As \(b_{i}  x_{i} \in V\) and \(\overline{V} + \overline{V} \subset U\), which imply that \(x_{i_{j}} + \overline{V} \subset b_{i_{j}} + U\), which means that \(T(x_{U}) \cap ((b_{i_{j}} + U) \neq \emptyset \), it follows that \(N \subset \{c \in C: T(x_{U}) \cap (c + U)\neq \emptyset \}\). By the fact that the subsets C, \(T(x_{U})\), and U are pconvex, we have that \(x_{U} \in \{c \in C: T(x_{U}) \cap (c+U)\neq \emptyset \}\), which means that \(T( x_{U}) \cap (x_{U} + U ) \neq \emptyset \).
Case 2, by assuming T is upper semicontinuous: We define \(F(b_{i}): = \{c \in C: T(c) \cap (x_{i} + \overline{V}) = \emptyset \}\), which is then open in C (as the subset \(x_{i} + \overline{V}\) is closed) for each \(i=0, 1, \ldots , n\). Then the argument is similar to the proof for the case T is USC, and by applying Lemma 3.1 and Lemma 3.2 again, it follows that there exists \(x_{U} \in A\) such that \(T(x_{U}) \cap (x_{U} + U) \neq \emptyset \). This completes the proof. □
By Theorem 4.1, we have the following Fan–Glicksberg fixed point theorems (Fan [40]) in locally pconvex vector spaces for (\(0 < p \leq 1\)), which also improve or generalize the corresponding results given by Yuan [143], Xiao, and Lu [134], Xiao and Zhu [135] into locally pconvex vector spaces.
Theorem 4.2
Let A be a pconvex compact subset of a locally pconvex vector space X, where \(0 < p \leq 1\). Suppose that \(T: A \rightarrow 2^{A}\) is upper semicontinuous with nonempty pconvex closed values. Then T has one fixed point.
Proof
We denote by \(\mathfrak{U}\) the family of neighborhoods of 0 in X, and \(U \in \mathfrak{U}\), by Theorem 4.1, there exists \(x_{U} \in A\) such that \(T(x_{U}) \cap (x_{U} + U) \neq \emptyset \). Then there exist \(a_{U}, b_{U} \in A\) for which \(b_{U} \in T(a_{U})\) and \(b_{U} \in a_{U} + U\). Now, two nets \(\{a_{U}\}\) and \(\{b_{U}\}\) in \(\mathrm{Graph} (T)\), which is a compact graph of mapping T as A is compact and T is semicontinuous, we may assume that \(a_{U}\) has a subnet converging to a and \(\{b_{U}\}\) has a subnet converging to b. As \(\mathfrak{U}\) is the family of neighborhoods for 0, we should have \(a=b\) (e.g., by the Hausdorff separation property) and \(a=b \in T(b)\) due to the fact that Graph(T) is close (e.g., see Lemma 3.1.1 in p. 40 of Yuan [142]). The proof is complete. □
In the next part of this section, we are going to establish fixed point theorems for quasi upper semicontinuous setvalued mappings in topological vector spaces, which include upper semicontinuous setvalued mappings as a special class. Now we recall the following definitions.
By following Repovs̆ et al. [111] (see also Ewert and Neubrunn [39] and Neubrunn [86]), we recall the following definition for quasi upper semicontinuous (QUSC) mappings, which are a generalization of upper semicontinuous (USC) mappings.
Definition 4.2
Let X and Y be two topological spaces and \(T: X \longrightarrow 2^{Y}\) be a setvalued mapping. The mapping T is said to be quasi upper semicontinuous (QUSC) at \(x \in X\) if, for each of its (\(x'\)) neighborhood \(W(x)\) and for each neighborhood V of the origin in Y, there exists a point \(q(x)\in W(x)\) such that \(x \in \operatorname{Int} T_{1}(T(q(x))+V))\), where \(T_{1}(T(q(x))+V))=\{z \in X: T(z) \subset T(q(x))+V\}\), and the notation \(\operatorname{Int} T_{1}(T(q(x))+V))\) denotes the (topological) interior of the set \(T_{1}(T(q(x))+V))\) in X. The mapping T is said to be quasi supper semicontinuous if it is quasi upper semicontinuous at each point of its domain.
Remark 4.1
It is clear that in Definition 4.2 for QUSC mappings, for each \(x \in X\), by taking \(q(x)\) just being x itself, then it is just the definition for upper semicontinuous mappings given by Definition 3.1. Therefore, a USC mapping is a QUSC one, but a QUSC mapping may not be a USC mapping as shown by the example in p. 1094 due to Repovs̆ et al. [111]. In addition, interested readers can see Ewert and Neubrunn [39] and Neubrunn [86] and the related references therein for the comprehensive study on the quasicontinuity for both single and setvalued mappings and related applications.
For a given set A in a vector space X, we denote by “\(\operatorname{lin}(A)\)” the “linear hull” of A in X, then we also recall the following definition.
Definition 4.3
Let A be a subset of a topological vector space X, and let Y be another topological vector space. We shall say that A can be linearly embedded in Y if there is a linear map \(L: \operatorname{lin}(A) \rightarrow Y\) (not necessarily continuous) whose restriction to A is a homeomorphism.
The following Lemma 4.1 is a significant embedded result for compact convex subsets in topological vector spaces, which is Theorem 1 of Kalton [62], which says that though not every compact convex set in TVS can be linearly imbedded in a locally convex space (e.g., see Roberts [112] and Kalton et al. [64]), but for pconvex sets when \(0 < p <1\), every compact pconvex set in topological vector spaces is considered as a subset of a locally pconvex vector space, hence every such set has sufficiently many pextreme points.
Secondly, by property (ii) of Lemma 2.1, each convex subset containing zero in a topological vector space is always pconvex for \(0 < p \leq 1\). Thus it is possible for us to transfer the problem involving pconvex subsets from topological vector spaces into the locally pconvex vector spaces, which indeed allows us to establish the existence of fixed points for compact singlevalued mappings for noncompact pconvex subsets in locally pconvex spaces and pvector spaces (\(0 < p \leq 1\)) to cover the case when the underlying is just a topological vector space, which provides the answer for Schauder’s conjecture in the affirmative for the general version of compact continuous (singlevalued) mappings in topological vector spaces (following the idea due to Ennassik and Taoudi [38]).
Lemma 4.1
Let K be a compact pconvex subset (\(0 < p < 1\)) of a topological vector space X. Then K can be linearly embedded in a locally pconvex topological vector space.
Proof
It is Theorem 1 of Kalton [62], which completes the proof. □
Remark 4.2
At this point, it is important to note that Lemma 4.1 does not hold for \(p = 1\). By Theorem 9.6 of Kalton et al. [64], it was shown that the spaces \(L_{p} = L_{p}(0, 1)\), where \(0 < p < 1\), contain compact convex sets with no extreme points, which thus cannot be linearly embedded in a locally convex space, see also Roberts [112].
Definition 4.4
We recall that for two given topological spaces X and Y, a setvalued mapping \(T: X \rightarrow 2^{Y}\) is said to be compact if there is a compact subset C in Y such that \(F(X) (=\{y \in F(x), x \in X\})\) is contained in C, i.e., \(F(X) \subset C\). Now we have the following noncompact versions of fixed point theorems for compact singlevalued mappings defined in locally pconvex and topological vector spaces for \(0 < p \leq 1\).
We now have the following result for a continuous singlevalued mapping in locally pconvex spaces or topological vector spaces.
Theorem 4.3
If K is a nonempty closed pconvex subset of either a Hausdorff locally pconvex space or a Hausdorff topological vector space X for \(p \in (0, 1]\), then the compact singlevalued continuous mapping \(T: K \rightarrow K\) has at least a fixed point.
Proof
As T is compact, there exists a compact subset A in K such that \(T(K)\subset A\). Let \(K_{0}: =\overline{C}_{p}(A)\) be the closure of the pconvex hull of the subset A in K. Then \(K_{0}\) is compact pconvex by Proposition 2.4, and the mapping \(T: K_{0} \rightarrow K_{0}\) is continuous.
First, if K is a nonempty closed pconvex subset of a locally pconvex space, where \(p \in (0, 1]\), the conclusion is obtained by considering the selfmapping T on \(K_{0}\) as an application of Theorem 3.1 by Ennassik and Taoudi [38].
Second, if K is a nonempty closed pconvex subset of a Hausdorff topological vector space X, we prove the conclusion by applying Lemma 4.1 in the following two cases.
Case 1: For \(0 < p < 1\), \(K_{0}\) is a nonempty compact pconvex subset of a topological vector space E for \(p \in (0, 1)\), by Lemma 4.1, it follows that \(K_{0}\) can be linearly embedded in a locally pconvex space E, which means that there exists a linear mapping \(L: \operatorname{lin}(K_{0}) \rightarrow E\) whose restriction to \(K_{0}\) is a homeomorphism. Define the mapping \(S: L(K_{0}) \rightarrow L(K_{0})\) by \(S(Lx): = L(Tx)\) for each \(x \in K_{0}\), then this mapping is easily checked to be well defined. The mapping S is continuous since L is a (continuous) homeomorphism and T is continuous on \(K_{0}\). Furthermore, the set \(L(K_{0})\) is compact, being the image of a compact set under a continuous mapping L, and \(L(K_{0})\) is also pconvex since it is the image of a pconvex set under a linear mapping. Then, by the conclusion given in the first part above, T has a fixed point \(x\in K_{0}\). Thus there exists \(x \in K_{0}\) such that \(Lx = S(Lx) = L(Tx)\), thus it implies that \(x = T(x)\) since L is a homeomorphism, which is the fixed point of T.
Case 2: For \(p=1\), taking any point \(x_{0} \in K_{0}\), let \(K'_{0}: =K_{0}  \{x_{0} \}\). Now define a new mapping \(T_{0}: K'_{0} \rightarrow K'_{0}\) by \(T_{0}(x  x_{0}): = T(x)x_{0}\) for each \(xx_{0} \in K'_{0}\). By the fact that now \(K'_{0}\) is compact and sconvex by Lemma 2.1(ii) for some \(s \in (0, 1)\), and \(T_{0}\) is also continuous and has a fixed point in \(K'_{0}\) by the proof in Case 1, so T has a fixed point in \(K_{0}\). The proof is complete. □
Before we establish the main results for the existence of fixed point theorem for quasi upper semicontinuous setvalued mappings in locally pconvex spaces, by following the idea for the proof of Theorem 1.10 by Repovs̆ et al. [111] for the graph approximation of quasi upper semicontinuous setvalued mappings, using the concept of the ``pconvexity” in locally pconvex spaces to replace the usual concept of “convexity” in LCS and TVS (see also related discussions by BenElMechaiekh [10], BenElMechaiekh and Saidi [11], Cellina [24], Kryszewsky [73], Repovs̆ et al. [111], and related applications), we have the following Lemma 4.2, which is then used to establish a general fixed point theorem for upper semicontinuous setvalued mappings in locally pconvex spaces for \(p \in (0, 1]\), which is actually an extension of Theorems 4.2 and 4.3.
We recall that if X and Y are two topological spaces and \(F: X \rightarrow 2^{Y}\) is a setvalued mapping, and we denote by either GraphF or \(\Gamma _{F}\) the graph of F in \(X \times Y\), and α is a given open cover of \(\Gamma _{F}\) in \(X \times Y\), then a (single or setvalued) mapping \(G: X \rightarrow Y\) is said to be an αapproximation (also called αgraph approximation) of F if for each point \(p \in \Gamma _{G}\) there exists a point \(q \in \Gamma _{F}\) such that p and q lie in some common element of the over α; and when G is a singlevalued (continuous), G is also called a selection (continuous) mapping. In the case Y is a topological vector space, if Ω is the open cover of X and V is an open neighborhood of their origin in Y, then \(\Omega \times \{y+V\}_{y \in Y}\) is one open cover of \(X \times Y\), which is denoted by \(\Omega \times V\) as used below. The following result was first given by Chang et al. [26], we provide the proof in detail here for the convenience of selfcontained reading.
Lemma 4.2
Let X be a paracompact space and Y be a topological vector space and \(p \in (0, 1]\). If \(F: X \rightarrow 2^{Y}\) is an upper semicontinuous mapping with pconvex values, then for each open cover Ω of X, and each pconvex open neighborhood V of the origin in Y, there exists a continuous singlevalued \((\Omega \times V)\)approximation for the setvalued mapping F. In particular, the conclusion holds if V is any convex open neighborhood of the origin in Y.
Proof
Let Ω be an open covering of X, and let V be a pconvex open neighborhood of the origin in Y. For each \(x \in X\), fix an arbitrary element \(W(x)\in \Omega \) such that \(x \in W(x)\), then we first claim the following statements:
(1) By the upper semicontinuity (USC) of the mapping F, for each \(x \in X\), there exists an open neighborhood \(U(x) \subset W(x)\) such that \(F(z) \subset F(x)+ V\) for all \(z \in U(x)\);
(2) As X is paracompact, by Theorem 3.5 of Dugundji [36] (see also Theorem 28 in Chap. 5 of Kelly [66]), without loss of generality, let the family \(\{G(x)\}_{x \in X}\) be a covering, which is a star refinement of the covering \(\{U(x)\}_{x \in X}\) of X (and see also the discussion on pp. 167–168 by Dugundji [36] for the concept of the star refinement for a given covering);
(3) Using the upper semicontinuity property again for the mapping F, for each \(x \in X\), there exists an open neighborhood \(U'(x) \subset G(x)\) such that \(F(z) \subset F(x) + V \) for all \(z \in U'(x)\);
(4) Let \(\{e_{\alpha}\}_{\alpha \in A}\) be a locally finite continuous partition of unity inscribed into the covering \(\{U'(x)\}_{x \in X}\) of X, where A is the index set, with \(\Sigma _{\alpha \in A} e_{\alpha (x)} = 1\) for each \(x \in X\); and for each \(\alpha \in A\), we can choose \(x_{\alpha} \in X\) such that \(\operatorname{supp} e_{\alpha} \subset U'(x_{\alpha})\) and one point \(y_{\alpha} \in F(x_{\alpha})\), where \(\operatorname{supp} e_{\alpha}\) is the support of \(e_{\alpha}\) (defined by \(\operatorname{supp} e_{\alpha}:=\overline{\{x\in X: e_{\alpha}(x) \neq 0\}}\)); and
(5) Finally, define a mapping \(f: X \rightarrow Y\) by \(f(x):= \Sigma _{\alpha \in A} e^{\frac{1}{p}}_{\alpha}(x) y_{\alpha}\) for each \(x \in X\), where \(y_{\alpha} \in F(x_{\alpha})\) as given by (4) above, then f is well defined, where the sum is taken over all \(\alpha \in A\) with \(e_{\alpha}(x) > 0\). By (3), it follows that \(\Sigma _{\alpha \in A} (e^{\frac{1}{p}}_{\alpha}(x))^{p} =\Sigma _{ \alpha \in A} e_{\alpha}(x)=1\).
Now we show that f is indeed the desired singlevalued continuous mapping, which is the \((\Omega \times V)\)approximation for the mapping F. Indeed, for any given \(x_{0} \in X\), we have that
for some \(x' \in X\), where \(St \{x_{0}, \{\operatorname{supp} e_{\alpha} \}_{\alpha \in A}\}\) denotes the star of the point \(\{x_{0}\}\) with respect to the family \(\{\operatorname{supp} e_{\alpha} \}_{\alpha \in A}\) and defined by \(St\{x_{0}, \{\operatorname{supp} e_{\alpha} \}_{\alpha \in A}\}:=\cup \{U: x_{0} \in U, U \in \{\operatorname{supp} e_{\alpha} \}_{\alpha \in A} \}\) (see also the corresponding discussion for the notation and concept on p. 349 given by Ageev and Repovs̆ [4]).
By the definition of upper semicontinuity, we have that \(x'\in W(x')\). Hence the points \(x_{0}\) and \(x'\) are Ωclose.
Secondly, if \(e_{\alpha}(x_{0}) > 0\) for \(\alpha \in A\), then \(x_{0} \in G(x_{\alpha})\) and \(x_{\alpha} \in G(x_{\alpha})\) by (3) above. Thus \(x_{\alpha}\in St\{x_{0}, \{G(x)\}_{x\in X} \} \subset U(x')\). Therefore, \(y_{\alpha} \in F(x_{\alpha}) \subset F(x') + V\), i.e., \(y_{\alpha}  v_{\alpha} \in V\) for some \(v_{\alpha} \in F(x')\) for \(\alpha \in A\). But then, for \(v: =\Sigma _{\alpha} e^{\frac{1}{p}}_{\alpha}(x_{0}) v_{\alpha} \in F(x')\) as F is pconvexvalued and we know that \(\Sigma _{\alpha \in A} (e^{\frac{1}{p}}_{\alpha}(x))^{p} =\Sigma _{ \alpha \in A} e_{\alpha}(x)=1\) as shown by (5) above, and \(y_{\alpha}  v_{\alpha} \in V\), too, for \(\alpha \in A\), thus we have that \(f(x_{0}) v =\Sigma e^{\frac{1}{p}}_{\alpha}(x_{0})(y_{\alpha}  v_{ \alpha}) \in V\) as V is pconvex. Hence, the point \((x_{0}, f(x_{0})) \in \operatorname{Graph}(f)\) is \((\Omega \times V)\)close to the point \((x', v) \in \operatorname{Graph}(F)\).
In particular, as each convex neighborhood of the origin in Y is also pconvex for each \(p \in (0, 1]\), the conclusion holds. The proof is complete. □
As an application of Lemma 4.2, we now have the following fixed point theorem for quasi upper semicontinuous setvalued mappings in locally pconvex spaces for \(p \in (0, 1]\), which was first initially discussed by Chang et al. [26].
Theorem 4.4
Let K be a compact sconvex subset of a Hausdorff locally pconvex space X, where \(p, s \in (0, 1]\). If \(T: K \rightarrow 2^{K}\) is a quasi upper continuous setvalued mapping with nonempty closed pconvex values and its graph is closed, then T has a fixed point in K.
Proof
We give the proof by using the graph approximation approach for upper semicontinuous setvalued mappings established in this section above. Let \(\mathfrak{U}\) be the family of absolutely pconvex open neighborhoods of the origin in X. By the fact the family \(\{x + u \}_{x \in K}\) is an open covering of K, we denote the family \(\{x + u \}_{x \in K}\) by Ω. Now, by Lemma 4.2, it follows that there exists one (singlevalued) continuous mapping \(f_{u}: K \rightarrow K\), which is \((\Omega \times u)\)approximation of the mapping T. By Theorem 4.3, \(f_{u}\) has a fixed point \(x_{u} = f_{u}(x_{u})\) in K for each \(u \in \mathfrak{U}\). Note that \((x_{u}, f_{u}(x_{u}))=(x_{u}, x_{u}) \in \operatorname{Graph}(f_{u})\), which is \((\Omega \times u)\)approximation of the Graph(T), and the graph of T is closed due to the assumption, we will prove T has a fixed point \(x^{*}\), which is indeed the limit of some subnet of the family \(\{x_{u}\}_{u \in \mathfrak{U}}\) in K, i.e., \(x^{*} \in T(x^{*})\), by using notations of language in general topology (for related references on the discussion for normed spaces or topological (vector) spaces, see Cellina [24], BenElMechaiekh [10], and Fan [40]).
Indeed, for any given open pconvex member u in \(\mathfrak{U}\), as the set \(\{x + u\}_{x \in K} \times \{y + u\}_{y \in K}\) is an open cover of \(K \times K\), by Lemma 4.2, there exists a singlevalued continuous mapping \(f_{u}: K \rightarrow K\), which is \((\Omega \times u)\)approximation of the \(\operatorname{Graph}(T)\), where \(\Omega : = \{x+ u\}_{x \in K}\) as mentioned above. By Theorem 4.3, \(f_{u}\) has a fixed point \(x_{u} = f_{u}(x_{u})\) in K for each \(u \in \mathfrak{U}\). Now, for \(x_{u} \in K\), by following the proof of Lemma 4.2, we observe that, firstly, there exists \(x'_{u} \in K\) such that \(x_{u} \in x'_{u} + u\); and secondly, there also exists \(v_{u} \in F(x'_{u})\) such that \(f_{u}(x_{u})v_{u} \in u\), which means that \(f_{u}(x_{u}) \in v_{u} + u\).
In summary, for any given \(u \in \mathfrak{U}\), there exists a continuous mapping \(f_{u}: K \rightarrow K\), which has at least one fixed point \(x_{u} \in K\) such that \(x_{u} = f_{u}(x_{u})\) with \((x_{u}, x_{u}) =(x_{u}, f_{u}(x_{u})) \in \operatorname{Graph}(f_{u})\), and we also have the following statements:
(1) There exists \(x'_{u} \in K\) such that \(x_{u} \in x'_{u} + u\); and
(2) There exists \(v_{u} \in F(x'_{u})\) such that \(f_{u}(x_{u})v_{u} \in u\), which means \(f_{u}(x_{u}) \in v_{u} + u\).
Since K is compact, without loss of generality, we may assume that there exists a subnet \((x_{u_{i}})_{u_{i} \in \mathfrak{U}}\) converging to \(x^{*}\) in K. Now we will show that \(x^{*}\) is the fixed point of T, i.e., \(x^{*} \in T(x^{*})\).
As K is compact, without loss of generality, we may assume that two nets \(\{x_{u}\}_{u \in \mathfrak{U}}\) and \(\{x'_{u}\}_{u \in \mathfrak{U}}\) in K have the subnet \(\{x_{u_{i}}\}_{u_{i} \in \mathfrak{U}}\) converging to \(x^{*}\), and the subnet \(\{x'_{u_{i}}\}_{u_{i} \in \mathfrak{U}}\) converges to \(x^{\prime *}\) respectively in K. By the statement of (1) above, it is clear that we must have \(x^{*} = x^{\prime *}\); otherwise, as the family \(\mathfrak{U}\) is the base of absolutely pconvex open neighborhoods of the origin in X, by (1) we will have the contradiction, and thus our claim that \(x^{*} = x^{\prime *}\) is true in a locally pconvex space X.
Now we prove that \(x^{*}\) is a fixed point of T by using the statement of (2) for all \(u \in \mathfrak{U}\). As the net \(\{v_{u}\}_{u \in \mathfrak{U}} \subset K\), we may assume its subnet \(\{v_{u_{i}}\}_{u_{i} \in \mathfrak{U}}\) converges to \(v^{*}\). Then, by the statement given by (2), it is clear that we have that \(\lim_{u_{i} \in \mathfrak{U}}v_{u_{i}}=v^{*} = \lim_{u_{i} \in \mathfrak{U}}f_{u_{i}}(x_{u_{i}})= \lim_{u_{i} \in \mathfrak{U}}x_{u_{i}} = x^{*}\). By the fact that \((v_{u_{i}}, x'_{u_{i}}) \in \operatorname{Graph}(T)\) and the graph of T is closed, it follows that \(x^{*} = v^{*} \in T(x^{*})\), which means that \(x^{*}\) is a fixed point of T. The proof is complete. □
We note that Theorem 4.4 improves or unifies corresponding results given by Cauty [22], Cauty [23], Chang et al. [27], Dobrowolski [35], Nhu [87], Park [101], Reich [110], Smart [126], Xiao and Lu [134], Xiao and Zhu [135], Yuan [143–145] under the framework of compact singlevalued or upper semicontinuous setvalued mappings.
Remark 4.3
Theorem 4.3 says that each compact singlevalued mapping defined on a closed pconvex subsets (\(0< p \leq 1\)) in topological vector spaces has the fixed point property, which does not only include or improve most available results for fixed point theorems in the existing literature as special cases (just to mention a few, BenElMechaiekh [10], BenElMechaiekh and Saidi [11], Ennassik and Taoudi [38], Mauldin [84], Granas and Dugundji [53], O’Regan and Precup [93], Reich [110], Park [101], and the references therein), but also provides an answer to Schauder conjecture in topological vector spaces in the affirmative for compact singlevalued mappings defined on noncompact convex pconvex subsets in locally pconvex spaces for \(0 < p \leq 1\) or topological vector spaces. In particular, we note that the answer to Schauder conjecture in the affirmative for a singlevalued continuous mapping recently was obtained by Ennaassik and Taoudi [38] defined on a nonempty compact pconvex subset in TVS. Actually, we will show that Schauder conjecture is also true for quasi upper semicontinuous setvalued mappings in locally pconvex spaces as discussed by Theorems 4.4 and 4.7.
In addition. we we would like to point out that it is not clear if the assumption \(``T(x)\) is with nonempty closed pconvex values” could be replaced with the condition \(``T(x)\) is with nonempty closed sconvex values” in Theorem 4.4. In fact, it seems that the proof of Theorem 4.3 given by Ennassik et al. [37] only goes through for the case \(s \leq p\), not for the general case when both \(s, p \in (0, 1]\) (please note that the letter p is denoted as the letter r by Ennassik et al. [37]). Thus, we are still looking for a proper way to prove if the conclusion of Theorem 4.4 is true under Hausdorff topological vector spaces instead of locally pconvex spaces for \(p \in (0, 1]\).
Now, as an immediate consequence of Theorem 4.4, we have the following fixed point result for QUSC mappings in a locally pconvex space X.
Corollary 4.1
If K is a nonempty compact sconvex subset of a locally convex space X, where \(s \in (0, 1]\), then any quasi upper semicontinuous setvalued mapping \(T: K \rightarrow 2^{K}\) with nonempty closed convex values and its graph being closed has at least one fixed point.
Proof
Apply Theorem 4.4 with \(p=1\), this completes the proof. □
Corollary 4.1 indeed improves or unifies the corresponding results given by Askoura and GodetThobie [6], Cauty [22], Cauty [23], Chang et al. [27], Chen [32], Theorem 3.1 and Theorem 3.3 of Ennssik and Taoudi [38], Theorem 3.14 of Gholizadeh et al. [46], Isac [60], Li [79], Nhu [87], Okon [89], Park [102], Reich [110], Smart [126], Xiao and Lu [134], Yuan [143] under the framework of locally pconvex spaces for setvalued (instead of singlevalued) mappings.
As an application of Theorem 4.4, we have the following fixed point theorem for quasi upper semicontinuous setvalued mappings in locally pconvex spaces, which could be regarded as the extension or a setvalued version of Theorem 3.1 and Theorem 3.3 of Ennassik and Taoudi [38].
Theorem 4.5
If K is a nonempty compact pconvex subset of a Hausdorff locally pconvex space X, where \(p \in (0, 1]\), then any quasi upper semicontinuous setvalued mapping \(T: K \rightarrow 2^{K}\) with nonempty pconvex values and with a closed graph, has at least one fixed point.
Proof
By taking \(s=p\) in Theorem 4.4, the conclusion follows. This completes the proof. □
By following the same idea used in the proof of Theorem 4.3, the conclusion of Theorem 4.4 still holds for compact quasi upper semicontinuous setvalued mappings as stated by Theorem 4.6 (and thus we omit its proof here).
Theorem 4.6
If K is a nonempty closed sconvex subset of a Hausdorff locally pconvex space X, where \(s, p \in (0, 1]\), then any compact quasi upper semicontinuous setvalued mapping \(T: K \rightarrow 2^{K}\) with nonempty pconvex values and with a closed graph has at least one fixed point.
Now, as a special case in Theorem 4.6 with \(p=1\), we have the following results for compact QUSC mappings defined on sconvex subsets in locally convex spaces, where \(s \in (0, 1]\).
Corollary 4.2
If K is a nonempty closed sconvex subset of a Hausdorff locally convex space X, then any compact quasi upper semicontinuous setvalued mapping \(T: K \rightarrow 2^{K}\) with nonempty convex values and with a closed graph has at least one fixed point.
Corollary 4.3
Let K be a closed convex compact subset of a Hausdorff locally convex space X. If \(T: K \rightarrow 2^{K}\) is a quasi upper continuous setvalued mapping with nonempty closed convex values and its graph is closed, then T has a fixed point in K.
Corollary 4.4
(Schauder fixed point theorem for USC mappings in LCS)
Let K be a closed convex compact subset of a Hausdorff locally convex space X. If \(T: K \rightarrow 2^{K}\) is an upper continuous setvalued mapping with nonempty closed convex values, then T has a fixed point in K.
So far in this section, as the application of graph approximation for quasi upper semicontinuous mappings, which is Lemma 4.2, we have established general fixed point theorems for general (compact) quasi upper semicontinuous setvalued mappings in locally pconvex spaces, which allows us not only to answer Schauder’s conjecture in the affirmative under the general framework of locally pconvex spaces, but also to unify or improve the corresponding results in the existing literature for nonlinear analysis, where \(p in (0, 1]\).
We would like to mention that by comparing with topological degree approach or other related methods used or developed by Cauty [22, 23], Nhu [87], and others, the arguments used in this section actually provide an accessible way for the study of nonlinear analysis for pconvex vector spaces for \(p \in (0, 1]\). The results given in this paper are new and may be easily understood and used by general readers in the mathematical community. In addition, the general fixed point theorems established for quasi upper semicontinuous setvalued mappings in locally pconvex spaces for \(p \in (0, 1]\) or in topological vector spaces would play important roles for the study in functional analysis as those by Agarwal et al. [1], BenElMechaiekh [10], BenElMechaiekh and Saidi [11], Browder [17], Cellina [24], Chang [25], Chang et al. [27], Ennassik et al. [37], Fan [40, 41], Górniewicz [51], Granas and Dugundji [53], Guo et al. [55], Nhu [87], Park [101], Reich [110], Smart [126], Tychonoff [130], Weber [132, 133], Xiao and Lu [134], Xiao and Zhu [135], Xu [137], Yuan [142–145], Zeidler [146], and the related references therein. We would also like to point out that the results given in this part are new, which is the continuation of the related work given by Yuan [144, 145] recently.
In order to establish fixed point theorems for the classes of USC 1set contractive and condensing mappings in locally pconvex spaces by using the concept of the measure of noncompactness (or saying, the noncompactness measures) that were introduced and widely accepted in mathematical community by Kuratowski [74], Darbo [33], and the related references therein, by following recent work due to Yuan [144, 145], we first need to have a brief introduction for the concept of noncompactness measures for the socalled Kuratowski or Hausdorff measures of noncompactness in normed spaces (see Alghamdi et al. [5], Machrafi and Oubbi [82], Nussbaum [88], Sadovskii [117], Silva et al. [123], Xiao and Lu [134] for the general concepts under the framework of pseminorm or locally convex pconvex settings for \(p \in (0, 1]\), which will be discussed below, too).
The same as those given by Yuan [144, 145], for a given metric space \((X, d)\) (or a pnormed space \((X, \\cdot \_{p})\)), here we recall some notions and concepts for the completeness, boundedness, relative compactness, and compactness, which will be used in what follows. Let \((X, d)\) and \((Y, d)\) be two metric spaces and \(T: X \rightarrow Y\) be a mapping (or, say, operator). Then: 1) T is said to be bounded if for each bounded set \(A\subset X\), \(T(A)\) is a bounded set of Y; 2) T is said to be continuous if for every \(x \in X\), \(\lim_{n \rightarrow \infty} x_{n} = x\) implies that \(\lim_{n\rightarrow \infty} T(x_{n})= T\); and 3) T is said to be completely continuous if T is continuous and \(T(A)\) is relatively compact for each bounded subset A of X.
Let \(A_{1}\), \(A_{2} \subset X\) be bounded of a metric space \((X, d)\), we also recall that the Hausdorff metric \(d_{H}(A_{1}, A_{2})\) between \(A_{1}\) and \(A_{2}\) is defined by
The Hausdorff and Kurotowskii measures of noncompactness (denoted by \(\beta _{H}\) and \(\beta _{K}\), respectively) for a nonempty bounded subset D in X are the nonnegative real numbers \(\beta _{H}(D)\) and \(\beta _{K}(D)\) defined by
and
here \(\operatorname{diam} D_{i}\) means the diameter of the set \(D_{i}\), and it is well known that \(\beta _{H} \leq \beta _{K} \leq 2 \beta _{H}\). We also point out that the notions above can be well defined under the framework of pseminorm spaces \((E, \\cdot \_{p})_{p \in \mathfrak{P}}\) by following a similar idea and method used by Chen and Singh [31], Ko and Tasi [71], and Kozlov et al. [72]; see the references therein for more details.
Let T be a mapping from \(D\subset X\) to X. Then we have that: 1) T is said to be a kset contraction with respect to \(\beta _{K}\) (or \(\beta _{H}\)) if there is a number \(k \in [0, 1)\) such that \(\beta _{K}(T(A)) \leq k \beta _{K}(A)\) (or \(\beta _{H}(T(A)) \leq k\beta _{H}(A)\)) for all bounded sets A in D; and 2) T is said to be \(\beta _{K}\)condensing (or \(\beta _{H}\)condensing) if \((\beta _{K}(T(A)) < \beta _{K}(A))\) (or \(\beta _{H} (T(A)) < \beta _{H}(A)\)) for all bounded sets A in D with \(\beta _{K}(A)> 0\) (or \(\beta _{H}(A)> 0\)).
For the convenience of our discussion, throughout the rest part of this paper, if a mapping “is \(\beta _{K}\)condensing (or \(\beta _{H}\)condensing)”, we simply say it is “a condensing mapping” unless specified otherwise.
Moreover, it is easy to see that: (1) if T is a compact operator, then T is a kset contraction; and (2) if T is a kset contraction for \(k \in (0, 1)\), then T is condensing.
To establish the fixed points of setvalued condensing mappings in locally pconvex spaces (and also pvector spaces) for \(p \in (0, 1]\), we need to recall some notions introduced by Machrafi and Oubbi [82] for the measure of noncompactness in locally pconvex vector spaces, which also satisfies some necessary (common) properties of the classical measures of noncompactness such as \(\beta _{K}\) and \(\beta _{H}\) mentioned above introduced by Kuratowski [74], Sadovskii [117](see also related discussion by Alghamdi et al. [5], Nussbaum [88], Silva et al. [123], Xiao and Lu [134], and the references therein). In particular, the measures of noncompactness in locally pvector spaces (for \(0 < p \leq 1\)) should have the stable property, which means the measure of noncompactness A is the same by transition to the (closure) for the pconvex hull of subset A.
For the convenience of discussion, we follow up to use α and β to denote the Kuratowski and the Hausdorff measures of noncompactness in topological vector spaces, respectively (see the same way used by Machrafi and Oubbi [82]), unless otherwise stated. The E is used to denote a Hausdorff topological vector space over the field \(\mathbb{K} \in \{\mathbb{R}, \mathbb{Q}\}\), here \(\mathbb{R}\) denotes all real numbers and \(\mathbb{Q}\) all complex numbers, and \(p \in (0, 1]\). Here, the base set of family of all balanced zero neighborhoods in E is denoted by \(\mathfrak{V}_{0}\).
We recall that \(U \in \mathfrak{V}_{0}\) is said to be shrinkable if it is absorbing, balanced, and \(r U \subset U\) for all \(r \in (0, 1)\), and we know that any topological vector space admits a local base at zero consisting of shrinkable sets (see Klee [69], or Jarchow [61] for details).
Recall that a topological vector space E is said to be a locally pconvex space if E has a local base at zero consisting of pconvex sets. The topology of a locally pconvex space is always given by an upward directed family P of pseminorms, where a pseminorm on E is any nonnegative realvalued and subadditive functional \(\\cdot \_{p}\) on E such that \(\ \lambda x\_{p}=\lambda ^{p}\x\_{p}\) for each \(x \in E\) and \(\lambda \in \mathbb{R}\) (i.e., the real number line). When E is Hausdorff, then for every \(x \neq 0\), there is some \(p \in P\) such that \(P(x) \neq 0\). Whenever the family P is reduced to a singleton, one says that \((E, \ \cdot \)\) is a pseminorm space. A pnormed space is a Hausdorff pseminorm space when \(p=1\), which is the usual locally convex case. Furthermore, a pnormed space is a metric vector space with the translation invariant metric \(d_{p}(x, y): = \ x y\_{p}\) for all \(x, y \in E\), which is the same notation as that used above.
By Remark 2.2, if P is a continuous pseminorm on E, then the ball \(B_{p}(0, s): = \{x \in E: P(x) < s \}\) is shrinkable for each \(s > 0\). Indeed, if \(r \in (0, 1)\) and \(x \in \overline{r B_{p}(0, s)}\), then there exists a net \((x_{i})_{i \in I} \subset B_{p}(0, s)\) such that \(r x_{i}\) converges to x. By the continuity of P, we get \(P(x) \leq r^{p} s < s\), which means that \(r \overline{B_{p}(0,s)} \subset B_{P}(0,s)\). In general, it can be shown that every pconvex \(U \in \mathfrak{V}_{0}\) is shrinkable.
We recall that given such a neighborhood U, a subset \(A \subset E\) is said to be Usmall if \(A  A \subset U\) (or, say, small of order U by Robertson [113]). Now, by following the idea of Kaniok [65] in the setting of a topological vector space E, we use zero neighborhoods in E instead of seminorms to define the measure of noncompactness in (local convex) pvector spaces (\(0< p \leq 1\)) as follows: For each \(A \subset E\), the Umeasures of noncompactness \(\alpha _{U}(A)\) and \(\beta _{U}(A)\) for A are defined by
and
here we set \(\inf \emptyset : = \infty \).
By the definition above, it is clear that when E is a normed space and U is the closed unit ball of E, \(\alpha _{U}\) and \(\beta _{U}\) are nothing else but the Kuratowski measure \(\beta _{K}\) and Hausdorff measure \(\beta _{H}\) of noncompactness, respectively. Thus, if \(\mathfrak{U}\) denotes a fundamental system of balanced and closed zero neighborhoods in E and \(\mathfrak{F}_{\mathfrak{U}}\) is the space of all functions \(\phi : \mathfrak{U} \rightarrow R\), endowed with the pointwise ordering, then the \(\alpha _{U}\) (resp., \(\beta _{U}\)) measures for noncompactness introduced by Kaniok [65] can be expressed by the Kuratowski (resp., the Hausdorf) measure of noncompact \(\alpha (A)\)(resp., \(\beta (A)\)) for a subset A of E as the function defined from \(\mathfrak{U}\) into \([0, \infty )\) by
By following Machrafi and Oubbi [82], to define the measure of noncompactness in (locally convex) pvector space E, we need the following notions of basic and sufficient collections for zero neighborhoods in a topological vector space. To do this, let us introduce an equivalence relation on \(V_{0}\) by saying that U is related to V, written \(U\mathfrak{R}V\), if and only if there exist \(r, s > 0\) such that \(r U \subset V \subset s U\). We now have the following definition.
Definition 4.5
(BCZN)
We say that \(\mathfrak{B} \subset \mathfrak{V}_{0}\) is a basic collection of zero neighborhoods (in short, BCZN) if it contains at most one representative member from each equivalence class with respect to \(\mathfrak{R}\). It is said to be sufficient (in short, SCZN) if it is basic and, for every \(V \in \mathfrak{V}_{0}\), there exist some \(U \in \mathfrak{B}\) and some \(r > 0 \) such that \(r U \subset V\).
Remark 4.4
By Remark 2.2, it follows that for a locally pconvex space E, its base set \(\mathfrak{U}\), the family of all open pconvex subsets for 0 is BCZB. We also note that: 1) In the case when E is a normed space, if f is a continuous functional on E, \(U: =\{x \in E: f(x) < 1\}\) and V is the open unit ball of E, then \(\{U\}\) is basic but not sufficient, but \(\{V\}\) is sufficient; 2) Secondly, if \((E, \tau )\) is a locally convex space, whose topology is given by an upward directed family P of seminorms so that no two of them are equivalent, then the collection \((B_{p})_{p \in \mathbb{P}}\) is SCZN, where \(B_{p}\) is the open unit ball of p. Further, if \(\mathfrak{W}\) is a fundamental system of zero neighborhoods in a topological vector space E, then there exists SCZN consisting of \(\mathfrak{W}\) members; and 3) By following Oubbi [94], we recall that a subset A of E is called uniformly bounded with respect to a sufficient collection \(\mathfrak{B}\) of zero neighborhoods if there exists \(r > 0 \) such that \(A \subset r V\) for all \(V \in \mathfrak{B}\). Note that in the locally convex space \(C_{c}(X): = C_{c}(X, \mathbb{K})\), the set \(B_{\infty}:=\{ f\in C(X): \f\_{\infty} \leq 1\}\) is uniformly bounded with respect to the SCZN \(\{B_{k}, k \in \mathbb{K}\}\), where \(B_{k}\) is the (closed or) open unit ball of the seminorm \(P_{k}\), where \(k \in \mathbb{K}\).
Now we are ready to give the definition for the measure of noncompactness in (locally pconvex) topological vector space E as follows.
Definition 4.6
Let \(\mathfrak{B}\) be SCZN in E. For each \(A \subset E\), we define the measure of noncompactness of A with respect to \(\mathfrak{B}\) by \(\alpha _{\mathfrak{B}}(A):=\sup_{U\in \mathfrak{B}}\alpha _{U}(A)\).
By the definition above, it is clear that: 1) The measure of noncompactness \(\alpha _{B}\) holds the semiadditivity, i.e., \(\alpha _{B}(A \cup B) = \max \{\alpha _{B}(A), \alpha _{B}(B)\}\); and 2) \(\alpha _{B}(A) = 0 \) if and only if A is a precompact subset of E (for more properties in detail, see Proposition 1 and related discussion by Machraf and Oubbi [94]).
As we know, under the normed spaces (and even seminormed spaces), Kuratowski [74], Darbo [33], and Sadovskii [117] introduced the notions of ksetcontractions for \(k \in (0, 1)\) and condensing mappings to establish fixed point theorems in the setting of Banach spaces, normed or seminorm spaces. By following the same idea, if E is a Hausdorff locally pconvex space, we have the following definition for general (nonlinear) mappings.
Definition 4.7
A mapping \(T: C \rightarrow 2^{C}\) is said to be a kset contraction (resp., condensing) if there is some SCZN \(\mathfrak{B}\) in E consisting of pconvex sets, such that (resp., condensing) for any \(U \in \mathfrak{B}\), there exists \(k \in (0,1)\) (resp., condensing) such that \(\alpha _{U}(T(A)) \leq k \alpha _{U}(A)\) for \(A \subset C\) (resp., \(\alpha _{U}(T(A)) < \alpha _{U}(A)\) for each \(A \subset C\) with \(\alpha _{U}(A) > 0\)).
It is clear that a contraction mapping on C is a kset contraction mapping (where we always mean \(k \in (0, 1)\)), and a kset contraction mapping on C is condensing; and they all reduce to the usual cases by the definitions for \(\beta _{K}\) and \(\beta _{H}\), which are the Kuratowski measure and the Hausdorff measure of noncompactness, respectively, in normed spaces (see Kuratowski [74]).
From now on, denote by \(\mathfrak{V}_{0}\) the set of all shrinkable zero neighborhoods in E, we then have the following result, which is Theorem 1 of Machrafi and Oubbi [82], saying that in the general setting of locally pconvex spaces, the measure of noncompactness α for U given by Definition 4.3 is stable from U to its pconvex hull \(C_{p}(A)\) of the subset A in E, which is key for us to establish fixed points for condensing mappings in locally pconvex spaces for \(0< p \leq 1\). This also means that the key property for the measures due to the Kurotowski and Hausdorff measures of noncompactness in normed (or pseminorm) spaces also holds for the measure of noncompactness by Definition 4.3 in the setting of locally pconvex spaces with (\(0 < p \leq 1\)) (for more details, see similar and related discussion by Alghamdi et al. [5] and Silva et al. [123]).
Lemma 4.3
If \(U \in \mathfrak{V}_{0}\) is pconvex for some \(0 < p \leq 1\), then \(\alpha (C_{p}(A)) = \alpha (A)\) for every \(A \subset E\).
Proof
It is Theorem 1 of Machrafi and Oubbi [82]. The proof is complete. □
Now, based on the definition for the measure of noncompactness given by Definition 4.3 (originally from Machrafi and Oubbi [82]), we have the following general extended version of Schauder, Darbo, and Sadovskii type fixed point theorems in the context of locally pconvex vector spaces for condensing mappings.
Theorem 4.7
Let \(C \subset E\) be a complete sconvex subset of a locally pconvex space E with \(s, p \in (0, 1]\). If \(T: C \rightarrow 2^{C}\) is quasi upper semicontinuous and (α) condensing setvalued mappings with nonempty pconvex values and with a closed graph, then T has a fixed point in C.
Proof
Let \(\mathfrak{B}\) be a sufficient collection of pconvex zero neighborhoods in E with respect to which T is condensing for any given \(U \in \mathfrak{B}\). We choose some \(x_{0} \in C\) and let \(\mathfrak{F}\) be the family of all closed pconvex subsets A of C with \(x_{0} \in A\) and \(T(A) \subset A\). Note that \(\mathfrak{F}\) is not empty since \(C \in \mathfrak{F}\). Let \(A_{0}=\bigcap_{A \in \mathfrak{F}} A\). Then \(A_{0}\) is a nonempty closed pconvex subset of C such that \(T(A_{0}) \subset A_{0}\). We shall show that \(A_{0}\) is compact. Let \(A_{1}=\overline{C_{p}(T(A_{0}) \cup \{x_{0}\})}\). Since \(T(A_{0})\subset A_{0}\) and \(A_{0}\) is closed and pconvex, \(A_{1}\subset A_{0}\). Hence, \(T(A_{1})\subset T(A_{0})\subset A_{1}\). It follows that \(A_{1} \in \mathfrak{F}\), and therefore \(A_{1}=A_{0}\). Now, by Proposition 1 of Machrafi and Oubbi [82] and Lemma 4.3 above (i.e., Theorem 1 and Theorem 2 in [82]), we get \(\alpha _{U}(T(A_{0})) = \alpha _{U}(A)\). Our assumption on T shows that \(\alpha _{U}(A_{0})=0\) since T is condensing. As U is arbitrary from the family \(\mathfrak{B}\), thus \(A_{0}\) is pconvex and compact (see Proposition 4 in [82]). Now, the conclusion follows by Theorem 4.4 (or Theorem 4.6) above. The proof is complete. □
As an application of Theorem 4.7, we have the following general result, which answers Schauder conjecture for quasi upper semicontinuous setvalued mappings defined on sconvex subsets in locally convex spaces, where \(p \in (0, 1]\).
Theorem 4.8
(Schauder fixed point theorem for QUSC condensing mappings in LCS)
Let K be a nonempty closed pconvex subset of a locally p convex space, where \(p \in (0, 1]\), then any quasi upper semicontinuous setvalued \((\alpha )\) condensing mapping \(T: K \rightarrow 2^{K}\) with nonempty convex values and with a closed graph has at least a fixed point.
Proof
By letting \(s=p\) in Theorem 4.7, the conclusion follows by Theorem 4.7. Thus we complete the proof. □
As a special case of Theorem 4.8, we have the following result.
Theorem 4.9
Let K be a closed pconvex subset of a Hausdorff locally pconvex space X, where \(p \in (0, 1]\). If \(T: K \rightarrow 2^{K}\) is an upper continuous condensing setvalued mapping with nonempty closed pconvex values, then T has a fixed point in K.
Proof
By the fact that each upper semicontinuous (USC) setvalued mapping is quasi upper semicontinuous and each USC with closed value has a closed graph, the conclusion follows by Theorem 4.7. This completes the proof. □
As applications of Theorem 4.9, we have a few theorems of fixed points for condensing mappings in locally pconvex spaces for \(p\in (0, 1]\) as follows.
Corollary 4.5
(Darbo type fixed point theorem)
Let C be a complete pconvex subset of a Hausdorff locally pconvex space E with \(0 < p \leq 1\). If \(T: C \rightarrow 2^{C}\) is a (k)setcontraction (where \(k \in (0,1)\)) with closed and pconvex values, then T has a fixed point.
Corollary 4.6
(Sadovskii type fixed point theorem)
Let \((E, \ \cdot \)\) be a complete pnormed space and C be a bounded, closed, and pconvex subset of E, where \(0 < p \leq 1\). Then every USC and condensing mapping \(T: C \rightarrow 2^{C}\) with closed and pconvex values has a fixed point.
Proof
In Theorem 4.7, let \(\mathfrak{B}: =\{B_{p}(0, 1) \}\), where \(B_{p}(0,1)\) stands for the closed unit ball of E, and by the fact that it is clear that \(\alpha (A)=(\alpha _{\mathfrak{B}}(A))^{p}\) for each \(A \subset E\). Then that T satisfies all conditions of Theorem 4.7. This completes the proof. □
Corollary 4.7
(Darbo type)
Let \((E, \ \cdot \)\) be a complete pnormed space and C be a bounded, closed, and pconvex subset of E, where \(0 < p \leq 1\). Then each mapping \(T: C \rightarrow C\) that is continuous and a setcontraction has a fixed point.
Theorem 4.7 and also Theorem 4.8 improve Theorem 5 of Machrafi and Oubbi [82] for general condensing mappings that are general upper semicontinuous mappings with closed pconvex values and also unify the corresponding results in the existing literature, e.g., see Alghamdi et al. [5], Górniewicz [51], Górniewicz et al. [52], Nussbaum [88], Silva et al. [123], Xiao and Lu [134], Xiao and Zhu [135], and the references therein.
Secondly, as an application of the KKM principle for abstract convex spaces with graph approximation Lemma 4.2 for quasi upper semicontinuous setvalued mappings in locally pconvex spaces, we establish general fixed point theorems for quasi upper semicontinuous setvalued mappings, which allow us to answer Schauder’s conjecture in the affirmative way under the framework of locally pconvex spaces for \(p \in (0, 1]\).
Before the ending of this section, we would also like to remark that by comparing with topological method or related arguments used by Askoura et al. [6], Cauty [22, 23], Dobrowolski [35], Nhu [87], Reich [110], the fixed points given in this section improve or unify the corresponding ones given by Alghamdi et al. [5], Darbo [33], Liu [81], Machrafi and Oubbi [82], Sadovskii [117], Silva et al. [123], Xiao and Lu [134], Yuan [144, 145], and those from the references therein.
5 Best approximation for the class of 1set contractive mappings in locally pconvex spaces
The goal of this section is first to establish one general best approximation result for 1set upper semicontinuous and hemicompact (see its definition below) nonself setvalued mappings, which in turn is used as a tool to derive the general principle for the existence of solutions for Birkhoff–Kellogg problems (see Birkhoff and Kellogg [14]) and fixed points for nonself 1set contractive setvalued mappings.
Here, we recall that since the Birkhoff–Kellogg theorem was first introduced and proved by Birkhoff and Kellogg [14] in 1922 in discussing the existence of solutions for the equation \(x = \lambda F(x)\), where λ is a real parameter and F is a general nonlinear nonself mapping defined on an open convex subset U of a topological vector space E, now the general form of the Birkhoff–Kellogg problem is to find the socalled invariant direction for nonlinear setvalued mappings F, i.e., to find \(x_{0} \in \overline{U}\) (or \(x_{0} \in \partial \overline{U}\)) and \(\lambda > 0\) such that \(\lambda x_{0} \in F(x_{0})\).
Since the Birkhoff and Kellogg theorem given by Birkhoff and Kellogg in 1920s, the study on Birkhoff–Kellogg problem has been received a lot of scholars’ attention. For example, one of the fundamental results in nonlinear functional analysis, called the Leray–Schauder alternative, was established via topological degree by Leray and Schauder [76] in 1934. Thereafter, certain other types of Leray–Schauder alternatives were proved using different techniques other than topological degree, see the work by Granas and Dugundji [53], Furi and Pera [44] in the Banach space setting and applications to the boundary value problems for ordinary differential equations, and a general class of mappings for nonlinear alternative of Leray–Schauder type in normal topological spaces, and also Birkhoff–Kellogg type theorems for general class mappings in TVS by Agarwal et al. [1], Agarwal and O’Regan [2, 3], Park [98]. In particular, recently O’Regan [91] used the Leray–Schauder type coincidence theory to establish some Birkhoff–Kellogg problem and Furi–Pera type results for a general class of setvalued mappings.
In this section, one best approximation result for 1set contractive mappings in pseminorm spaces is first established, which is then used to the general principle for solutions of Birkhoff–Kellogg problems and related nonlinear alternatives, then it allows us to give general existence results for the Leray–Schauder type and related fixed point theorems of nonself mappings in pseminorm spaces for \(p\in (0, 1]\). The new results given in this part not only include the corresponding results in the existing literature as special cases, but are also expected to be useful tools for the study of nonlinear problems arising from theory to practice for 1set contractive mappings.
We also note that the general nonlinear alternative related to Leray–Schauder alternative under the framework of pseminorm spaces for \(p \in (0, 1]\) given in this section would be a useful tool for the study of nonlinear problems. In addition, we also note that corresponding results in the existing literature for Birkhoff–Kellogg problems and the Leray–Schauder alternative have been studied comprehensively by Granas and Dugundji [53], Isac [60], Park [99–101], Carbone and Conti [21], Chang and Yen [30], Chang et al. [28, 29], Kim et al. [67], Shahzad [120–122], Singh [125]; and in particular, many general forms have been recently obtained by O’Regan [92] (see also the references therein).
To study the existence of fixed points for nonself mappings in pvector spaces, we need the following definitions.
Definition 5.1
(Inward and outward sets in pvector spaces)
Let C be a subset of a pvector space E and \(x \in E\) for \(0 < p \leq 1\). Then the pinward set \(I^{p}_{C}(x)\) and poutward set \(O^{p}_{C}(x)\) are defined by

\(I^{p}_{C}(x): =\{ x + r(yx): y \in C \text{ for any } r \geq 0 \text{ (1) if } 0 \leq r \leq 1 \text{ with } (1r)^{p} + r^{p} =1; \text{ or (2) if } r \geq 1 \text{ with } (\frac{1}{r})^{p} + (1 \frac{1}{r})^{p} = 1 \}\); and

\(O^{p}_{C}(x): =\{x + r(yx): y \in C \text{ for any } r \leq 0 \text{ (1) if } 0 \leq r \leq 1 \text{ with } (1r)^{p} + r^{p} = 1; \text{ or (2) if } r \geq 1 \text{ with } (\frac{1}{r})^{p} + (1 \frac{1}{r})^{p} =1 \}\).
From the definition, it is obvious that when \(p=1\), both the inward and outward sets \(I^{p}_{C}(x)\), \(O^{p}_{C}(x)\) are reduced to the definition for the inward set \(I_{C}(x)\) and the outward set \(O_{C}(x)\), respectively, in topological vector spaces introduced by Halpern and Bergman [56] and used for the study of nonself mappings related to nonlinear functional analysis in the literature. In this paper, we mainly focus on the study of the pinward set \(I_{U}^{p}(x)\) for the best approximation related to the boundary condition for the existence of fixed points in pvector spaces. By the special property of pconvex concept when \(p \in (0, 1)\) and \(p=1\), we have the following fact.
Lemma 5.1
Let C be a subset of a pvector space E and \(x \in E\), where \(0 < p \leq 1\). Then for both pinward and outward sets \(I^{p}_{C}(x)\) and \(O^{p}_{C}(x)\) defined above, we have

(I)
when \(p \in (0, 1)\), \(I^{p}_{C}(x)= [\{x\}\cup C]\) and \(O^{p}_{C}(x)=[\{x \} \cup \{2x\} \cup  C ]\),

(II)
when \(p=1\), in general \([\{x \}\cup C] \subset I^{p}_{C}(x)\) and \([\{ x \} \cup \{2x\} \cup C] \subset O^{p}_{C}(x)\).
Proof
First, when \(p\in (0, 1)\), by the definitions of \(I^{p}_{C}(x)\), the only real number \(r \geq 0\) satisfying the equation \((1r)^{p} + r^{p} =1\) for \(r\in [0,1]\) is \(r=0\) or \(r=1\), and when \(r \geq 1\), the equation \((\frac{1}{r})^{p} + (1 \frac{1}{r})^{p} = 1\) implies that \(r=1\). The same reason for \(O^{p}_{C}(x)\), it follows that \(r=0\) and \(r= 1\).
Secondly when \(p=1\), all \(r\geq 0\) and all \(r\leq 0\) satisfy the requirement of definition for \(I^{p}_{C}(x)\) and \(O^{p}_{C}(x)\), respectively, thus the proof is complete. □
By following the original idea by Tan and Yuan [129] for hemicompact mappings in metric spaces, we introduce the following definition for a mapping being hemicompact in pseminorm spaces for \(p \in (0,1]\), which is indeed the “(H) condition” used in Theorem 5.1 to prove the existence of best approximation results for 1set contractive setvalued mappings in pseminorm vector spaces for \(p \in (0, 1]\).
Definition 5.2
(Hemicompact mapping)
Let E be a pvector space with pseminorm for \(1 < p \leq 1\). For a given bonded (closed) subset D in E, a mapping \(F: D \rightarrow 2^{E}\) is said to be hemicompact if each sequence \(\{x_{n}\}_{n\in N}\) in D has a convergent subsequence with limit \(x_{0}\) such that \(x_{0} \in F(x_{0})\), whenever \(\lim_{n \rightarrow \infty} d_{P_{U}}(x_{n}, F(x_{n})) =0\) for each \(U \in \mathfrak{U}\), where \(d_{P_{U}}(x, C):= \inf \{P_{U}(xy): y \in C\}\) is the distance of a single point x with the subset C in E based on \(P_{U}\), \(P_{U}\) is the Minkowski pfunctional in E for \(U \in \mathfrak{U}\), which is the base of the family consisting of all subsets of 0neighborhoods in E.
Remark 5.1
We would like to point out that Definition 5.2 is indeed an extension for a “hemicompact mapping” defined from a metric space to a pvector space with the pseminorm, where \(p \in (0, 1]\) (see Tan and Yuan [129]). By the monotonicity of Minkowski pfunctionals, i.e., the bigger 0neighborhoods, the smaller Minkowski pfunctionals’ values (see also p. 178 of Balachandran [7]), Definition 5.2 describes the convergence for the distance between \(x_{n}\) and \(F(x_{n})\) by using the language of seminorms in terms of Minkowski pfunctionals for each 0neighborhood in \(\mathfrak{U}\) (the base), which is the family consisting of its 0neighborhoods in pvector space E.
Now we have the following Schauder fixed point theorem for 1set contractive mappings in locally pconvex spaces for \(p \in (0, 1]\).
Theorem 5.1
(Schauder fixed point theorem for 1set contractive mappings)
Let U be a nonempty bounded open pconvex subset of a (Hausdorff) locally pconvex space E and its zero \(0 \in U\), and let \(C \subset E\) be a closed pconvex subset of E such that \(0 \in C\) with \(0 < p \leq 1\). If \(F: C \cap \overline{U} \rightarrow 2^{C \cap \overline{U}}\) is a quasi upper semicontinuous and 1set contractive setvalued mapping with nonempty pconvex values and with a closed graph and satisfying the following (H) or (H1) condition:
(H) Condition: The sequence \(\{x_{n}\}_{n\in \mathbb{N}}\) in U̅ has a convergent subsequence with limit \(x_{0} \in \overline{U}\) such that \(x_{0} \in F(x_{0})\), whenever \(\lim_{n \rightarrow \infty} d_{P_{U}}(x_{n}, F(x_{n})) =0\), where \(d_{P_{U}}(x_{n}, F(x_{n})):=\inf \{P_{U}(x_{n} z): z \in F(x_{n})\}\), where \(P_{U}\) is the Minkowski pfunctional for any \(U \in \mathfrak{U}\), which is the family of all nonempty open pconvex subset containing zero in E.
(H1) Condition: There exists \(x_{0}\) in U̅ with \(x_{0} \in F(x_{0})\) if there exists \(\{x_{n}\}_{n\in \mathbb{N}}\) in U̅ such that \(\lim_{n \rightarrow \infty} d_{P_{U}}(x_{n}, F(x_{n})) =0\), where \(P_{U}\) is the Minkowski pfunctional for any \(U \in \mathfrak{U}\), which is the family of all nonempty open pconvex subsets containing zero in E.
Then F has at least one fixed point in \(C \cap \overline{U}\).
Proof
Let \(\mathfrak{U}\) be a family of all nonempty open pconvex subset containing zero in E, and let U be any element in \(\mathfrak{U}\). As the mapping T is 1set contractive, take an increasing sequence \(\{\lambda _{n}\}\) such that \(0 < \lambda _{n} < 1\) and \(\lim_{n \rightarrow \infty} \lambda _{n} =1\), where \(n \in \mathbb{N}\). Now we define a mapping \(F_{n}: C \rightarrow 2^{C}\) by \(F_{n}(x): = \lambda _{n} F(x)\) for each \(x \in C\) and \(n\in \mathbb{N}\). Then it follows that \(F_{n}\) is a \(\lambda _{n}\)setcontractive mapping with \(0 < \lambda _{n} < 1\), and it is also quasi upper semicontinuous with pconvex values, and its graph is also closed. Now, by Theorem 4.8 on the condensing mapping \(F_{n}\) in locally pconvex spaces with pseminorm \(P_{U}\) (which is the Minkowski pfunctional for \(U \in \mathfrak{U}\)), for each \(n \in \mathbb{N}\), there exists \(x_{n} \in C \) such that \(x_{n} \in F_{n}(x_{n})=\lambda _{n} F(x_{n})\). Thus there exists \(y_{n} \in F(x_{n})\) such that \(x_{n}=\lambda _{n} y_{n}\). As \(P_{U}\) is the Minkowski pfunctional of U in E, it follows that \(P_{U}\) is continuous as \(0 \in \operatorname{int}(U)=U\). Note that for each \(n \in \mathbb{N}\), \(\lambda _{n} x_{n} \in \overline{U} \cap C\), which implies that \(x_{n} = r(\lambda _{n} y_{n}) = \lambda _{n} y_{n}\), thus \(P_{U}(\lambda _{n} y_{n}) \leq 1\) by Lemma 2.2. Note that
which implies that \(\lim_{n\rightarrow \infty} P_{U}(y_{n}x_{n})=0\) for all \(U \in \mathfrak{U}\).
Now (1) if F satisfies the (H) condition, it implies that the consequence \(\{x_{n}\}_{n \in \mathbb{N}}\) has a convergent subsequence that converges to \(x_{0}\) such that \(x_{0} \in F(x_{0})\). Without loss of generality, we assume that \(\lim_{n \rightarrow \infty} x_{n}=x_{0}\), here \(y_{n} \in F(x_{n})\) is with \(x_{n}=\lambda _{n} y_{n}\), and \(\lim_{n \rightarrow \infty} \lambda _{n}=1\), it implies that \(x_{0}=\lim_{n \rightarrow \infty} (\lambda _{n} y_{n})\), which means \(y_{0}:=\lim_{n\rightarrow \infty} y_{n}= x_{0}\). There exists \(y_{0} (= x_{0}) \in F(x_{0})\).
(ii) If F satisfies the (H1) condition, then by the (H1) condition, it follows that there exists \(x_{0}\) in U̅ such that \(x_{0} \in F(x_{0})\), which is a fixed point of F. We complete the proof. □
Theorem 5.2
(Best approximation for 1setcontractive mappings)
Let U be a bounded open pconvex subset of a locally pconvex space E (\(0 < p \leq 1\)), zero \(0 \in U\), and C be a (bounded) closed convex subset of E with also zero \(0\in C\). Assume that \(F: \overline{U}\cap C \rightarrow 2^{C}\) is a 1set contractive and quasi upper semicontinuous mapping with nonempty pconvex values and with a closed graph, and for each \(x \in \partial _{C} U\) with \(y \in F(x) \cap (C \diagdown \overline{U})\)), \((P^{\frac{1}{p}}_{U}(y) 1)^{p} \leq P_{U} (yx)\) for \(0< p \leq 1\) (this is trivial when \(p=1\)). In addition, if F satisfies the following (H) or (H1) condition:
(H) Condition: The sequence \(\{x_{n}\}_{n\in \mathbb{N}}\) in U̅ has a convergent subsequence with limit \(x_{0} \in \overline{U}\) such that \(x_{0} \in F(x_{0})\), whenever \(\lim_{n \rightarrow \infty} d_{P_{U}}(x_{n}, F(x_{n})) =0\), where \(d_{P_{U}}(x_{n}, F(x_{n})):=\inf \{P_{U}(x_{n} z): z \in F(x_{n})\}\), where \(P_{U}\) is the Minkowski pfunctional for any \(U \in \mathfrak{U}\), which is the family of all nonempty open pconvex subsets containing zero in E.
(H1) Condition: There exists \(x_{0}\) in U̅ with \(x_{0} \in F(x_{0})\) if there exists \(\{x_{n}\}_{n\in \mathbb{N}}\) in U̅ such that \(\lim_{n \rightarrow \infty} d_{P_{U}}(x_{n}, F(x_{n})) =0\), where \(P_{U}\) is the Minkowski pfunctional for any \(U \in \mathfrak{U}\), which is the family of all nonempty open pconvex subsets containing zero in E.
Then we have that there exist \(x_{0} \in C \cap \overline{U}\) and \(y_{0} \in F(x_{0})\) such that
where \(P_{U}\) is the Minkowski pfunctional of U. More precisely, we have that either (I) or (II) holds:

(I)
F has a fixed point \(x_{0} \in \overline{U} \cap C\), i.e., \(0=P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, \overline{I^{p}_{\overline{U}}( x_{0})} \cap C)\);

(II)
There exist \(x_{0} \in \partial _{C}(U)\) and \(y_{0} \in F(x_{0}) \diagdown \overline{U}\) with
$$ P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p} \bigl(y_{0}, \overline{I^{p}_{\overline{U}}( x_{0})} \cap C \bigr)= \bigl(P^{\frac{1}{p}}_{U}(y_{0})1 \bigr)^{p} > 0. $$
Proof
As E is a pconvex space and U is a bounded open pconvex subset of E, it suffices to prove that there exists a sequence \((x_{n})_{n \in \mathbb{N}}\) in U̅ and \(y_{n} \in F(x_{n})\) such that \(\lim_{n\rightarrow \infty} P_{U}(y_{n}x_{n})=0\), and the conclusion follows by applying the (H) condition.
Let \(r: E \rightarrow U\) be a retraction mapping defined by \(r(x): = \frac{x}{\max \{1, (P_{U}(x))^{\frac{1}{p}}\}}\) for each \(x \in E\), where \(P_{U}\) is the Minkowski pfunctional of U. Since the space E’s zero \(0 \in U\)(\(=\operatorname{int}U\) as U is open), it follows that r is continuous by Lemma 2.2. As the mapping F is 1set contractive, take an increasing sequence \(\{\lambda _{n}\}\) such that \(0 < \lambda _{n} < 1\) and \(\lim_{n \rightarrow \infty} \lambda _{n} =1\), where \(n \in \mathbb{N}\). Now, for each \(n\in \mathbb{N}\), we define a mapping \(F_{n}: C \cap \overline{U} \rightarrow 2^{C}\) by \(F_{n}(x): = \lambda _{n} F\circ r(x)\) for each \(x \in C \cap \overline{U}\). By the fact that C and U̅ are pconvex, it follows that \(r(C) \subset C\) and \(r(\overline{U}) \subset \overline{U}\), thus \(r( C \cap \overline{U}) \subset C \cap \overline{U}\). Therefore \(F_{n}\) is a mapping from \(\overline{U}\cap C\) to itself. For each \(n \in \mathbb{N}\), by the fact that \(F_{n}\) is a \(\lambda _{n}\)setcontractive mapping with \(0 < \lambda _{n} < 1\), it is also QUSC with nonempty pconvex and its graph is also closed. Then it follows by Theorem 4.8 for the condensing mapping that there exists \(z_{n} \in C \cap \overline{U}\) such that \(z_{n} \in F_{n}(z_{n})=\lambda _{n} F \circ r(z_{n})\). As \(r( C \cap \overline{U}) \subset C \cap \overline{U}\), let \(x_{n}= r(z_{n})\). Then we have that \(x_{n} \in C\cap \overline{U}\), and there exists \(y_{n} \in F(x_{n})\) with \(x_{n} = r(\lambda _{n} y_{n})\) such that the following (1) or (2) holds for each \(n \in \mathbb{N}\): (1) \(\lambda _{n} y_{n} \in C\cap \overline{U}\); or (2) \(\lambda _{n} y_{n} \in C \diagdown \overline{U}\).
Now we prove the conclusion by considering the following two cases under (H) condition and (H1) condition.
Case (I) For each \(n \in N\), \(\lambda _{n} y_{n} \in C \cap \overline{U}\); or
Case (II) There exists a positive integer n such that \(\lambda _{n} y_{n} \in C \diagdown \overline{U}\).
First, by case (I), for each \(n \in \mathbb{N}\), \(\lambda _{n} y_{n} \in \overline{U} \cap C\), which implies that \(x_{n} = r(\lambda _{n} y_{n}) = \lambda _{n} y_{n}\), thus \(P_{U}(\lambda _{n} y_{n}) \leq 1\) by Lemma 2.2. Note that
which implies that \(\lim_{n\rightarrow \infty} P_{U}(y_{n}x_{n})=0\). Now, for any \(V \in \mathbb{U}\), without loss of generality, let \(U_{0} = V \cap U\). Then we have the following conclusion:
which implies that \(\lim_{n\rightarrow \infty} P_{U_{0}}(y_{n}x_{n})=0\), where \(P_{U_{0}}\) is the Minkowski pfunctional of \(U_{0}\) in E.
Now, if F satisfies the (H) condition, then it follows that the consequence \(\{x_{n}\}_{n \in \mathbb{N}}\) has a convergent subsequence that converges to \(x_{0}\) such that \(x_{0} \in F(x_{0})\). Without loss of generality, we assume that \(\lim_{n \rightarrow \infty} x_{n}=x_{0}\), where \(y_{n} \in F(x_{n})\) is with \(x_{n}=\lambda _{n} y_{n}\) and \(\lim_{n \rightarrow \infty} \lambda _{n}=1\), and as \(x_{0}=\lim_{n \rightarrow \infty} (\lambda _{n} y_{n})\), which implies that \(y_{0}=\lim_{n\rightarrow \infty} y_{n}= x_{0}\). Thus there exists \(y_{0} (= x_{0}) \in F(x_{0})\), we have \(0 = d_{p}(x_{0}, F(x_{0})) = d(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, \overline{I^{p}_{\overline{U}}(x_{0})} \cap C))\) as indeed \(x_{0} =y_{0} \in F(x_{0}) \in \overline{U}\cap C \subset \overline{I^{p}_{\overline{U}}( x_{0})} \cap C)\).
If F satisfies the (H1) condition, then it follows that there exists \(x_{0} \in \overline{U} \cap C\) with \(x_{0} \in F(x_{0})\). Then we have \(0=P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, \overline{I^{p}_{\overline{U}}( x_{0})} \cap C)\).
Second, by case (II) there exists a positive integer n such that \(\lambda _{n} y_{n} \in C \diagdown \overline{U}\). Then we have that \(P_{U}(\lambda _{n} y_{n})> 1\), and also \(P_{U}(y_{n})> 1\) as \(\lambda _{n} < 1\). As \(x_{n} = r(\lambda _{n} y_{n}) = \frac{\lambda _{n} y_{n}}{(P_{U}(\lambda _{n} y_{n}))^{\frac{1}{p}}}\), which implies that \(P_{U}(x_{n})=1\), thus \(x_{n} \in \partial _{C}(U)\). Note that
By the assumption, we have \((P^{\frac{1}{p}}_{U}(y_{n})1)^{p} \leq P_{U}(y_{n} x)\) for \(x \in C \cap \partial \overline{U}\), it follows that
Thus we have the best approximation: \(P_{U}(y_{n}  x_{n})=d_{P}(y_{n}, \overline{U} \cap C) = (P^{ \frac{1}{p}}_{U}(y_{n})1)^{p} > 0\).
Now we want to show that \(P_{U}(y_{n}x_{n})= d_{P}(y_{n}, \overline{U} \cap C) = d_{p}(y_{n}, \overline{I^{p}_{\overline{U}}( x_{0})} \cap C) > 0\).
By the fact that \((\overline{U}\cap C) \subset I^{p}_{\overline{U}}(x_{n})\cap C\), let \(z \in I^{p}_{\overline{U}}(x_{n})\cap C \diagdown (\overline{U}\cap C)\), we first claim that \(P_{U}(y_{n}  x_{n}) \leq P_{U}(y_{n}z)\). If not, we have \(P_{U}(y_{n}  x_{n}) > P_{U}(y_{n}z)\). As \(z \in I^{p}_{\overline{U}}(x_{n}) \cap C \diagdown (\overline{U} \cap C)\), there exist \(y \in \overline{U}\) and a nonnegative number c (actually \(c\geq 1\) as shown soon below) with \(z = x_{n} + c (y  x_{n})\). Since \(z \in C\), but \(z \notin \overline{U} \cap C\), it implies that \(z \notin \overline{U}\). By the fact that \(x_{n}\in \overline{U}\) and \(y \in \overline{U}\), we must have the constant \(c \geq 1\); otherwise, it implies that \(z ( = (1 c )x_{n} + c y) \in \overline{U}\), this is impossible by our assumption, i.e., \(z\notin \overline{U}\). Thus we have that \(c\geq 1\), which implies that \(y =\frac{1}{c} z + (1\frac{1}{c}) x_{n} \in C\) (as both \(x_{n} \in C\) and \(z\in C\)). On the other hand, as \(z \in I^{p}_{\overline{U}}(x_{n}) \cap C \diagdown (\overline{U} \cap C)\), and \(c\geq 1\) with \((\frac{1}{c})^{p}+ (1\frac{1}{c})^{p} = 1 \), combining with our assumption that for each \(x \in \partial _{C} \overline{U}\) and \(y \in F(x_{n})\diagdown \overline{U}\), \(P^{\frac{1}{p}}_{U}(y) 1 \leq P^{\frac{1}{p}}_{U} (yx)\) for \(0< p \leq 1\), it then follows that
which contradicts that \(P_{U} (y_{n}  x_{n}) = d_{P}(y_{n}, \overline{U}\cap C)\) as shown above, we know that \(y \in \overline{U}\cap C\), we should have \(P_{U}(y_{n} x_{n})\leq P_{U}(y_{n}  y)\)! This helps us to complete the claim: \(P_{U}(y_{n}  x_{n}) \leq P_{U}(y_{n}  z)\) for any \(z \in I^{p}_{\overline{U}}(x_{n})\cap C \diagdown (\overline{U}\cap C)\), which means that the following best approximation of Fan type (see [42, 43]) holds:
Now, by the continuity of \(P_{U}\), it follows that the following best approximation of Fan type is also true:
The proof is complete. □
Remark 5.2
Based on the proof of Theorem 5.2, we have that (1): For the condition “\(x \in \partial _{C} U\) with \(y \in F(x)\), \(P^{\frac{1}{p}}_{U}(y) 1 \leq P^{\frac{1}{p}}_{U} (yx)\) for \(0< p \leq 1\)”, indeed we only need that for “\(x \in \partial _{C} U\) with \(y \in F(x) \cap (C \diagdown \overline{U})\)), \(P^{\frac{1}{p}}_{U}(y) 1 \leq P^{\frac{1}{p}}_{U} (yx)\) for \(0< p \leq 1\)”; (2): Theorem 5.2 also improves the corresponding best approximation for 1set contractive mappings given by Li et al. [78], Liu [81], Xu [139], Xu et al. [140], and the results from the references therein; and (3): When \(p=1\), we have a similar best approximation result for the mapping F in the locally convex spaces with outward set boundary condition below (see Theorem 3 of Park [97] and related discussion by the references therein).
For the pvector space with \(p=1\) being a topological vector space E, we have the following best approximation for the outward set \(\overline{O_{\overline{U}}(x_{0})}\) based on the point \(\{x_{0}\}\) with respect to the convex subset U in E.
Theorem 5.3
(Best approximation for outward sets)
Let U be a bounded open convex subset of a locally convex space E (i.e., \(p=1\)) with zero \(0 \in \operatorname{int}U=U\) (the interior \(\operatorname{int}U=U\) as U is open), and let C be a closed pconvex subset of E with also zero \(0\in C\). Assume that \(F: \overline{U}\cap C \rightarrow 2^{C}\) is a 1setcontractive quasi upper semicontinuous mapping with nonempty pconvex values and with a closed graph that satisfies condition (H) or (H1) above. Then there exist \(x_{0} \in \overline{U} \cap X\) and \(y_{0} \in F(x_{0})\) such that \(P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, \overline{O_{\overline{U}}( x_{0})} \cap C)\), where \(P_{U}\) is the Minkowski pfunctional of U. More precisely, we have that either (I) or (II) holds:

(I)
F has a fixed point \(x_{0} \in U \cap C\), i.e., \(x_{0} \in F(x_{0})\) such that \(P_{U} (y_{0}  x_{0}) = P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, \overline{O_{\overline{U}}( x_{0})} \cap C))=0\);

(II)
There exist \(x_{0} \in \partial _{C}(U)\) and \(y_{0} \in F(x_{0})\diagdown \overline{U}\) with
$$\begin{aligned} P_{U} (y_{0}  x_{0}) &= d_{P}(y_{0}, \overline{U}\cap C) \\ &= d_{p} \bigl(y_{0}, O_{\overline{U}}(x_{0}) \cap C \bigr) =d_{p} \bigl(y_{0}, \overline{O_{\overline{U}}( x_{0})} \cap C \bigr) > 0. \end{aligned}$$
Proof
We define a new mapping \(F_{1}: \overline{U}\cap C \rightarrow 2^{C}\) by \(F_{1} (x): = \{2x\}  F(x)\) for each \(x \in \overline{U}\cap C\), then \(F_{1}\) is also compact and upper semicontinuous mapping with nonempty closed convex values, and \(F_{1}\) satisfies all hypotheses of Theorem 5.2 with \(p=1\). If follows by Theorem 5.2 that there exist \(x_{0} \in \overline{U} \cap X\) and \(y_{1} \in F_{1}(x_{0})\) such that \(P_{U} (y_{1}  x_{0}) = d_{P}(y_{1}, \overline{U}\cap C) = d_{p}(y_{1}, \overline{I_{\overline{U}}(x_{0})} \cap C)\). More precisely, we have the following either (I) or (II) holding:
(I) \(F_{1}\) has a fixed point \(x_{0} \in U \cap C\) (so \(0= P_{U} (y_{1}  x_{0}) = P_{U} (y_{1}  x_{0}) = d_{P}(y_{1}, \overline{U}\cap C) = d_{p}(y_{1}, \overline{I_{\overline{U}}( x_{0})} \cap C)\));
(II) There exist \(x_{0} \in \partial _{C}(U)\) and \(y_{1} \in F_{1}(x_{0})\diagdown \overline{U}\) with
Now, for any \(x \in O_{\overline{U}}(x_{0})\), there exist \(r < 0\), \(u \in \overline{U}\) such that \(x=x_{0} + r (u  x_{0})\). Let \(x_{1}=2x_{0}  x\), then \(x_{1} = 2x_{0}  x_{0}  r(u  x_{0})= x_{0} +(r) (u  x_{0}) \in I_{ \overline{U}}(x_{0})\). Let \(y_{1} = 2 x_{0}  y_{0}\) for some \(y_{0}\in F(x_{0})\). As we have \(P_{U} (y_{1}  x_{0}) = d_{P}(y_{1}, \overline{U}\cap C) = d_{p}(y_{1}, \overline{I_{\overline{U}}(x_{0})} \cap C)\), it follows that \(P_{U} (y_{1}  x_{0}) \leq P_{U} (y_{1}  x_{1})\), which implies that
for all \(x \in O_{\overline{U}}(x_{0})\). Thus we have \(P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, O_{\overline{U}}( x_{0}) \cap C)\), and by the continuity of \(P_{U}\), it follows that
This completes the proof. □
Now, by the application of Theorem 5.2, Theorem 5.3, Remark 5.2, and the argument used in Theorem 5.2, we have the following general principle for the existence of solutions for Birkhoff–Kellogg problems in pseminorm spaces, where (\(0 < p \leq 1\)).
Theorem 5.4
(Principle of Birkhoff–Kellogg alternative)
Let U be a bounded open pconvex subset of a locally pconvex space E (\(0 < p \leq 1\)) with zero \(0 \in \operatorname{int}U=U\), and let C be a closed pconvex subset of E with also zero \(0\in C\). Assume that \(F: \overline{U}\cap C \rightarrow 2^{C}\) is a 1setcontractive quasi upper semicontinuous mapping with nonempty pconvex values and with a closed graph, satisfying the (H) or (H1) condition above. Then F has at least one of the following two properties:

(I)
F has a fixed point \(x_{0} \in U \cap C\) such that \(x_{0} \in F(x_{0})\);

(II)
There exist \(x_{0} \in \partial _{C}(U)\), \(y_{0} \in F(x_{0})\diagdown \overline{U}\), and \(\lambda = \frac{1}{(P_{U}(y_{0}))^{\frac{1}{p}}} \in (0, 1)\) such that \(x_{0} = \lambda y_{0} \in \lambda F(x_{0})\); In addition if for each \(x \in \partial _{C} U\), \(P^{\frac{1}{p}}_{U}(y) 1 \leq P^{\frac{1}{p}}_{U} (yx)\) for \(0< p \leq 1\) (this is trivial when \(p=1\)), then the best approximation between \(x_{0}\) and \(y_{0}\) is given by
$$ P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p} \bigl(y_{0}, \overline{I^{p}_{\overline{U}}( x_{0})} \cap C \bigr) = \bigl(P^{\frac{1}{p}}_{U}(y_{0})1 \bigr)^{p} > 0. $$
Proof
If (I) is not the case, then (II) is proved by Remark 5.2 and by following the proof in Theorem 5.2 for case (ii): \(y_{0} \in C \diagdown \overline{U}\) with \(y_{0}: = f(x_{0}) \in F(x_{0})\). Indeed, as \(y_{0} \notin \overline{U}\), it follows that \(P_{U}(y_{0}) > 1\) and \(x_{0}= f(y_{0}) = y_{0} \frac{1}{(P_{U}(y_{0}))^{\frac{1}{p}}}\). Now let \(\lambda = \frac{1}{(P_{U}(y_{0}))^{\frac{1}{p}}}\), we have \(\lambda < 1\) and \(x_{0} = \lambda y_{0}\) with \(y_{0} \in F(x_{0})\). Finally, the additionally assumption in (II) allows us to have the best approximation between \(x_{0}\) and \(y_{0}\) obtained by following the proof of Theorem 5.2 as \(P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, \overline{I^{p}_{\overline{U}}( x_{0})} \cap C) > 0\). This completes the proof. □
As an application of Theorem 5.2 for the nonself setvalued mappings discussed in Theorem 5.3 with outward set condition, we have the following general principle of Birkhoff–Kellogg alternative in topological vector spaces.
Theorem 5.5
(Principle of Birkhoff–Kellogg alternative in TVS)
Let U be a bounded open pconvex subset of a locally pconvex space E with zero \(0 \in U\), and let C be a closed convex subset of E with also zero \(0\in C\). Assume that \(F: \overline{U}\cap C \rightarrow 2^{C}\) is a 1set contractive and quasi upper semicontinuous mapping with nonempty pconvex values and with a closed graph, satisfying the (H) or (H1) condition (H) above. Then it has at least one of the following two properties:

(I)
F has a fixed point \(x_{0} \in U \cap C\) such that \(x_{0} \in F(x_{0})\);

(II)
There exist \(x_{0} \in \partial _{C}(U)\) and \(y_{0} \in F(x_{0}) \diagdown \overline{U}\) and \(\lambda \in (0, 1)\) such that \(x_{0} = \lambda y_{0}\), and the best approximation between \(x_{0}\) and \(y_{0}\) is given by \(P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, \overline{I^{p}_{\overline{U}}( x_{0})} \cap C) > 0\).
On the other hand, by the proof of Theorem 5.2, we note that for case (II) of Theorem 5.2, the assumption “each \(x \in \partial _{C} U\) with \(y \in F(x)\), \(P^{\frac{1}{p}}_{U}(y) 1 \leq P^{\frac{1}{p}}_{U} (yx)\)” is only used to guarantee the best approximation \(``P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, \overline{I^{p}_{\overline{U}}( x_{0})} \cap C) > 0\)”, thus we have the following Leray–Schauder alternative in pvector spaces, which, of course, includes the corresponding results in locally convex spaces as special cases.
Theorem 5.6
(The Leray–Schauder nonlinear alternative)
Let C be a closed pconvex subset of pseminorm space E with \(0 \leq p \leq 1\) and zero \(0 \in C\). Assume that \(F: C \rightarrow 2^{C}\) is a 1set contractive and quasi upper semicontinuous mapping with nonempty pconvex values and with a closed graph, satisfying the (H) or (H1) condition above. Let \(\varepsilon (F): =\{x \in C: x\in \lambda F(x) \textit{ for some } 0 < \lambda < 1\}\). Then either F has a fixed point in C or the set \(\varepsilon (F)\) is unbounded.
Proof
We prove the conclusion by assuming that F has no fixed point, then we claim that the set \(\varepsilon (F)\) is unbounded. Otherwise, assume the set \(\varepsilon (F)\) is bounded, and assume that P is the continuous pseminorm for E, then there exists \(r>0\) such that the set \(B(0, r):=\{x \in E: P(x) < r\}\), which contains the set \(\varepsilon (F)\), i.e., \(\varepsilon (F) \subset B(0, r)\), which means for any \(x \in \varepsilon (F)\), \(P(x) < r\). Then \(B(0. r)\) is an open pconvex subset of E and zero \(0 \in B(0, r)\) by Lemma 2.2 and Remark 2.4. Now let \(U:=B(0, r)\) in Theorem 5.4, it follows that the mapping \(F: B(0, r) \cap C \rightarrow 2^{C}\) satisfies all general conditions of Theorem 5.4, and we have that any \(x_{0} \in \partial _{C} B(0, r)\), no any \(\lambda \in (0, 1)\) such that \(x_{0}=\lambda y_{0}\), where \(y_{0} \in F(x_{0})\). Indeed, for any \(x \in \varepsilon (F)\), it follows that \(P(x) < r\) as \(\varepsilon (F) \subset B(0, r)\), but for any \(x_{0} \in \partial _{C} B(0, r)\), we have \(P(x_{0})=r\), thus conclusion (II) of Theorem 5.4 does not hold. By Theorem 5.4 again, F must have a fixed point, but this contradicts our assumption that F is fixed point free. This completes the proof. □
Now assume a given pvector space E equipped with the Pseminorm (by assuming it is continuous at zero) for \(0< p \leq 1\), then we know that \(P: E \rightarrow \mathbb{R}^{+}\), \(P^{1}(0)=0\), \(P(\lambda x) = \lambda ^{p} P(x)\) for any \(x\in E\) and \(\lambda \in \mathbb{R}\). Then we have the following useful result for fixed points due to Rothe and Altman types in locally pconvex spaces, which plays important roles for optimization problem, variational inequality, complementarity problems (see Isac [60] or Yuan [143] and the references therein for related study in detail).
Corollary 5.1
Let U be a bounded open pconvex subset of a locally pconvex space E and zero \(0 \in U\), plus C is a closed pconvex subset of E with \(U \subset C\), where \(0< p \leq 1\). Assume that \(F: \overline{U} \rightarrow 2^{C}\) is a 1set contractive quasi upper semicontinuous mapping with nonempty pconvex values and with a closed graph, satisfying the (H) or (H1) condition above. If one of the following is satisfied:

(1)
(Rothe type condition): \(P_{U}(y) \leq P_{U}(x)\) for \(y \in F(x)\), where \(x \in \partial U\);

(2)
(Petryshyn type condition): \(P_{U}(y) \leq P_{U}(yx)\) for \(y \in F(x)\), where \(x \in \partial U\);

(3)
(Altman type condition): \(P_{U}(y)^{\frac{2}{p}} \leq [P_{U}(y) x)]^{\frac{2}{p}} + [P_{U}(x)]^{ \frac{2}{p}}\) for \(y \in F(x)\), where \(x \in \partial U\),
then F has at least one fixed point.
Proof
By conditions (1), (2), and (3), it follows that the conclusion of (II) in Theorem 5.4 “there exist \(x_{0} \in \partial _{C}(U)\) and \(\lambda \in (0, 1)\) such that \(x_{0} \notin \lambda F(x_{0})\)” does not hold, thus by the alternative of Theorem 5.4, F has a fixed point. This completes the proof. □
By the fact that when \(p=1\), each locally pconvex space is a locally convex space, we have the following classical Fan’s best approximation (see [42]) as a powerful tool for the study in the optimization, mathematical programming, games theory, and mathematical economics, and other related topics in applied mathematics.
Corollary 5.2
(Fan’s best approximation)
Let U be a bounded open convex subset of a locally convex space E with zero \(0 \in U\), let C be a closed convex subset of E with also zero \(0\in C\), and assume that \(F: \overline{U}\cap C \rightarrow 2^{C}\) is a 1set contractive and quasi upper semicontinuous mapping with nonempty closed convex values satisfying the (H) or (H1) condition above. Assume that \(P_{U}\) is the Minkowski pfunctional of U in E. Then there exist \(x_{0} \in \overline{U} \cap X\) and \(y_{0} \in T(x_{0})\) such that \(P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, \overline{I_{\overline{U}}( x_{0})} \cap C)\). More precisely, we have the following either (I) or (II) holding, where \(W_{\overline{U}}(x_{0})\) is either the inward set \(I_{\overline{U}}(x_{0})\) or the outward set \(O_{\overline{U}}(x_{0})\):

(I)
F has a fixed point \(x_{0} \in U \cap C\), \(0= P_{U} (y_{0}  x_{0}) = P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, \overline{W_{\overline{U}}( x_{0})} \cap C))\);

(II)
There exist \(x_{0} \in \partial _{C}(U)\) and \(y_{0} \in F(x_{0}) \diagdown \overline{U}\) with
$$ P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p} \bigl(y_{0}, \overline{W_{\overline{U}}( x_{0})} \cap C \bigr) = P_{U}(y_{0})  1 > 0. $$
Proof
When \(p=1\), it automatically satisfies the inequality \(P^{\frac{1}{p}}_{U}(y) 1 \leq P^{\frac{1}{p}}_{U} (yx)\), and indeed we have that for \(x_{0}\in \partial _{C}(U)\), with \(y_{0} \in F(x_{0})\), we have \(P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, \overline{W_{\overline{U}}( x_{0})} \cap C)= P_{U}(y_{0})1\). The conclusions are given by Theorem 5.2 (or Theorem 5.3). The proof is complete. □
We would like to point out that similar results on Rothe and Leray–Schauder alternative have been developed by Isac [60], Park [96], Potter [108], Shahzad [120–122], Xiao and Zhu [135], and the related references therein as tools of nonlinear analysis in locally pconvex spaces. As mentioned above, when \(p=1\) and take F as a continuous mapping, then we obtain the version of Leray–Schauder in locally convex spaces, and thus we omit its statement in detail.
6 Fixed points for the class of nonself semiclosed 1set contractive mappings
In this section, based on the best approximation Theorem 5.2 for classes of semiclosed 1set contractive mappings developed in Sect. 5, we show how it can be used as a useful tool to establish fixed point theorems for nonself upper semicontinuous mappings in locally pconvex spaces for \(p \in (0, 1]\), including norm spaces and uniformly convex Banach spaces as special classes.
By following Browder [18], Li [77], Goebel and Kirk [48], Petryshyn [104, 105], Tan and Yuan [129], Xu [139], and the references therein, we recall some definitions as follows for pseminorm spaces, where \(p \in (0, 1]\).
Definition 6.1
Let D be a nonempty (bounded) closed subset of pvector spaces \((E, \\cdot \_{p})\) with pseminorm, where \(p \in (0, 1]\). Suppose that \(f: D \rightarrow X\) is a (singlevalued) mapping, then: (1) f is said to be nonexpansive if for each \(x, y \in D\), we have \(\f(x) f(y)\_{p} \leq \xy\_{p}\); (2) f (actually, \((If)\)) is said to be demiclosed (see Borwder [18]) at \(y \in X\) if for any sequence \(\{x_{n}\}_{n \in \mathbb{N}}\) in D, the conditions \(x_{n} \rightarrow x_{0}\in D\) weakly, and \((If)(x_{n}) \rightarrow y_{0}\) strongly imply that \((If)(x_{0})=y_{0}\), where I is the identity mapping; (3) f is said to be hemicompact (see p. 379 of Tan and Yuan [129]) if each sequence \(\{x_{n}\}_{n \in \mathbb{N}}\) in D has a convergent subsequence with the limit \(x_{0}\) such that \(x_{0} = f(x_{0})\), whenever \(\lim_{n \rightarrow \infty}d_{p}(x_{n}, f(x_{n}))=0\), here \(d_{P}(x_{n}, f(x_{n})):=\inf \{P_{U}(x_{n} z): z \in f(x_{n})\}\), and \(P_{U}\) is the Minkowski pfunctional for any \(U \in \mathfrak{U}\), which is the family of all nonempty open pconvex subset containing zero in E; (4) f is said to be demicompact (by Petryshyn [104]) if each sequence \(\{x_{n}\}_{n \in \mathbb{N}}\) in D has a convergent subsequence whenever \(\{x_{n} f(x_{n})\}_{n \in \mathbb{N}}\) is a convergent sequence in X; (5) f is said to be a semiclosed 1set contractive mapping if f is 1set contractive mapping and \((If)\) is closed, where I is an identity mapping (by Li [77]); and (6) f is said to be semicontractive (see Petryshyn [105] and Browder [18]) if there exists a mapping \(V: D \times D \rightarrow 2^{X}\) such that \(f(x) = V(x, x)\) for each \(x \in D\), with (a) for each fixed \(x \in D\), \(V(\cdot , x)\) is nonexpansive from D to X; and (b) for each fixed \(x\in D\), \(V(x, \cdot )\) is completely continuous from D to X, uniformly for u in a bounded subset of D (which means if \(v_{j}\) converges weakly to v in D and \(u_{j}\) is a bounded sequence in D, then \(V(u_{j}, v_{j})  V(u_{j}, v) \rightarrow 0\), strongly in D).
From the definition above, we first observe that definitions (1) to (6) for setvalued mappings can be given in a similar way with the Hausdorff metric H (we omit their definitions here in detail to save space); Secondly, if f is a continuous demicompact mapping, then \((I  f)\) is closed, where I is the identity mapping on X. It is also clear from the definitions that every demicompact map is hemicompact in seminorm spaces, but the converse is not true by the example on p. 380 by Tan and Yuan [129]. It is evident that if f is demicompact, then \(If\) is demiclosed. It is known that for each condensing mapping f, when D or \(f(D)\) is bounded, then f is hemicompact; and also f is demicompact in metric spaces by Lemma 2.1 and Lemma 2.2 of Tan and Yuan [129], respectively. In addition, it is known that every nonexpansive map is a 1setcontractive mapping; and also if f is a hemicompact 1setcontractive mapping, then f is a 1setcontractive mapping satisfying the following (H1) condition (which is the same as “condition (H1)” in Sect. 5, but slightly different from condition (H) used there in Sect. 5):
(H1) condition: Let D be a nonempty bounded subset of a space E, and assume that \(F: \overline{D} \rightarrow 2^{E}\) a setvalued mapping. If \(\{x_{n}\}_{n \in \mathbb{N}}\) is any sequence in D such that for each \(x_{n}\), there exists \(y_{n} \in F(x_{n})\) with \(\lim_{n \rightarrow \infty} (x_{n} y_{n})=0\), then there exists a point \(x\in \overline{D}\) such that \(x \in F(x)\).
We first note that the “(H1) condition” above is actually the same one as the “condition (C)” used in Theorem 1 by Petryshyn [105]. Secondly, it was shown by Browder [18] that indeed the nonexpansive mapping in a uniformly convex Banach X enjoys condition (H1) as shown below.
Lemma 6.1
Let D be a nonempty bonded convex subset of a uniformly convex Banach space E. Assume that \(F: \overline{D} \rightarrow E\) is a nonexpansive (singlevalued) mapping, then the mapping \(P: = I  F\) defined by \(P(x): = (xF(x)) \) for each \(x \in \overline{D}\) is demiclosed, and in particular, the “(H1) condition” holds.
Proof
By following the argument given on p. 329 (see the proof of Theorem 2.2 and Corollary 2.1) by Petryshyn [105], the mapping F is demiclosed (which actually is called Browder’s demiclosedness principle), which says that by the assumption of (H1) condition, if \(\{x_{n}\}_{n \in \mathbb{N}}\) is any sequence in D such that for each \(x_{n}\) there exists \(y_{n} \in F(x_{n})\) with \(\lim_{n \rightarrow \infty} (x_{n} y_{n})=0\), then we have \(0 \in (I  F) (\overline{D})\), which means that there exists \(x_{0} \in \overline{D}\) with \(0 \in (IF)(x_{0})\), this implies that \(x_{0} \in F(x_{0})\). The proof is complete. □
Remark 6.1
When a pvector space E is with a pnorm, then “(H) condition” satisfies the “(H1) condition”. The (H1) condition is mainly supported by the socalled demiclosedness principle after the work by Browder [18].
Lemma 6.1 above shows that s singlevalued nonexpansive mapping defined in a uniformly convex Banach space satisfied the (H1) condition. Actually, the nonexpansive setvalued mappings defined on a special class of Banach spaces with the socalled the “Opial’s condition” do not only satisfy condition (H1), but also belong to the classes of semiclosed 1set contractive mappings, as shown below.
The notion of the socalled “Opial’s condition” first given by Opial [90] says that a Banach space X is said to satisfy Opial’s condition if \(\liminf_{n \rightarrow \infty} \ w_{n}  w \ < \liminf_{n \rightarrow \infty} \w_{n}p\\) whenever \((w_{n})\) is a sequence in X weakly convergent to w and \(p\neq w\). We know that Opial’s condition plays an important role in the fixed point theory, e.g., see Lami Dozo [75], Goebel and Kirk [49], Xu [137], and the references therein. The following result shows that there are nonexpansive setvalued mappings in Banach spaces with Opial’s condition (see Lami Dozo [75] satisfying the condition (H1).
Lemma 6.2
Let C be convex weakly compact of a Banach space X that satisfies Opial’s condition. Let \(T: C \rightarrow K(C)\) be a nonexpansive setvalued mapping with nonempty compact values. Then the graph of \((IT)\) is closed in \((X, \sigma (X, X^{*}) \times (X, \\cdot \))\), thus T satisfies the “(H1) condition”, where I denotes the identity on X, \(\sigma (X, X^{*})\) is the weak topology, and \(\\cdot \\) is the norm (or strong) topology.
Proof
By following Theorem 3.1 of Lami Dozo [75], it follows that the mapping T is demiclosed, thus T satisfies the “(H1) condition”. The proof is complete. □
For the convenience of our study, for the fixed point theory for a class of semiclosed 1set contractive mappings in pseminorm spaces, we also need to introduce the following definition, which is a setvalued generalization of singlevalue semiclosed 1set mappings first discussed by Li [77], Xu [139] (see also Li et al. [78], Xu et al. [140], and the references therein).
Definition 6.2
Let D be a nonempty (bounded) closed subset of pvector spaces \((E, \\cdot \_{p})\) with pseminorm, where \(p \in (0, 1]\) (which includes norm spaces or Banach spaces as special classes), and suppose that \(T: D \rightarrow X\) is a setvalued mapping. Then F is said to be a semiclosed 1set contraction mapping if T is 1set contraction, and \((IT)\) is closed, which means that for a given net \(\{x_{n}\}_{i \in I}\), for each \(i \in I\), there exists \(y_{i} \in T(x_{i})\) with \(\lim_{i \in I} (x_{i}  y_{i})=0\), then \(0 \in (IT)(\overline{D})\), i.e., there exists \(x_{0} \in \overline{D}\) such that \(x_{0} \in T(x_{0})\).
By Lemmas 6.1 and 6.2, it follows that each nonexpansive (singlevalued) mapping defined on a subset of uniformly convex Banach spaces and each nonexpansive setvalued mapping defined on a subset of Banach spaces satisfying Opial’s condition is a semiclosed 1set contractive mapping (see also Goebel [47], Goebel and Kirk [48], Petrusel et al. [103], Xu [137], Yangai [141], and the references therein for related discussion). In particular, under the setting of metric spaces or Banach spaces with certain property, it is clear that each semiclosed 1set contractive mapping satisfies condition (H1) above.
We know that compared to the singlevalued case, based on the study in the literature about the approximation of fixed points for multivalued mappings, a wellknown counterexample due to Pietramala [106] (see also Muglia and Marino [85]) proved in 1991 that Browder approximation Theorem 1 given by Browder [16] cannot be extended to the genuine multivalued case even on a finite dimensional space \(\mathbb{R}^{2}\). Moreover, if a Banach space X satisfies Opial’s property (see Opial [90]) that is, if \(x_{n}\) weakly converges to x, then we have that \(\limsup \x_{n}x\ < \limsup \x_{n} y\\) for all \(x \in X\) and \(y \neq x\)), then \(I  f\) is demiclosed at 0 (see Lami Dozo [75], Yanagi [141], and the related references therein) provided \(f: C: \rightarrow K(C)\) is nonexpansive (here \(K(C)\) denotes a family of nonempty compact subsets of C). We know that all Hilbert spaces and \(L^{p}\) spaces \(p \in (1, \infty )\) have Opial’s property, but it seems that whether \(If\) is demiclosed at zero 0 if f is a nonexpansive setvalued mapping defined on the space X which is uniformly convex (e.g., \(L[0, 1]\), \(1 < p < \infty \), ≠2) and \(f: C \rightarrow K(C)\) is nonexpansive. Here we remark that for a singlevalued nonexpansive mapping f is yes, which is the famous theorem of Browder [15]. A remarkable fixed point theorem for multivalued mappings is Lim’s result in [80], which says that: If C is a nonempty closed bounded convex subset of a uniformly convex Banach space X and \(f: C \rightarrow K(C)\) is nonexpansive, then f has a fixed point.
Now, based on the concept for the semiclosed 1set contractive mappings, we give the existence results for their best approximation, fixed points, and related nonlinear alterative under the framework of pseminorm spaces for \(p \in (0, 1]\).
Theorem 6.1
(Schauder fixed point theorem for semiclosed 1set contractive mappings)
Let U be a nonempty bounded open psubset of a (Hausdorff) locally pconvex space E and its zero \(0 \in U\), and let \(C \subset E\) be a closed pconvex subset of E such that \(0 \in C\) with \(0 < p \leq 1\). If \(F: C \cap \overline{U} \rightarrow 2^{C \cap \overline{U}}\) is a quasi upper semicontinuous and semiclosed 1set contractive setvalued mappings with nonempty convex pconvex values and with a closed graph, then T has at least one fixed point in \(C \cap \overline{U}\).
Proof
As the mapping T is 1set contractive, take an increasing sequence \(\{\lambda _{n}\}\) such that \(0 < \lambda _{n} < 1\) and \(\lim_{n \rightarrow \infty} \lambda _{n} =1\), where \(n \in \mathbb{N}\). Now we define a mapping \(F_{n}: C \rightarrow 2^{C}\) by \(F_{n}(x): = \lambda _{n} F(x)\) for each \(x \in C\) and \(n\in \mathbb{N}\). Then it follows that \(F_{n}\) is a \(\lambda _{n}\)setcontractive mapping with \(0 < \lambda _{n} < 1\), quasi upper semicontinuous with nonempty pconvex, and its graph is closed. Now, by Theorem 4.8 on the condensing mapping \(F_{n}\) in pvector space with pseminorm \(P_{U}\) for each \(n \in \mathbb{N}\), there exists \(x_{n} \in C \) such that \(x_{n} \in F_{n}(x_{n})=\lambda _{n} F(x_{n})\). Thus there exists \(y_{n} \in F(x_{n})\) such that \(x_{n}=\lambda _{n} y_{n}\). Let \(P_{U}\) be the Minkowski pfunctional of U in E, it follows that \(P_{U}\) is continuous as \(0 \in \operatorname{int}(U)=U\). Note that for each \(n \in \mathbb{N}\), \(\lambda _{n} x_{n} \in \overline{U} \cap C\), which implies that \(x_{n} = r(\lambda _{n} y_{n}) = \lambda _{n} y_{n}\), thus \(P_{U}(\lambda _{n} y_{n}) \leq 1\) by Lemma 2.2. Note that
which implies that \(\lim_{n\rightarrow \infty} P_{U}(y_{n}x_{n})=0\). Now, by the assumption that F is semiclosed, which means that \((IF)\) is closed at zero, there exists one point \(x_{0} \in \overline{C}\) such that \(0 \in (IF)(\overline{C})\), thus we have \(x_{0} \in F(x_{0})\).
Indeed, without loss of generality, we assume that \(\lim_{n \rightarrow \infty} x_{n} = x_{0}\), here \(y_{n} \in F(x_{n})\) is with \(x_{n}=\lambda _{n} y_{n}\), and \(\lim_{n \rightarrow \infty} \lambda _{n}=1\), it implies that \(x_{0}=\lim_{n \rightarrow \infty} (\lambda _{n} y_{n})\), which means \(y_{0}:=\lim_{n\rightarrow \infty} y_{n}= x_{0}\). There exists \(y_{0} (= x_{0}) \in F(x_{0})\). We complete the proof. □
Theorem 6.2
(Best approximation for semiclosed 1set contractive mappings)
Let U be a bounded open pconvex subset of a locally pconvex space E (\(0 < p \leq 1\)) zero \(0 \in U\), and let C be a (bounded) closed pconvex subset of E with also zero \(0\in C\). Assume that \(F: \overline{U}\cap C \rightarrow 2^{C}\) is a semiclosed 1set contractive and quasi upper semicontinuous mapping with nonempty pconvex values and with a closed graph, and for each \(x \in \partial _{C} U\) with \(y \in F(x) \cap (C \diagdown \overline{U})\)), \((P^{\frac{1}{p}}_{U}(y) 1)^{p} \leq P_{U} (yx)\) for \(0< p \leq 1\) (this is trivial when \(p=1\)). Then we have that there exist \(x_{0} \in C \cap \overline{U}\) and \(y_{0} \in F(x_{0})\) such that \(P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, \overline{I^{p}_{\overline{U}}( x_{0})} \cap C)\), where \(P_{U}\) is the Minkowski pfunctional of U. More precisely, we have that either (I) or (II) holds:

(I)
F has a fixed point \(x_{0} \in U \cap C\), i.e., \(0=P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, \overline{I^{p}_{\overline{U}}( x_{0})} \cap C)\);

(II)
There exist \(x_{0} \in \partial _{C}(U)\) and \(y_{0} \in F(x_{0}) \diagdown \overline{U}\) with
$$ P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p} \bigl(y_{0}, \overline{I^{p}_{\overline{U}}( x_{0})} \cap C \bigr)= \bigl(P^{\frac{1}{p}}_{U}(y_{0})1 \bigr)^{p} > 0. $$
Proof
Let \(r: E \rightarrow U\) be a retraction mapping defined by \(r(x): = \frac{x}{\max \{ 1, (P_{U}(x))^{\frac{1}{p}}\}}\) for each \(x \in E\), where \(P_{U}\) is the Minkowski pfunctional of U. Since the space E’s zero \(0 \in U\)(\(=\operatorname{int}U\) as U is open), it follows that r is continuous by Lemma 2.2. As the mapping F is 1set contractive, take an increasing sequence \(\{\lambda _{n}\}\) such that \(0 < \lambda _{n} < 1\) and \(\lim_{n \rightarrow \infty} \lambda _{n} =1\), where \(n \in \mathbb{N}\). Now we define a mapping \(F_{n}: C \cap \overline{U} \rightarrow 2^{C}\) by \(F_{n}(x): = \lambda _{n} F \circ r(x)\) for each \(x \in C \cap \overline{U}\) and \(n\in \mathbb{N}\). Then it follows that \(F_{n}\) is a \(\lambda _{n}\)setcontractive mapping with \(0 < \lambda _{n} < 1\) for each \(n \in \mathbb{N}\). As C and U̅ are pconvex, we have \(r(C) \subset C\) and \(r(\overline{U}) \subset \overline{U}\), so \(r( C \cap \overline{U}) \subset C \cap \overline{U}\). Thus \(F_{n}\) is a selfmapping defined on \(C \cap \overline{U}\), and we can also show that \(F_{n}\) satisfies all conditions of Theorem 4.8. By Theorem 4.8 for condensing mapping \(F_{n}\), for each \(n \in \mathbb{N}\), there exists \(z_{n} \in C \cap \overline{U}\) such that \(z_{n} \in F_{n}(z_{n})=\lambda _{n} F \circ r(z_{n})\). Let \(x_{n}= r(z_{n})\), then we have \(x_{n} \in C\cap \overline{U}\), and there exists \(y_{n} \in F(x_{n})\) with \(x_{n} = r(\lambda _{n} y_{n})\) such that the following (1) or (2) holds for each \(n \in \mathbb{N}\):
(1): \(\lambda _{n} y_{n} \in C\cap \overline{U}\); or (2): \(\lambda _{n} y_{n} \in C \diagdown \overline{U}\).
Now we prove the conclusion by considering the following two cases:
Case (I): For each \(n \in N\), \(\lambda _{n} y_{n} \in C \cap \overline{U}\); or
Case (II): There exists a positive integer n such that \(\lambda _{n} y_{n} \in C \diagdown \overline{U}\).
First, by case (I), for each \(n \in \mathbb{N}\), \(\lambda _{n} y_{n} \in \overline{U} \cap C\), which implies that \(x_{n} = r(\lambda _{n} y_{n}) = \lambda _{n} y_{n}\), thus \(P_{U}(\lambda _{n} y_{n}) \leq 1\) by Lemma 2.2. Note that
which implies that \(\lim_{n\rightarrow \infty} P_{U}(y_{n}x_{n})=0\). Now by the fact that F is semiclosed, it implies that there exists a point \(x_{0} \in \overline{U}\) (i.e., the consequence \(\{x_{n}\}_{n \in \mathbb{N}}\) has a convergent subsequence with the limit \(x_{0}\)) such that \(x_{0} \in F(x_{0})\). Indeed, without loss of generality, we assume that \(\lim_{n \rightarrow \infty} x_{n}=x_{0}\), where \(y_{n} \in F(x_{n})\) is with \(x_{n}=\lambda _{n} y_{n}\) and \(\lim_{n \rightarrow \infty} \lambda _{n}=1\), and as \(x_{0}=\lim_{n \rightarrow \infty} (\lambda _{n} y_{n})\), it implies that \(y_{0}=\lim_{n\rightarrow \infty} y_{n}= x_{0}\). Thus there exists \(y_{0} (= x_{0}) \in F(x_{0})\), we have \(0 = d_{p}(x_{0}, F(x_{0})) = d(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, \overline{I^{p}_{\overline{U}}(x_{0})} \cap C))\) as indeed \(x_{0} =y_{0} \in F(x_{0}) \in \overline{U}\cap C \subset \overline{I^{p}_{\overline{U}}( x_{0})} \cap C)\).
Second, by case (II), there exists a positive integer n such that \(\lambda _{n} y_{n} \in C \diagdown \overline{U}\). Then we have that \(P_{U}(\lambda _{n} y_{n})> 1\), and also \(P_{U}(y_{n})> 1\) as \(\lambda _{n} < 1\). As \(x_{n} = r(\lambda _{n} y_{n}) = \frac{\lambda _{n} y_{n}}{(P_{U}(\lambda _{n} y_{n}))^{\frac{1}{p}}}\), it implies that \(P_{U}(x_{n})=1\), thus \(x_{n} \in \partial _{C}(U)\). Note that
By the assumption, we have \((P^{\frac{1}{p}}_{U}(y_{n})1)^{p} \leq P_{U}(y_{n} x)\) for \(x \in C \cap \partial \overline{U}\), it follows that
Thus we have the best approximation: \(P_{U}(y_{n}  x_{n})=d_{P}(y_{n}, \overline{U} \cap C) = (P^{ \frac{1}{p}}_{U}(y_{n})1)^{p} > 0\).
Now we want to show that \(P_{U}(y_{n}x_{n})= d_{P}(y_{n}, \overline{U} \cap C) = d_{p}(y_{n}, \overline{I^{p}_{\overline{U}}( x_{0})} \cap C) > 0\).
By the fact that \((\overline{U}\cap C) \subset I^{p}_{\overline{U}}(x_{n})\cap C\), let \(z \in I^{p}_{\overline{U}}(x_{n})\cap C \diagdown (\overline{U}\cap C)\), we first claim that \(P_{U}(y_{n}  x_{n}) \leq P_{U}(y_{n}z)\). If not, we have \(P_{U}(y_{n}  x_{n}) > P_{U}(y_{n}z)\). As \(z \in I^{p}_{\overline{U}}(x_{n}) \cap C \diagdown (\overline{U} \cap C)\), there exist \(y \in \overline{U}\) and a nonnegative number c (actually \(c\geq 1\) as shown soon below) with \(z = x_{n} + c (y  x_{n})\). Since \(z \in C\), but \(z \notin \overline{U} \cap C\), it implies that \(z \notin \overline{U}\). By the fact that \(x_{n}\in \overline{U}\) and \(y \in \overline{U}\), we must have the constant \(c \geq 1\); otherwise, it implies that \(z ( = (1 c )x_{n} + c y) \in \overline{U}\), this is impossible by our assumption, i.e., \(z\notin \overline{U}\). Thus we have that \(c\geq 1\), which implies that \(y =\frac{1}{c} z + (1\frac{1}{c}) x_{n} \in C\) (as both \(x_{n} \in C\) and \(z\in C\)). On the other hand, as \(z \in I^{p}_{\overline{U}}(x_{n}) \cap C \diagdown (\overline{U} \cap C)\) and \(c\geq 1\) with \((\frac{1}{c})^{p}+ (1\frac{1}{c})^{p} = 1 \), combining with our assumption that for each \(x \in \partial _{C} \overline{U}\) and \(y \in F(x_{n})\diagdown \overline{U}\), \(P^{\frac{1}{p}}_{U}(y) 1 \leq P^{\frac{1}{p}}_{U} (yx)\) for \(0< p \leq 1\), it then follows that
which contradicts that \(P_{U} (y_{n}  x_{n}) = d_{P}(y_{n}, \overline{U}\cap C)\) as shown above. We know that \(y \in \overline{U}\cap C\), and we should have \(P_{U}(y_{n} x_{n})\leq P_{U}(y_{n}  y)\)! This helps us to complete the claim: \(P_{U}(y_{n}  x_{n}) \leq P_{U}(y_{n}  z)\) for any \(z \in I^{p}_{\overline{U}}(x_{n})\cap C \diagdown (\overline{U}\cap C)\), which means that the following best approximation of Fan’s type (see [42, 43]) holds:
Now, by the continuity of \(P_{U}\), it follows that the following best approximation of Fan type is also true:
The proof is complete. □
For a pvector space when \(p=1\), it is a (Hausdorff) topological vector space E, we have the following best approximation for the outward set \(\overline{O_{\overline{U}}(x_{0})}\) based on the point \(\{x_{0}\}\) with respect to the convex subset U in E.
Theorem 6.3
(Best approximation for outward sets)
Let U be a bounded open convex subset of a locally convex space E (i.e., \(p=1\)) with zero \(0 \in \operatorname{int}U=U\) (the interior \(\operatorname{int}U=U\) as U is open), and C be a closed convex subset of E with also zero \(0\in C\). Assume that \(F: \overline{U}\cap C \rightarrow 2^{C}\) is a semiclosed 1setcontractive quasi upper semicontinuous mapping with nonempty pconvex values and with a closed graph, then there exist \(x_{0} \in \overline{U} \cap X\) and \(y_{0} \in F(x_{0})\) such that \(P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, \overline{O_{\overline{U}}( x_{0})} \cap C)\), where \(P_{U}\) is the Minkowski pfunctional of U. More precisely, we have that either (I) or (II) holds:

(I)
F has a fixed point \(x_{0} \in U \cap C\), i.e., \(P_{U} (y_{0}  x_{0}) = P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, \overline{O_{\overline{U}}( x_{0})} \cap C))=0\);

(II)
There exist \(x_{0} \in \partial _{C}(U)\) and \(y_{0} \in F(x_{0})\diagdown \overline{U}\) with
$$ P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p} \bigl(y_{0}, O_{\overline{U}}(x_{0}) \cap C \bigr) =d_{p} \bigl(y_{0}, \overline{O_{\overline{U}}( x_{0})} \cap C \bigr) > 0. $$
Proof
We define a new mapping \(F_{1}: \overline{U}\cap C \rightarrow 2^{C}\) by \(F_{1} (x): = \{2x\}  F(x)\) for each \(x \in \overline{U}\cap C\), then \(F_{1}\) is also compact and upper semicontinuous mapping with nonempty closed convex values, and \(F_{1}\) satisfies all hypotheses of Theorem 5.2 with \(p=1\). It follows by Theorem 5.2 that there exist \(x_{0} \in \overline{U} \cap X\) and \(y_{1} \in F_{1}(x_{0})\) such that \(P_{U} (y_{1}  x_{0}) = d_{P}(y_{1}, \overline{U}\cap C) = d_{p}(y_{1}, \overline{I_{\overline{U}}(x_{0})} \cap C)\). More precisely, we have that either (I) or (II) holds:

(I)
\(F_{1}\) has a fixed point \(x_{0} \in U \cap C\) (so \(0= P_{U} (y_{1}  x_{0}) = P_{U} (y_{1}  x_{0}) = d_{P}(y_{1}, \overline{U}\cap C) = d_{p}(y_{1}, \overline{I_{\overline{U}}( x_{0})} \cap C)\));

(II)
There exist \(x_{0} \in \partial _{C}(U)\) and \(y_{1} \in F_{1}(x_{0})\diagdown \overline{U}\) with
$$ P_{U} (y_{1}  x_{0}) = d_{P}(y_{1}, \overline{U}\cap C) = d_{p} \bigl(y_{1}, \overline{O_{\overline{U}}( x_{0})} \cap C \bigr) >0. $$
Now, for any \(x \in O_{\overline{U}}(x_{0})\), there exist \(r < 0\), \(u \in \overline{U}\) such that \(x=x_{0} + r (u  x_{0})\). Let \(x_{1}=2x_{0}  x\), then \(x_{1} = 2x_{0}  x_{0}  r(u  x_{0})= x_{0} +(r) (u  x_{0}) \in I_{ \overline{U}}(x_{0})\). Let \(y_{1} = 2 x_{0}  y_{0}\) for some \(y_{0}\in F(x_{0})\). As we have \(P_{U} (y_{1}  x_{0}) = d_{P}(y_{1}, \overline{U}\cap C) = d_{p}(y_{1}, \overline{I_{\overline{U}}(x_{0})} \cap C)\), it follows that \(P_{U} (y_{1}  x_{0}) \leq P_{U} (y_{1}  x_{1})\), which implies that
for all \(x \in O_{\overline{U}}(x_{0})\). Thus we have \(P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, O_{\overline{U}}( x_{0}) \cap C)\) and by the continuity of \(P_{U}\), it follows that
This completes the proof. □
Now, by the application of Theorems 6.2 and 6.3, we have the following general principle for the existence of solutions for Birkhoff–Kellogg problems in pseminorm spaces, where (\(0 < p \leq 1\)).
Theorem 6.4
(Principle of Birkhoff–Kellogg alternative)
Let U be a bounded open pconvex subset of a locally pconvex space E (\(0 < p \leq 1\)) with zero \(0 \in \operatorname{int}U=(U)\) (the interior intU as U is open), and let C be a closed pconvex subset of E with also zero \(0\in C\). Assume that \(F: \overline{U}\cap C \rightarrow 2^{C}\) is a semiclosed 1setcontractive quasi upper semicontinuous mapping with nonempty pconvex values and with a closed graph, then F has at least one of the following two properties:

(I)
F has a fixed point \(x_{0} \in U \cap C\) such that \(x_{0} \in F(x_{0})\);

(II)
There exist \(x_{0} \in \partial _{C}(U)\), \(y_{0} \in F(x_{0})\diagdown \overline{U}\), and \(\lambda = \frac{1}{(P_{U}(y_{0}))^{\frac{1}{p}}} \in (0, 1)\) such that \(x_{0} = \lambda y_{0} \in \lambda F(x_{0})\); In addition, if for each \(x \in \partial _{C} U\), \(P^{\frac{1}{p}}_{U}(y) 1 \leq P^{\frac{1}{p}}_{U} (yx)\) for \(0< p \leq 1\) (this is trivial when \(p=1\)), then the best approximation between \(x_{0}\) and \(y_{0}\) is given by
$$ P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p} \bigl(y_{0}, \overline{I^{p}_{\overline{U}}( x_{0})} \cap C \bigr) = \bigl(P^{\frac{1}{p}}_{U}(y_{0})1 \bigr)^{p} > 0. $$
Proof
If (I) is not the case, then (II) is proved by Remark 5.2 and by following the proof in Theorem 6.2 for case (ii): \(y_{0} \in C \diagdown \overline{U}\) with \(y_{0}: = f(x_{0}) \in F(x_{0})\). Indeed, as \(y_{0} \notin \overline{U}\), it follows that \(P_{U}(y_{0}) > 1\) and \(x_{0}= f(y_{0}) = y_{0} \frac{1}{(P_{U}(y_{0}))^{\frac{1}{p}}}\). Now let \(\lambda = \frac{1}{(P_{U}(y_{0}))^{\frac{1}{p}}}\), we have \(\lambda < 1\) and \(x_{0} = \lambda y_{0}\) with \(y_{0} \in F(x_{0})\). Finally, the additional assumption in (II) allows us to have the best approximation between \(x_{0}\) and \(y_{0}\) obtained by following the proof of Theorem 6.2 as \(P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, \overline{I^{p}_{\overline{U}}( x_{0})} \cap C) > 0\). This completes the proof. □
As an application of Theorem 6.2 for the nonself setvalued mappings discussed in Theorem 6.3 with the outward set condition, we have the following general principle of Birkhoff–Kellogg alternative in locally pconvex spaces.
Theorem 6.5
(Principle of Birkhoff–Kellogg alternative in LCS)
Let U be a bounded open pconvex subset of a locally pconvex space E (\(0 < p \leq 1\)) with zero \(0 \in U\), and let C be a closed convex subset of E with also zero \(0\in C\). Assume that \(F: \overline{U}\cap C \rightarrow 2^{C}\) is a semiclosed 1set contractive and quasi upper semicontinuous mapping with nonempty pconvex values and with a closed graph, then it has at least one of the following two properties:

(I)
F has a fixed point \(x_{0} \in U \cap C\) such that \(x_{0} \in F(x_{0})\);

(II)
There exist \(x_{0} \in \partial _{C}(U)\) and \(y_{0} \in F(x_{0}) \diagdown \overline{U}\) and \(\lambda \in (0, 1)\) such that \(x_{0} = \lambda y_{0}\), and the best approximation between \(x_{0}\) and \(y_{0}\) is given by \(P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, \overline{I^{p}_{\overline{U}}( x_{0})} \cap C) > 0\).
On the other hand, by the proof of Theorem 6.2, we note that for case (II) of Theorem 6.2, the assumption “each \(x \in \partial _{C} U\) with \(y \in F(x)\), \(P^{\frac{1}{p}}_{U}(y) 1 \leq P^{\frac{1}{p}}_{U} (yx)\)” is only used to guarantee the best approximation \(``P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, \overline{I^{p}_{\overline{U}}( x_{0})} \cap C) > 0\)”, thus we have the following Leray–Schauder alternative in pvector spaces, which, of course, includes the corresponding results in locally convex spaces as special cases.
Theorem 6.6
(Leray–Schauder nonlinear alternative)
Let C be a closed pconvex subset of pseminorm space E with \(0 < p \leq 1\) and zero \(0 \in C\). Assume that \(F: C \rightarrow 2^{C}\) is a semiclosed 1set contractive and quasi upper semicontinuous mapping with nonempty pconvex values and with a closed graph. Let \(\varepsilon (F): =\{x \in C: x\in \lambda F(x) \textit{ for some } 0 < \lambda < 1\}\). Then either F has a fixed point in C or the set \(\varepsilon (F)\) is unbounded.
Proof
By assuming that case (I) is not true, i.e., F has no fixed point, we claim that the set \(\varepsilon (F)\) is unbounded. Otherwise, assume that the set \(\varepsilon (F)\) is bounded, and assume that P is the continuous pseminorm for E, then there exists \(r>0\) such that the set \(B(0, r):=\{x \in E: P(x) < r\}\), which contains the set \(\varepsilon (F)\), i.e., \(\varepsilon (F) \subset B(0, r)\), which means for any \(x \in \varepsilon (F)\), \(P(x) < r\). Then \(B(0. r)\) is an open pconvex subset of E and zero \(0 \in B(0, r)\) by Lemma 2.2 and Remark 2.4. Now, let \(U:=B(0, r)\) in Theorem 6.4, it follows that the mapping \(F: B(0, r) \cap C \rightarrow 2^{C}\) satisfies all general conditions of Theorem 6.4, and we have that any \(x_{0} \in \partial _{C} B(0, r)\), no any \(\lambda \in (0, 1)\) such that \(x_{0}=\lambda y_{0}\), where \(y_{0} \in F(x_{0})\). Indeed, for any \(x \in \varepsilon (F)\), it follows that \(P(x) < r\) as \(\varepsilon (F) \subset B(0, r)\), but for any \(x_{0} \in \partial _{C} B(0, r)\), we have \(P(x_{0})=r\), thus conclusion (II) of Theorem 6.4 does not hold. By Theorem 6.4 again, F must have a fixed point, but this contradicts with our assumption that F is fixed point free. This completes the proof. □
Now assume a given pvector space E equipped with the Pseminorm (by assuming it is continuous at zero) for \(0< p \leq 1\), then we know that \(P: E \rightarrow \mathbb{R}^{+}\), \(P^{1}(0)=0\), \(P(\lambda x) = \lambda ^{p} P(x)\) for any \(x\in E\) and \(\lambda \in \mathbb{R}\). Then we have the following useful result for fixed points due to Rothe and Altman types in pvector spaces, which plays important roles for optimization problems, variational inequalities, complementarity problems.
Corollary 6.1
Let U be a bounded open pconvex subset of a locally pconvex space E and zero \(0 \in U\), plus C is a closed pconvex subset of E with \(U \subset C\), where \(0< p \leq 1\). Assume that \(F: \overline{U} \rightarrow 2^{C}\) is a semiclosed 1set contractive quasi upper semicontinuous mapping with nonempty pconvex values and with a closed graph, and one of the following conditions is satisfied:

(1)
(Rothe type condition): \(P_{U}(y) \leq P_{U}(x)\) for \(y \in F(x)\), where \(x \in \partial U\);

(2)
(Petryshyn type condition): \(P_{U}(y) \leq P_{U}(yx)\) for \(y \in F(x)\), where \(x \in \partial U\);

(3)
(Altman type condition): \(P_{U}(y)^{\frac{2}{p}} \leq [P_{U}(y) x)]^{\frac{2}{p}} + [P_{U}(x)]^{ \frac{2}{p}}\) for \(y \in F(x)\), where \(x \in \partial U\),
then F has at least one fixed point.
Proof
By conditions (1), (2), and (3), it follows that the conclusion of (II) in Theorem 6.4 “there exist \(x_{0} \in \partial _{C}(U)\) and \(\lambda \in (0, 1)\) such that \(x_{0} \notin \lambda F(x_{0})\)” does not hold, thus by the alternative of Theorem 6.4, F has a fixed point. This completes the proof. □
By the fact that when \(p=1\), each pvector space is a topological vector space, we have the following classical Fan’s best approximation (see [42]) as a powerful tool for the study in the optimization, mathematical programming, games theory, mathematical economics, and other related topics in applied mathematics.
Corollary 6.2
(Fan’s best approximation in LCS)
Let U be a bounded open convex subset of a locally convex space E with zero \(0 \in U\), let C be a closed convex subset of E with also zero \(0\in C\), and assume that \(F: \overline{U}\cap C \rightarrow 2^{C}\) is a semiclosed 1set contractive and quasi upper semicontinuous mapping with nonempty convex values and with a closed graph. Then there exist \(x_{0} \in \overline{U} \cap X\) and \(y_{0} \in T(x_{0})\) such that \(P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, \overline{I_{\overline{U}}( x_{0})} \cap C)\), where \(P_{U}\) is the Minkowski pfunctional of U in E. More precisely, we have that either (I) or (II) holds, where \(W_{\overline{U}}(x_{0})\) is either the inward set \(I_{\overline{U}}(x_{0})\) or the outward set \(O_{\overline{U}}(x_{0})\):

(I)
F has a fixed point \(x_{0} \in U \cap C\), \(0= P_{U} (y_{0}  x_{0}) = P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, \overline{W_{\overline{U}}( x_{0})} \cap C))\);

(II)
There exist \(x_{0} \in \partial _{C}(U)\) and \(y_{0} \in F(x_{0}) \diagdown \overline{U}\) with
$$ P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p} \bigl(y_{0}, \overline{W_{\overline{U}}( x_{0})} \cap C \bigr) = P_{U}(y_{0})  1 > 0. $$
Proof
When \(p=1\), it automatically satisfies that the inequality: \(P^{\frac{1}{p}}_{U}(y) 1 \leq P^{\frac{1}{p}}_{U} (yx)\), and indeed we have that for \(x_{0}\in \partial _{C}(U)\) with \(y_{0} \in F(x_{0})\), we have \(P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, \overline{W_{\overline{U}}( x_{0})} \cap C)= P_{U}(y_{0})1\). The conclusions are given by Theorem 6.2 (or Theorem 6.3). The proof is complete. □
We would like to point out that similar results on Rothe and Leray–Schauder alternative have been developed by Isac [60], Park [96], Potter [108], Shahzad [120–122], Xiao and Zhu [135], and the related references therein as tools of nonlinear analysis in topological vector spaces. As mentioned above, when \(p=1\) and take F as a continuous mapping, then we obtain a version of Leray–Schauder in general local convex spaces, and thus we omit its statement in detail.
7 Principle of nonlinear alternatives for nonself semiclosed 1set contractive mappings
As applications of results in Sect. 6, we now establish general results for the existence of solutions for Birkhoff–Kellogg problem and the principle of Leray–Schauder alternatives for semiclosed 1set contractive mappings in locally pconvex spaces for \(0 < p \leq 1\).
Theorem 7.1
(Birkhoff–Kellogg alternative in pvector spaces)
Let U be a bounded open pconvex subset of a locally pconvex space E (where, \(0 < p \leq 1\)) with zero \(0 \in U\), let C be a closed pconvex subset of E with also zero \(0\in C\), and assume that \(F: \overline{U}\cap C \rightarrow 2^{C}\) is a semiclosed 1set contractive and quasi upper semicontinuous mapping with nonempty pconvex values and with a closed graph. In addition, for each \(x \in \partial _{C}(U)\) with \(y \in F(x)\), \(P^{\frac{1}{p}}_{U}(y) 1 \leq P^{\frac{1}{p}}_{U} (yx)\) for \(0< p \leq 1\) (this is trivial when \(p=1\)), where \(P_{U}\) is the Minkowski pfunctional of U. Then we have that either (I) or (II) holds:

(I)
There exists \(x_{0} \in \overline{U}\cap C\) such that \(x_{0} \in F(x_{0})\);

(II)
There exists \(x_{0} \in \partial _{C}(U)\) with \(y_{0} \in F(x_{0})\diagdown \overline{U}\) and \(\lambda >1\) such that \(\lambda x_{0} = y_{0} \in F(x_{0})\), i.e., \(F(x_{0}) \cap \{\lambda x_{0}: \lambda > 1 \} \neq \emptyset \).
Proof
By following the argument and notations used in Theorem 6.2, we have that either
(1) F has a fixed point \(x_{0} \in U \cap C\); or
(2) there exist \(x_{0} \in \partial _{C}(U)\) and \(y_{0} \in F(x_{0})\) with \(x_{0}=f(y_{0})\) such that
where \(\partial _{C}(U)\) denotes the boundary of U relative to C in E and f is the restriction of the continuous retraction r with respect to the set U in E.
If F has no fixed point, then (2) above holds and \(x_{0} \notin F(x_{0})\). As given in the proof of Theorem 6.2, we have that \(y_{0} \in F(x_{0})\) and \(y_{0}\notin \overline{U}\), thus \(P_{U}(y_{0}) > 1\) and \(x_{0}= f(y_{0})=\frac{y_{0}}{(P_{U}(y_{0}))^{\frac{1}{p}}}\), which means \(y_{0} =(P_{U}(y_{0}))^{\frac{1}{p}} x_{0}\). Let \(\lambda = (P_{U}(y_{0}))^{\frac{1}{p}}\), then \(\lambda > 1\), and we have \(\lambda x_{0} = y_{o} \in F(x_{0})\). This completes the proof. □
Theorem 7.2
(Birkhoff–Kellogg alternative in LCS)
Let U be a bounded open convex subset of a locally convex space E with zero \(0 \in U\), let C be a closed convex subset of E with also zero \(0\in C\), and assume that \(F: \overline{U}\cap C \rightarrow 2^{C}\) is a semiclosed 1set contractive and quasi upper semicontinuous mapping with nonempty pconvex values and with a closed graph. Then we have that either (I) or (II) holds, where \(W_{\overline{U}}(x_{0})\) is either the inward set \(I_{\overline{U}}(x_{0})\) or the outward set \(O_{\overline{U}}(x_{0})\):

(I)
There exists \(x_{0} \in \overline{U}\cap C\) such that \(x_{0} \in F(x_{0})\);

(II)
There exists \(x_{0} \in \partial _{C}(U)\) with \(y_{0} \in F(x_{0})\diagdown \overline{U}\) and \(\lambda >1\) such that \(\lambda x_{0} = y_{0} \in F(x_{0})\), i.e., \(F(x_{0}) \cap \{\lambda x_{0}: \lambda > 1 \} \neq \emptyset \).
Proof
When \(p=1\), it automatically satisfies that the inequality \(P^{\frac{1}{p}}_{U}(y) 1 \leq P^{\frac{1}{p}}_{U} (yx)\), and indeed we have that for \(x_{0}\in \partial _{C}(U)\) with \(y_{0} \in F(x_{0})\), we have \(P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, \overline{W_{\overline{U}}( x_{0})} \cap C)= P_{U}(y_{0})1\). The conclusions are given by Theorems 6.3 and 6.4. The proof is complete. □
Indeed, we have the following fixed points for nonself mappings in locally pconvex spaces for \(0 < p \leq 1\) under various boundary conditions.
Theorem 7.3
(Fixed points of nonself mappings)
Let U be a bounded open pconvex subset of a locally pconvex space E (where \(0 < p \leq 1\)) with zero \(0 \in U\), let C be a closed pconvex subset of E with also zero \(0\in C\), and assume that \(F: \overline{U}\cap C \rightarrow 2^{C}\) is a semiclosed 1set contractive and quasi upper semicontinuous mapping with nonempty pconvex values and with a closed graph. In addition, for each \(x \in \partial _{C}(U)\) with \(y \in F(x)\), \(P^{\frac{1}{p}}_{U}(y) 1 \leq P^{\frac{1}{p}}_{U} (yx)\) for \(0< p \leq 1\) (this is trivial when \(p=1\)), where \(P_{U}\) is the Minkowski pfunctional of U. If F satisfies any one of the following conditions for any \(x \in \partial _{C}(U) \diagdown F(x)\):

(i)
For each \(y \in F(x)\), \(P_{U}(yz) < P_{U}(yx)\) for some \(z \in \overline{I_{\overline{U}}(x)}\cap C\);

(ii)
For each \(y \in F(x)\), there exists λ with \(\lambda  < 1\) such that \(\lambda x + (1\lambda )y \in \overline{I_{\overline{U}}(x)}\cap C\);

(iii)
\(F(x) \subset \overline{I_{\overline{U}}(x)}\cap C\);

(iv)
\(F(x) \cap \{\lambda x: \lambda > 1 \} =\emptyset \);

(v)
\(F(\partial U) \subset \overline{U} \cap C\);

(vi)
For each \(y \in F(x)\), \(P_{U}(yx) \neq ((P_{U}(y))^{\frac{1}{p}}1)^{p}\);
then F must have a fixed point.
Proof
By following the argument and symbols used in the proof of Theorem 6.2 (see also Theorem 6.4), we have that either
(1) F has a fixed point \(x_{0} \in U \cap C\); or
(2) There exist \(x_{0} \in \partial _{C}(U)\) and \(y_{0} \in F(x_{0})\) with \(x_{0}=f(y_{0})\) such that
where \(\partial _{C}(U)\) denotes the boundary of U relative to C in E and f is the restriction of the continuous retraction r with respect to the set U in E.
First, suppose that F satisfies condition (i), if F has no fixed point, then (2) above holds and \(x_{0} \notin F(x_{0})\). Then, by condition (i), it follows that \(P_{U}(y_{0}z) < P_{U}(y_{0}x_{0})\) for some \(z \in \overline{I_{\overline{U}}(x)}\cap C\), this contradicts the best approximation equations given by (2) above, thus F mush have a fixed point.
Second, suppose that F satisfies condition (ii), if F has no fixed point, then (2) above holds and \(x_{0} \notin F(x_{0})\). Then, by condition (ii), there exists \(\lambda >1\) such that \(\lambda x_{0} + (1  \lambda ) y_{0} \in \overline{I_{\overline{U}}(x)}\cap C\). It follows that
this is impossible, and thus F must have a fixed point in \(\overline{U}\cap C\).
Third, suppose that F satisfies condition (iii), i.e., \(F(x) \subset \overline{I_{\overline{U}}(x)} \cap C\); then (2), we have that \(P_{U} (y_{0}  x_{0})\) and thus \(x_{0}= y_{0} \in F(x_{0})\), which means F has a fixed point.
Forth, suppose that F satisfies condition (iv), if F has no fixed point, then (2) above holds and \(x_{0} \notin F(x_{0})\). As given by the proof of Theorem 6.2, we have that \(y_{0} \notin \overline{U}\), thus \(P_{U}(y_{0}) > 1\) and \(x_{0}= f(y_{0})=\frac{y_{0}}{(P_{U}(y_{0}))^{\frac{1}{p}}}\), which means \(y_{0}=(P_{U}(y_{0}))^{\frac{1}{p}} x_{0}\), where \((P_{U}(y_{0}))^{\frac{1}{p}} > 1\), this contradicts the assumption (iv), thus F must have a fixed point in \(\overline{U} \cap C\).
Fifth, suppose that F satisfies condition (v), then \(x_{0} \notin F(x_{0})\). As \(x_{0} \in \partial _{C}{U}\), now by condition (v), we have that \(F(\partial U) \subset \overline{U} \cap C\). It follows that for any \(y_{0} \in F(x_{0})\), we have \(y_{0}\in \overline{U}\cap C\), thus \(y\notin \overline{U} \diagdown \cap C\), which implies that \(0 < P_{U}(y_{0} x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = 0\), this is impossible, thus F must have a fixed point. Here, as pointed out by Remark 5.2, we know that based on condition (v) the mapping F has a fixed point by applying \(F(\partial U) \subset \overline{U} \cap C\) is enough, we do not need the general hypothesis: “for each \(x \in \partial _{C}(U)\) with \(y \in F(x)\), \(P^{\frac{1}{p}}_{U}(y) 1 \leq P^{\frac{1}{p}}_{U} (yx)\) for \(0< p \leq 1\)”.
Finally, suppose that F satisfies condition (vi), if F has no fixed point, then (2) above holds and \(x_{0} \notin F(x_{0})\). Then condition (v) implies that \(P_{U}(y_{0} x_{0}) \neq ((P_{U}(y))^{\frac{1}{p}}1)^{p}\), but our proof in theorem shows that \(P_{U}(y_{0} x_{0})=((P_{U}(y))^{\frac{1}{p}}1)^{p}\), this is impossible, thus F must have a fixed point. Then the proof is complete. □
Now by taking the set C in Theorem 7.1 as the whole pvector space E itself, we have the following general results for nonself upper semicontinuous setvalued mappings, which include the results of Rothe, Petryshyn, Altman, and Leray–Schauder type fixed points as special cases.
Taking \(p=1\) and \(C =E\) in Theorem 7.3, we have fixed points for nonself upper semicontinuous setvalued mappings associated with inward or outward sets in locally convex spaces (LCS) as follows.
Theorem 7.4
(Fixed point theorem of nonself mappings with boundary conditions)
Let U be a bounded open convex subset of a locally convex spaces E with zero \(0 \in U\), and assume that \(F: \overline{U} \rightarrow 2^{E}\) is a semiclosed 1set contractive and quasi upper semicontinuous mapping with nonempty pconvex values and with a closed graph. If F satisfies any one of the following conditions for any \(x \in \partial (U) \diagdown F(x)\):

(i)
For each \(y \in F(x)\), \(P_{U}(yz) < P_{U}(yx)\) for some \(z \in \overline{I_{\overline{U}}(x)}\) (or \(z \in \overline{O_{\overline{U}}(x)}\));

(ii)
For each \(y \in F(x)\), there exists λ with \(\lambda  < 1\) such that \(\lambda x + (1\lambda )y \in \overline{I_{\overline{U}}(x)}\) (or \(\overline{O_{\overline{U}}(x)}\));

(iii)
\(F(x) \subset \overline{I_{\overline{U}}(x)}\) (or \(\overline{O_{\overline{U}}(x)}\));

(iv)
\(F(x) \cap \{\lambda x: \lambda > 1 \} =\emptyset \);

(v)
\(F(\partial (U) \subset \overline{U}\);

(vi)
For each \(y \in F(x)\), \(P_{U}(yx) \neq P_{U}(y)1\);
then F must have a fixed point.
In what follows, based on the best approximation theorem in a pseminorm space, we will also give some fixed point theorems for nonself setvalued mappings with various boundary conditions, which are related to the study for the existence of solutions for PDE and differential equations with boundary problems (see Browder [18], Petryshyn [104, 105], Reich [110]), which would play roles in nonlinear analysis for pseminorm space as shown below.
First, as discussed by Remark 5.2, the proof of Theorem 7.2 with the strongly boundary condition “\(F(\partial (U)) \subset \overline{U} \cap C\)” only, we can prove that F has a fixed point, thus we have the following fixed point theorem of Rothe type in pvector spaces.
Theorem 7.5
(Rothe type)
Let U be a bounded open pconvex subset of a locally pconvex space E (where \(0 < p \leq 1\)) with zero \(0 \in U\). Assume \(F: \overline{U}\rightarrow 2^{E}\) is a semiclosed 1set contractive and quasi upper semicontinuous mapping with nonempty pconvex values, with a closed graph, and such that \(F(\partial (U)) \subset \overline{U}\), then F must have a fixed point.
Now, as applications of Theorem 7.5, we give the following Leray–Schauder alternative in pvector spaces for nonself setvalued mappings associated with the boundary condition, which often appears in the applications (see Isac [60] and the references therein for the study of complementary problems and related topics in optimization).
Theorem 7.6
(Leray–Schauder alternative in pvector spaces)
Let E be a locally pconvex space E, where \(0 < p \leq 1\), \(B \subset E\) is bounded closed pconvex such that \(0 \in \operatorname{int} B\). Let \(F: [0, 1] \times B \rightarrow 2^{E}\) be a semiclosed 1set contractive and quasi upper semicontinuous setvalued mapping with nonempty pconvex values, with a closed graph, and such that the set \(F([0, 1] \times B)\) is relatively compact in E. If the following assumptions are satisfied:

(1)
\(x \notin F(t, x)\) for all \(x \notin \partial B\) and \(t \in [0, 1]\);

(2)
\(F(\{0\} \times \partial B) \subset B\),
then there is an element \(x^{*} \in B\) such that \(x^{*} \in F(1, x^{*})\).
Proof
For any \(n \in N\), we consider the mapping
where \(P_{B}\) is the Minkowski pfunctional of B and \(\{\epsilon _{n}\}_{n \in N}\) is a sequence of real numbers such that \(\lim_{n \rightarrow \infty} \epsilon _{n}=0\) and \(0 < \epsilon _{n} < \frac{1}{2}\) for any \(n \in N\). We observe that for each \(n \in N\), the mapping \(F_{n}\) is 1set contractive upper semicontinuous with nonempty closed pconvex values on B. From assumption (2), we have that \(F_{n}(\partial B) \subset B\), and the assumptions of Theorem 7.5 are satisfied, then for each \(n \in N\), there exists an element \(u_{n} \in B\) such that \(u_{n} \in F_{n}(u_{n})\).
We first prove the following statement: “It is impossible to have an infinite number of the elements \(u_{n}\) satisfying the following inequality: \(1  \epsilon _{n} \leq P_{B}(u_{n}) \leq 1\).”
If not, we assume to have an infinite number of elements \(u_{n}\) satisfying the following inequality:
As \(F_{n}(B)\) is relatively compact and by the definition of mappings \(F_{n}\), we have that \(\{u_{n}\}_{n \in N}\) is contained in a compact set in E. Without loss of generality (indeed, each compact set is also countably compact), we define the sequence \(\{t_{n}\}_{n\in N}\) by \(t_{n}: =\frac{1P_{B}(u_{n})}{\epsilon}\) for each \(n \in N\). Then we have that \(\{t_{n}\}_{n\in N}\subset [0, 1]\), and we may assume that \(\lim_{n \rightarrow \infty}t_{n} = t \in [0, 1]\). The corresponding subsequence of \(\{u_{n}\}_{n \in N}\) is denoted again by \(\{u_{n}\}_{n\in N}\), and it also satisfies the inequality \(1\epsilon _{n} \leq P_{B}(u_{n}) \leq 1\), which implies that \(\lim_{n\rightarrow \infty} P_{B} (u_{n})=1\).
Now let \(u^{*}\) be an accumulation point of \(\{u_{n}\}_{n\in N}\), thus have \(\lim_{n \rightarrow \infty}(t_{n},\frac{u_{n}}{P_{B}(u_{n})}, u_{n}) = (t, u^{*}, u^{*})\). By the fact that F is compact, we assume that \(u_{n}\in F(t_{n}, \frac{u_{n}}{P_{B}(u_{n})})\) for each \(n \in N\), it follows that \(u^{*} \in F(t, u^{*})\), this contradicts assumption (1) as we have \(\lim_{n \rightarrow \infty}P_{B}(u_{n})=1\) (which means that \(u^{*} \in \partial B\), this is impossible).
Thus it is impossible “to have an infinite number of elements \(u_{n}\) satisfying the inequality \(1  \epsilon _{n} \leq P_{B}(u_{n}) \leq 1\)”, which means that there is only a finite number of elements of sequence \(\{u_{n}\}_{n \in N}\) satisfying the inequality \(1  \epsilon _{n} \leq P_{B}(u_{n}) \leq 1\). Now, without loss of generality, for \(n \in N\), we have the following inequality:
By the fact that \(\lim_{n \rightarrow} (1\epsilon _{n})=1\), \(u_{n} \in F(1, \frac{u_{n}}{1\epsilon})\) for all \(n \in N\) and assuming that \(\lim_{n\rightarrow} u_{n} = u^{*}\), the upper semicontinuity of F with nonempty closed values implies that the graph of F is closed, and by the fact \(u_{n} \in F(1, \frac{u_{n}}{1\epsilon})\), it implies that \(u^{*} \in F(1, u^{*})\). This completes the proof. □
As a special case of Theorem 7.6, we have the following principle for the implicit form of Leray–Schauder type alternative for setvalued mappings in pvector spaces for \(0< p \leq 1\).
Corollary 7.1
(The implicit Leray–Schauder alternative)
Let E be a locally pconvex space E, where \(0 < p \leq 1\), \(B \subset E\) be bounded closed pconvex such that \(0 \in \operatorname{int} B\). Let \(F: [0, 1] \times B \rightarrow 2^{E}\) be semiclosed 1set contractive and quasi upper semicontinuous with nonempty pconvex values and with a closed graph, and let the set \(F([0, 1] \times B)\) be relatively compact in E. If the following assumptions are satisfied:

(1)
\(F(\{0\} \times \partial B) \subset B\),

(2)
\(x \notin F(0, x)\) for all \(x \in \partial B\),
then at least one of the following properties is satisfied:

(i)
there exists \(x^{*} \in B\) such that \(x^{*} \in F(1, x^{*})\); or

(ii)
there exists \((\lambda ^{*}, x^{*}) \in (0, 1) \times \partial B\) such that \(x^{*} \in F(\lambda ^{*}, x^{*})\).
Proof
The result is an immediate consequence of Theorem 7.6, this completes the proof. □
We would like to point out that similar results on Rothe and Leray–Schauder alternative have been developed by Furi and Pera [44], Granas and Dugundji [53], Górniewicz [51], Górniewicz et al. [52], Isac [60], Li et al. [78], Liu [81], Park [96], Potter [108], Shahzad [120–122], Xu [139], Xu et al. [140] (see also the related references therein) as tools of nonlinear analysis in the Banach space setting and applications to the boundary value problems for ordinary differential equations in noncompact problems and a general class of mappings for nonlinear alternative of Leray–Schauder type in normal topological spaces. Some Birkhoff–Kellogg type theorems for general class mappings in topological vector spaces have also been established by Agarwal et al. [1], Agarwal and O’Regan [2, 3], and Park [98] (see the references therein for more details); and in particular, recently O’Regan [91] used the Leray–Schauder type coincidence theory to establish some Birkhoff–Kellogg problems, Furi–Pera type results for a general class of mappings.
Before closing this section, we would like to share that as the application of the best approximation result for 1set contractive mappings we can establish fixed point theorems and the general principle of Leray–Schauder alternative for nonself mappings, which would seen to play important roles for the nonlinear analysis under the framework of pseminorm spaces, as the achievement of nonlinear analysis for the underling being locally convex spaces, normed spaces, or in Banach spaces.
8 Fixed points for nonself semiclosed 1set contractive mappings with various boundary conditions
In this section, based on the best approximation Theorem 6.2 established for the 1set contractive mappings in Sect. 6, we will show how it is used as a useful tool for us to develop fixed point theorems for semiclosed 1set contractive nonself upper semicontinuous mappings in pseminorm spaces, where \(p \in (0, 1]\), by including seminorm, norm spaces, and uniformly convex Banach spaces as special cases.
By following Definitions 6.1 and 6.2 above, we first observe that if f is a continuous demicompact mapping, then \((I  f)\) is closed, where I is the identity mapping on X. It is also clear from definitions that every demicompact map is hemicompact in seminorm spaces, but the converse is not true in general (e.g., see the example in p. 380 by Tan and Yuan [129]). It is evident that if f is demicompact, then \(If\) is demiclosed. It is known that for each condensing mapping f, when D or \(f(D)\) is bounded, then f is hemicompact; and also f is demicompact in metric spaces by Lemma 2.1 and Lemma 2.2 of Tan and Yuan [129], respectively. In addition, it is known that every nonexpansive map is a 1setcontractive map; and also if f is a hemicompact 1setcontractive mapping, then f is a 1setcontractive mapping satisfying the following “Condition (H1)” (the same as (H1), and slightly different from condition (H) used in Sect. 5):
(H1) Condition: Let D be a nonempty bounded subset of a space E and assume that \(F: \overline{D} \rightarrow 2^{E}\) is a setvalued mapping. If \(\{x_{n}\}_{n \in \mathbb{N}}\) is any sequence in D such that for each \(x_{n}\), there exists \(y_{n} \in F(x_{n})\) with \(\lim_{n \rightarrow \infty} (x_{n} y_{n})=0\), then there exists a point \(x\in \overline{D}\) such that \(x \in F(x)\).
We first note that the “(H1) Condition” above is actually “Condition (C)” used by Theorem 1 of Petryshyn [105]. Indeed, by following Goebel and Kirk [49] (see also Xu [137] and the references therein), Browder [18] (see also [19], p. 103) proved that if K is a closed and convex subset of a uniformly convex Banach space X, and if \(T: K \rightarrow X\) is nonexpansive, then the mapping \(f: = I  T\) is demiclosed on X. This result, known as Browder’s demiclosedness principle (Browder’s proof, which was inspired by the technique of Göhde in [50]), is one of the fundamental results in the theory of nonexpansive mappings that satisfies the “(H1) condition”.
The following is Browder’s demiclosedness principle proved by Browder [18] that says that a nonexpansive mapping in a uniformly convex Banach X enjoys condition (H1) as shown below.
Lemma 8.1
Let D be a nonempty bonded convex subset of a uniformly convex Banach space E. Assume that \(F: \overline{D} \rightarrow E\) is a nonexpansive singlevalued mapping, then the mapping \(P: =I  F\) defined by \(P(x): = (xF(x)) \) for each \(x \in \overline{D}\) is demiclosed, and in particular, the “(H1) condition” holds.
Proof
By following the argument given in p. 329 (see also the proof of Theorem 2.2 and Corollary 2.1) by Petryshyn [105], by the Browder demiclosedness principle (see Goebel and Kirk [49] or Xu [137]), \(P=(IF)\) is closed at zero, thus there exists \(x_{0} \in \overline{U}\) such \(0 \in (IF)x_{0})\), which means that \(x_{0} \in F(x_{0})\). The proof is complete. □
On the other hand, by following the notion called “Opial’s condition” given by Opial [90], which says that a Banach space X is said to satisfy Opial’s condition if \(\liminf_{n \rightarrow \infty} \ w_{n}  w \ < \liminf_{n \rightarrow \infty} \w_{n}p\\) whenever \((w_{n})\) is a sequence in X weakly convergent to w and \(p\neq w\), we know that Opial’s condition plays an important role in the fixed point theory, e.g., see Lami Dozo [75], Goebel and Kirk [49], Xu [137], and the references therein. Actually, the following result shows that there exists a class of nonexpansive setvalued mappings in Banach spaces with Opial’s condition (see Lami Dozo [75] satisfying the “(H1) Condition”.
Lemma 8.2
Let C be a nonempty convex weakly compact subset of a Banach space X that satisfies Opial’s condition. Let \(T: C \rightarrow K(C)\) be a nonexpansive setvalued mapping with nonempty compact values. Then the graph of \((IT)\) is closed \((X, \sigma (X, X^{*}) \times (X, \\cdot \))\), thus T satisfies the “(H1) condition”, where I denotes the identity on X, \(\sigma (X, X^{*})\)is the weak topology, and \(\\cdot \\) is the norm (or strong) topology.
Proof
By following Theorem 3.1 of Lami Dozo [75], it follows that the mapping T is demiclosed, thus T satisfies the “(H1) condition”. The proof is complete. □
By Theorem 3.1 of Lami Dozo [75], indeed, we have the following statement, which is another version by using the term of “distance convergence” for Lemma 8.2.
Lemma 8.3
Let C be a nonempty closed convex subset of a Banach space \((X, d)\) that satisfies the Opial condition. Let \(T: C \rightarrow K(C)\) be a multivalued nonexpansive mapping (with fixed points). Let \((y_{n})_{n \in \mathbb{N}}\) be a bounded sequence such that \(_{n \rightarrow \infty}d(y_{,} T(y_{n}))=0\), then the weak cluster points of \((y_{n})\), \(n \in \mathbb{N}\) is a fixed point of T.
Proof
It is Theorem 3.1 of Lami Dozo [75] (see also Lemma 3.2 of Xu and Muglia [138]). □
We note that another class of setvalued mappings, called “∗nonexpansive mappings in Banach spaces (introduced by Husain and Tarafdar [59], see also Husain and Latif [58]), was proved to hold the demiclosedness principle in reflexive Banach spaces satisfying Opial’s condition by Muglia and Marino (i.e., Lemma 3.4 in [85]), thus the demiclosedness principle also holds in reflexive Banach spaces with duality mapping that is weakly sequentially continuous since these satisfy Opial’s condition.
Let E denote a Hausdorff locally convex topological vector space and \(\mathfrak{F}\) denote the family of continuous seminorms generating the topology of E. Also \(C(E)\) will denote the family of nonempty compact subsets of E. For each \(p\in \mathfrak{F}\) and \(A, B \in C(E)\), we can define \(\delta (A, B): = \sup \{p(a  b): a\in A, b \in B\}\) and \(D_{p}(A,B):= \max \{\sup_{a \in A}\inf_{b\in B} P( a b), \sup_{b \in B}\inf_{a \in A} P(ab) \}\). Though p is only a seminorm, \(D_{p}\) is a Hausdorff metric on \(C(E)\) (e.g., see Ko and Tsai [71]).
Definition 8.1
Let K be a nonempty subset of E. A mapping \(T: K \rightarrow C(E)\) is said to be a multivalued contraction if there exists a constant \(k_{p} \in (0, 1)\) such that \(D_{p}(T(x), T(y)) \leq k_{p} P(xy)\). T is said to be nonexpansive if for any \(x, y \in K\), we have \(P_{p}(T(x), T(y)))\leq P(xy)\).
By Chen and Singh [31], we now have the following definition of Opial’s condition in locally convex spaces.
Definition 8.2
The locally convex space E is said to satisfy Opial’s condition if for each \(x \in E\) and every net \((x_{\alpha})\) converging weakly to x, for each \(P \in \mathfrak{F}\), we have \(\liminf P(x_{\alpha}  y ) > \liminf P(x_{\alpha} x)\) for any \(y\neq x\).
Now we have the following demiclosedness principle for nonexpansive setvalued mappings in (Hausdorff) local convex spaces E, which is indeed Theorem 1 of Chen and Singh [31].
Lemma 8.4
Let K be a nonempty, weakly, compact, and convex subset of E. Let \(T: K \rightarrow C(E)\) be nonexpansive. If E satisfies Opial’s condition, then the graph \((IG)\) is closed in \(E_{w} \times E\), where \(E_{w}\) is E with its weak topology and I is the identity mapping.
Proof
The conclusion follows by Theorem 1 of Chen and Singh [31]. □
Remark 8.1
When a pvector space E is with a pnorm, then both (H1) and (H) conditions for their convergence can be described by the convergence weakly and strongly by the weak topology and strong topology induced by pnorm for \(p \in (0, 1]\). Secondly, if a given pvector space E has a nonempty open pconvex subset U containing zero, then any mapping satisfying the “(H) condition” is a hemicompact mapping (with respect \(P_{U}\) for a given bounded open pconvex subset U containing zero of pvector space E), thus satisfying the “(H) condition” used in Theorem 5.1.
By the fact that each semiclosed 1set mapping satisfies the “(H1) condition”, we have the existence of fixed points for the class of semiclosed 1set mappings. First, as an application of Theorem 8.2, we have the following result for nonself mappings in pseminorm spaces for \(p \in (0, 1]\).
Theorem 8.1
Let U be a bounded open pconvex subset of a pseminorm space E (\(0 < p \leq 1\)) zero \(0 \in U\). Assume that \(F: \overline{U} \rightarrow 2^{E}\) is a semiclosed 1set contractive and quasi upper semicontinuous mapping with nonempty pconvex values and with a closed graph. In addition, for any \(x\in \partial \overline{U}\) and \(y \in F(x)\), we have \(\lambda x \neq y\) for any \(\lambda > 1\) (i.e., the “Leray–Schauder boundary condition”). Then F has at least one fixed point.
Proof
By the proof of Theorem 6.2 with \(C= E\), we actually have thaat either (I) or (II) holds:

(I)
F has a fixed point \(x_{0} \in U \), i.e., \(P_{U} (y_{0}  x_{0}) = 0\);

(II)
There exist \(x_{0} \in \partial (U)\) and \(y_{0} \in F(x_{0})\) with \(P_{U} (y_{0}  x_{0}) = (P^{\frac{1}{p}}_{U}(y_{0})1)^{p} > 0\).
If F has no fixed point, then (II) above holds and \(x_{0} \notin F(x_{0})\). By the proof of Theorem 6.2, we have that \(x_{0}=f(y_{0})\) and \(y_{0} \notin \overline{U}\). Thus \(P_{U}(y_{0}) > 1\) and \(x_{0}= f(y_{0})=\frac{y_{0}}{(P_{U}(y_{0}))^{\frac{1}{p}}}\), which means \(y_{0}=(P_{U}(y_{0}))^{\frac{1}{p}} x_{0}\), where \((P_{U}(y_{0}))^{\frac{1}{p}} > 1\), this contradicts the assumption. Thus F must have a fixed point. The proof is complete. □
By following the idea used and developed by Browder [18], Li [77], Li et al. [78], Goebel and Kirk [48], Petryshyn [104, 105], Tan and Yuan [129], Xu [139], Xu et al. [140] and the references therein, we have the following existence theorems for the principle of Leray–Schauder type alternatives in pseminorm spaces \((E, \\cdot \_{p})\) for \(p \in (0, 1]\).
Theorem 8.2
Let U be a bounded open pconvex subset of a pseminorm space \((E, \\cdot \_{p})\) (\(0 < p \leq 1\)) zero \(0 \in U\). Assume that \(F: \overline{U} \rightarrow 2^{E}\) is a semiclosed 1set contractive and quasi upper semicontinuous mapping with nonempty pconvex values and with a closed graph. In addition, there exist \(\alpha >1\), \(\beta \geq 0\) such that for each \(x \in \partial \overline{U}\), we have that for any \(y \in F(x)\), \(\y x\_{p}^{\alpha /p}\geq \y\_{p}^{(\alpha +\beta )/p}\x\_{p}^{ \beta /p}  \x\_{p}^{\alpha /p}\). Then F has at least one fixed point.
Proof
By assuming F has no fixed point, we prove the conclusion by showing that the Leray–Schauder boundary condition in Theorem 8.1 does not hold. If we assume that F has no fixed point, by the boundary condition of Theorem 8.1, there exist \(x_{0}\in \partial \overline{U}\), \(y_{0} \in F(x_{0})\), and \(\lambda _{0} >1\) such that \(y_{0} = \lambda _{0} x_{0}\).
Now, consider the function f defined by \(f(t): =(t1)^{\alpha}  t^{\alpha + \beta}+1\) for \(t\geq 1\). We observe that f is a strictly decreasing function for \(t \in [1, \infty )\) as the derivative of \(f'(t) =\alpha (t1)^{\alpha 1}  (\alpha + \beta ) t^{\alpha +\beta 1} < 0\) by the differentiation, thus we have \(t^{\alpha + \beta} 1 > (t1)^{\alpha}\) for \(t \in (1, \infty )\). By combining the boundary condition, we have that \(\y_{0}x_{0}\_{p}^{\alpha /p}=\\lambda _{0}x_{0}x_{0}\_{p}^{ \alpha /p}=(\lambda _{0}1)^{\alpha}\x_{0}\_{p}^{\alpha /p} < ( \lambda _{0}^{\alpha +\beta}1)\x_{0}\_{p}^{(\alpha +\beta )/p}\x_{0} \_{p}^{\beta /p}=\y_{0}\_{p}^{(\alpha +\beta )/p}\x_{0}\_{p}^{ \beta /p} \x_{0}\_{p}^{\alpha /p}\), which contradicts the boundary condition given by Theorem 8.2. Thus, the conclusion follows. □
Theorem 8.3
Let U be a bounded open pconvex subset of a pseminorm space \((E, \\cdot \_{p})\) (\(0 < p \leq 1\)) zero \(0 \in U\). Assume that \(F: \overline{U} \rightarrow 2^{E}\) is a semiclosed 1set contractive and quasi upper semicontinuous mapping with nonempty pconvex values and with a closed graph. In addition, there exist \(\alpha >1\), \(\beta \geq 0\) such that for each \(x \in \partial \overline{U}\), we have that for any \(y \in F(x)\), \(\y + x\_{p}^{(\alpha +\beta )/p} \leq \y\_{p}^{\alpha /p}\x\_{p}^{ \beta /p} + \x\_{p}^{(\alpha +\beta )/p}\). Then F has at least one fixed point.
Proof
We prove the conclusion by showing that the Leray–Schauder boundary condition in Theorem 8.1 does not hold. If we assume that F has no fixed point, by the boundary condition of Theorem 8.1, there exist \(x_{0}\in \partial \overline{U}\), \(y_{0} \in F(x_{0})\), and \(\lambda _{0} >1\) such that \(y_{0} = \lambda _{0} x_{0}\).
Now, consider the function f defined by \(f(t): =(t+1)^{\alpha +\beta}  t^{\alpha}  1 \) for \(t\geq 1\). We then can show that f is a strictly increasing function for \(t \in [1, \infty )\), thus we have \(t^{\alpha}+1 < (t + 1)^{\alpha +\beta}\) for \(t \in (1, \infty )\). By the boundary condition given in Theorem 8.3, we have that
which contradicts the boundary condition given by Theorem 8.3. Thus, the conclusion follows and the proof is complete. □
Theorem 8.4
Let U be a bounded open pconvex subset of a pseminorm space \((E, \\cdot \_{p})\) (\(0 < p \leq 1\)) zero \(0 \in U\). Assume that \(F: \overline{U} \rightarrow 2^{E}\) is a semiclosed 1set contractive and quasi upper semicontinuous mapping with nonempty pconvex values and with a closed graph. In addition, there exist \(\alpha >1\), \(\beta \geq 0\) (or alternatively, \(\alpha >1\), \(\beta \geq 0\)) such that for each \(x \in \partial \overline{U}\), we have that for any \(y \in F(x)\), \(\y  x\_{p}^{\alpha /p} \x\_{p}^{\beta /p} \geq \y\_{p}^{ \alpha /p}\y+x\_{p}^{\beta /p} \x\_{p}^{(\alpha +\beta )/p}\). Then F has at least one fixed point.
Proof
The same as above, we prove the conclusion by showing that the Leray–Schauder boundary condition in Theorem 8.1 does not hold. If we assume that F has no fixed point, by the boundary condition of Theorem 8.1, there exist \(x_{0}\in \partial \overline{U}\), \(y_{0} \in F(x_{0})\), and \(\lambda _{0} >1\) such that \(y_{0} = \lambda _{0} x_{0}\).
Now, consider the function f defined by \(f(t): =(t1)^{\alpha}  t^{\alpha}(t1)^{\beta}+1\) for \(t\geq 1\). We then can show that f is a strictly decreasing function for \(t \in [1, \infty )\), thus we have \((t1)^{\alpha} < t^{\alpha} (t+1)^{\beta}1\) for \(t \in (1, \infty )\). By the boundary condition given in Theorem 8.4, we have that
which contradicts the boundary condition given by Theorem 8.4. Thus, the conclusion follows and the proof is complete. □
Theorem 8.5
Let U be a bounded open pconvex subset of a pseminorm space \((E, \\cdot \_{p})\) (\(0 < p \leq 1\)) zero \(0 \in U\). Assume that \(F: \overline{U} \rightarrow 2^{E}\) is a semiclosed 1set contractive and quasi upper semicontinuous mapping with nonempty pconvex values and with a closed graph. In addition, there exist \(\alpha >1\), \(\beta \geq 0\), we have that for any \(y \in F(x)\), \(\y + x\_{p}^{(\alpha +\beta )/p} \leq \yx\_{p}^{\alpha /p}\x\_{p}^{ \beta /p} +\y\_{p}^{\beta /p} \x\^{\alpha /p}\). Then F has at least one fixed point.
Proof
The same as above, we prove the conclusion by showing that the Leray–Schauder boundary condition in Theorem 8.1 does not hold. If we assume F has no fixed point, by the boundary condition of Theorem 8.1, there exist \(x_{0}\in \partial \overline{U}\), \(y_{0} \in F(x_{0})\), and \(\lambda _{0} >1\) such that \(y_{0} = \lambda _{0} x_{0}\).
Now, consider the function f defined by \(f(t): =(t+1)^{\alpha +\beta}  (t1)^{\alpha}t^{\beta}\) for \(t\geq 1\). We then can show that f is a strictly increasing function for \(t \in [1, \infty )\), thus we have \((t+1)^{\alpha +\beta} > (t1)^{\alpha} +t^{\beta}\) for \(t \in (1, \infty )\).
By the boundary condition given in Theorem 8.5, we have that \(\y_{0} +x_{0}\_{p}^{(\alpha +\beta )/p}=(\lambda _{0} +1)^{\alpha + \beta}\x_{0}\_{p}^{(\alpha +\beta )/p} > ((\lambda _{0}1)^{\alpha}+ \lambda _{0}^{\beta})\x_{0}\_{p}^{(\alpha +\beta )/p}=\\lambda _{0} x_{0} x_{0}\_{p}^{\alpha /p}\x_{0}\_{p}^{\beta /p} + \\lambda _{0} x_{0}\_{p}^{\beta /p}\x_{0}\_{p}^{\alpha /p} = \y_{0}x_{0}\_{p}^{ \beta /p}\x_{0}\_{p}^{\alpha /p} +\y_{0}\_{p}^{\beta /p}\x_{9}\^{ \alpha /p}\), which implies that
this contradicts the boundary condition given by Theorem 8.5. Thus, the conclusion follows and the proof is complete. □
As an application of Theorem 8.1, by testing the Leray–Schauder boundary condition, we have the following conclusion for each special case, and thus we omit their proofs in detail here.
Corollary 8.1
Let U be a bounded open pconvex subset of a pseminorm space \((E, \\cdot \_{p})\) (\(0 < p \leq 1\)) zero \(0 \in U\). Assume that \(F: \overline{U} \rightarrow 2^{E}\) is a semiclosed 1set contractive and quasi upper semicontinuous mapping with nonempty pconvex values and with a closed graph. Then F has at least one fixed point if one of the following (strong) conditions holds for \(x \in \partial \overline{U}\) and \(y \in F(x)\):

(i)
\(\y\_{p} \leq \x\_{p}\),

(ii)
\(\y\_{p} \leq \yx\_{p}\),

(iii)
\(\y+x_{p} \leq \y\_{p}\),

(iv)
\(\y+ x\_{p} \leq \x\_{p}\),

(v)
\(\y+x\_{p} \leq \y x\_{p}\),

(vi)
\(\y\_{p} \cdot \y+x\_{p} \leq \x\_{p}^{2}\),

(vii)
\(\y\_{p} \cdot \y +x\_{p} \leq \y x\_{p} \cdot \x\_{p}\).
If the p(semi)norm space E is a uniformly convex Banach space \((E, \ \cdot \)\) (for pnorm space with \(p=1\)), then we have the following general existence result, which can apply to general nonexpansive (singlevalued) mappings, too.
Theorem 8.6
Let U be a bounded open convex subset of a uniformly convex Banach space \((E, \\cdot \)\) (with \(p=1\)) with zero \(0 \in U\). Assume that \(F: \overline{U} \rightarrow E\) is a semicontractive and continuous (singlevalued) mapping. In addition, for any \(x\in \partial \overline{U}\), we have \(\lambda x \neq F(x)\) for any \(\lambda > 1\) (i.e., the “Leray–Schauder boundary condition”). Then F has at least one fixed point.
Proof
By Lemma 8.1, F is a semiclosed 1set contractive mapping. Moreover, by the assumption that E is a uniformly convex Banach, the mapping \((IF)\) is closed at zero, and thus F is semiclosed at zero (see Browder [18] or Goebel and Kirk [48]). Thus all assumptions of Theorem 8.2 are satisfied. The conclusion follows by Theorem 8.2. The proof is complete. □
Now we have the following results for nonexpansive setvalued mappings in a Banach space X with Opial’s condition.
Theorem 8.7
Let C be a nonempty convex weakly compact subset of a Banach space X that satisfies Opial’s condition and \(0 \in \operatorname{int}C\). Let \(T: C \rightarrow K(X)\) be a nonexpansive setvalued mapping with nonempty compact convex values. In addition, for any \(x\in \partial \overline{C}\), we have \(\lambda x \neq F(x)\) for any \(\lambda > 1\) (i.e., the “Leray–Schauder boundary condition”). Then F has at least one fixed point.
Proof
As T is nonexpansive, it is 1set contractive. By Lemma 8.2, it is then semicontractive and continuous. Then all conditions of Theorem 8.1 are satisfied, the conclusion follows by Theorem 8.1, and the proof is complete. □
By using Lemma 8.4, we have the following result in local convex spaces for nonexpansive setvalued mappings.
Theorem 8.8
Let C be a nonempty convex weakly compact subset of a local convex space X that satisfies Opial’s condition and \(0 \in \operatorname{int}C\). Let \(T: C \rightarrow K(X)\) be a nonexpansive setvalued mapping with nonempty compact convex values. In addition, for any \(x\in \partial \overline{C}\), we have \(\lambda x \neq F(x)\) for any \(\lambda > 1\) (i.e., the “Leray–Schauder boundary condition”). Then F has at least one fixed point.
Proof
As T is nonexpansive, it is 1set contractive. By Lemma 8.4, it is then semicontractive and continuous. Then all conditions of Theorem 8.1 are satisfied, the conclusion follows by Theorem 10.1, and the proof is complete. □
By considering a pseminorm space \((E, \\cdot \)\) with a seminorm for \(p=1\), the following corollary is a special case of the corresponding results from Theorem 8.2 to Theorem 8.5, and thus we omit its proof.
Corollary 8.2
Let U be a bounded open convex subset of a norm space \((E, \\cdot \)\). Assume that \(F: \overline{U} \rightarrow 2^{E}\) is a semiclosed 1set contractive and quasi upper semicontinuous mapping with nonempty pconvex values and with a closed graph. Then F has at least one fixed point if there exist \(\alpha >1\), \(\beta \geq 0\) such that any one of the following conditions is satisfied:

(i)
For each \(x \in \partial \overline{U}\) and any \(y \in F(x)\), \(\y x\^{\alpha}\geq \y\^{(\alpha +\beta )}\x\^{\beta}  \x\^{ \alpha}\);

(ii)
For each \(x \in \partial \overline{U}\) and any \(y \in F(x)\), \(\y + x\^{(\alpha +\beta )} \leq \y\^{\alpha}\x\^{\beta} + \x \^{(\alpha +\beta )}\);

(iii)
For each \(x \in \partial \overline{U}\) and any \(y \in F(x)\), \(\y  x\^{\alpha} \x\^{\beta} \geq \y\^{\alpha}\y+x\^{\beta}  \x\^{(\alpha +\beta )}\);

(iv)
For each \(x \in \partial \overline{U}\) and any \(y \in F(x)\), \(\y + x\^{(\alpha +\beta )} \leq \yx\^{\alpha}\x\^{\beta} +\y \^{\beta} \x\^{\alpha}\).
Remark 8.2
As discussed by Lemma 8.1 and the proof of Theorem 8.6, when the pvector space is a uniformly convex Banach space, the semicontractive or nonexpansive mappings automatically satisfy the conditions (see (H1)) required by Theorem 8.1, that is, the mappings are indeed semiclosed. Moreover, our results from Theorem 8.1 to Theorem 8.6, Corollary 8.1 and Corollary 8.2 also improve or unify corresponding results given by Browder [18], Li [77], Li et al. [78], Goebel and Kirk [48], Petryshyn [104, 105], Reich [110], Tan and Yuan [129], Xu [136], Xu [139], Xu et al. [140], and the results from the references therein by extending the nonself mappings to the classes of semiclosed 1set contractive setvalued mappings in pseminorm spaces with \(p \in (0, 1]\), including the norm space or Banach space when \(p=1\) for pseminorm spaces.
Before the ending of this paper, we would like to share with readers that the main goal of this paper was to develop new fixed point theorems and tools in nonlinear analysis for 1set contractive upper semicontinuous setvalued mappings in locally pconvex spaces for \(p\in (0, 1]\).
Actually, the corresponding theory in nonlinear functional analysis could be developed by applying Theorem 4.3 as a tool in locally pconvex, pvector and topological vector spaces for singevalued mappings for \(p in \in (0, 1]\), and we do not discuss them in detail here due to the limited space.
In addition, we do expect that results established in this paper would become useful tools for the study on optimization, nonlinear programming, variational inequality, complementarity, game theory, mathematical economics, and other related social science areas.
Finally, we would like to share that the results established in this paper do not only unify or improve the corresponding results in the existing literature for nonlinear analysis, but they can also be regarded as the continuation of (or) related work established recently by Yuan [144, 145].
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Acknowledgements
The author thanks Professor S.S. Chang (ShiSheng Zhang), Professor Jianzhong Xiao, Professor Jinlu Li, Professor K.K. Tan, Professor B. Sims, Professor Bevan Thompson, Professor Bruce Smith, Professor Jian Yu, Professor Hong Ma, Professor Jinlu Li, Professor L.S. Liu, Professor Y.J. Cho, Professor S. Park, and Professor M. Nashed for their encouragements in the past for more than two decades. The author’s thanks also go to Professor HongKun Xu, Professor Tiexin Guo, Professor XiaoLong Qin, Professor Ganshan Yang, Professor Xian Wu, Professor Nanjing Huang, Professor Shiqing Zhang, Professor Mohamed Ennassik, Professor Yonglu Shu, Professor Bo Li, and my colleagues and friends across China, Australia, Canada, UK, USA, and elsewhere. In particular, the author thanks anonymous referees for their comments and suggestions, which led to the present version of the paper. This research is also partially supported by the National Natural Science Foundation of China [grant numbers 71971031 and U1811462].
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This research is partially supported by the National Natural Science Foundation of China [grant numbers 71971031 and U1811462].
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This paper is dedicated to the memory of Professor Wataru Takahashi (1944–2020), Professor Kazimierz Goebel (1940–2022), and Professor William Art Kirk (1936–2022) for their significant contribution on the development of fixed point theory and applications for nonlinear functional analysis in mathematics and related disciplines.
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Yuan, G.X. Fixed point theorems and applications in pvector spaces. Fixed Point Theory Algorithms Sci Eng 2023, 10 (2023). https://doi.org/10.1186/s1366302300747w
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DOI: https://doi.org/10.1186/s1366302300747w
MSC
Keywords
 Nonlinear analysis
 pvector space
 Locally pconvex space
 Schauder conjecture
 Fixed point theorem
 Graphapproximation
 Quasi upper semicontinuous
 Measure of noncompactness
 1set contractive mapping
 Condensing mapping
 Best approximation
 Nonlinear alternative
 LeraySchauder alternative
 Nonexpansive mapping
 pinward and poutward set
 Uniform convex space