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Nonlinear analysis by applying best approximation method in pvector spaces
Fixed Point Theory and Algorithms for Sciences and Engineering volume 2022, Article number: 20 (2022)
Abstract
It is known that the class of pvector spaces \((0 < p \leq 1)\) is an important generalization of the usual norm spaces with rich topological and geometrical structure, but most tools and general principles with nature in nonlinearity have not been developed yet. The goal of this paper is to develop some useful tools in nonlinear analysis by applying the best approximation approach for the classes of 1set contractive setvalued mappings in pvector spaces. In particular, we first develop general fixed point theorems of compact (singlevalued) continuous mappings for closed pconvex subsets, which also provide an answer to Schauder’s conjecture of 1930s in the affirmative way under the setting of topological vector spaces for \(0 < p \leq 1\). Then one best approximation result for upper semicontinuous and 1set contractive setvalued mappings is established, which is used as a useful tool to establish fixed points of nonself setvalued mappings with either inward or outward set conditions and related various boundary conditions under the framework of locally pconvex spaces for \(0 < p \leq 1\). In addition, based on the framework for the study of nonlinear analysis obtained for setvalued mappings with closed pconvex values in this paper, we conclude that development of nonlinear analysis and related tools for singevalued mappings in locally pconvex spaces for \(0 < p \leq 1\) seems even more important, and can be done by the approach established in this paper.
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 study has received a lot of attention (e.g., see Agarwal et al. [1], Alghamdi et al. [5], Balaj [8], Balachandran [7], Bayoumi [9], Bayoumi et al. [10], Bernuées and Pena [12], Ding [31], Ennassik and Taoudi [34], Ennassik et al. [33], Gal and Goldstein [40], Gholizadeh et al. [41], Jarchow [54], Kalton [55, 56], Kalton et al. [57], Machrafi and Oubbi [73], Park [90], Qiu and Rolewicz [99], Rolewicz [103], Silva et al. [112], Simons [113], Tabor et al. [116], Tan [117], Wang [120], Xiao and Lu [123], Xiao and Zhu [124, 125], Yuan [133], 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 for nonlinear analysis for 1set contractive mappings under the framework of pvector spaces, and in particular, for locally pconvex spaces with \(1 < p \leq 1\).
In particular, we first develop general fixed point theorems for compact continuous mappings, which provide an answer to Schauder’s conjecture of 1930s in the affirmative way under the general framework of topological vector spaces (with \(p=1\) for pvector spaces). Then, the one best approximation result for upper semicontinuous and 1set contractive mappings is given with the boundary condition, which is used as a tool to establish fixed points for nonself setvalued mappings with either inward or outward set conditions in locally pconvex spaces. 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 in locally pconvex spaces. The results given in this paper do 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 social science, engineering, applied mathematics, and related topics and areas under the framewkro of locally pconvex spaces for \(0 < p \leq 1\).
Before discussing the study of best approximation and related nonlinear analysis tools under the framework of locally pconvex spaces, we first would like to share with readers that although most of the results in nonlinear analysis are normally highly associated with convexity hypotheses under local convex topological vector spaces (of course, including normed spaces and Banach spaces, nice metric spaces), it seems to be a surprise that pvector spaces, which in general do not have the local convex structure comparing with locally convex spaces, provide some nice properties in the nature way with some kinds of nice approximation and better (i.e., the bigger) structures for the socalled convexities of pconvex subset play very important roles for us to describe Birkhoff and Kellogg problems and related nonlinear problems (such as fixed point problem and so on) in topological vector spaces (TVS) or locally convex spaces (LCS) based on pvector space behaviors for p in \((0, 1]\) (a pvector space reduces to TVS when \(p = 1\)), and also see the corresponding results and properties as pointed by Remark 2.1 (1), Lemma 2.1(ii), and Lemma 2.3.
Here, we would also like to recall that since the first Birkhoff–Kellogg problem was introduced and an associated theorem was proved by Birkhoff and Kellogg [13] 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 the nonlinear setvalued mappings F, i.e., to find \(x_{0} \in \overline{U}\) and \(\lambda > 0\) such that \(\lambda x_{0} \in F(x_{0})\) (one may also consider if \(x_{0}\) is from the boundary U̅).
On the other hand, after the Birkhoff and Kellogg theorem given by Birkhoff and Kellogg in 1920s, the study on Birkhoff–Kellogg problem has received a lot of scholars’ attention since then. For example, in 1934, one of the fundamental results in nonlinear functional analysis, famously called Leray–Schauder alternative first establoished by Leray and Schauder [67] was also studied via topological degree and some other advanced approaches by Agarwal et al. [1], Alghamdi et al. [5], Balaj [8], and many others. Thereafter, certain other types of Leray–Schauder alternatives were proved using techniques other than topological degree, see the works given by Granas and Dugundji [48], Furi and Pera [39] in the Banach space setting and applications to the boundary value problems for ordinary differential equations in noncompact problems, a general class of mappings for nonlinear alternative of Leray–Schauder type in normal topological spaces, and some Birkhoff–Kellogg type theorems for general class mappings in TVS or LCS by Agarwal et al. [2], Agarwal and O’Regan [3, 4], Park [88], O’Regan [81] by using the Leray–Schauder type coincidence theory applying to establish Birkhoff–Kellogg problem, the Furi–Pera type result for a general class of setvalued mappings. In this paper, based on the application of our best approximation as a tool for general 1set contractive setvalued mappings, we develop the general principle 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 locally pconvex spaces for \((0< p \leq 1)\). These new results given 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.
Now we give a brief introduction on the best approximation theorem related to the development of the nonlinear analysis as a powerful tool with some background.
We all know that the best approximation in nature is related to fixed points for nonself mappings, which tightly link with the classical Leray–Schauder alternative based on the Leray–Schauder continuation theorem by Leray and Schauder [67], which is a remarkable result in nonlinear analysis; and in addition, there exist several continuation theorems, which have many applications to the study of nonlinear functional equations (see Agarwal et al. [1], Alghamdi et al. [5], Balaj [8], O’Regan and Precup [83]). Historically, it seems that the continuation theorem is based on the idea of obtaining a solution of a given equation, starting from one of the solutions of a simpler equation. The essential part of this theorem is the Leray–Schauder boundary condition. It seems that the continuation method was initiated by Poincare [97] and Bernstein [11]. Certainly, Leray and Schauder [67] in 1934 gave the first abstract formulation of the continuation principle using the topological degree (see also Granas and Dugundji [48], Isac [53], Rothe [104, 105], Zeidler [134]). 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 for \(0 < p \leq 1\), which as a basic tool will help us to develop the principle of nonlinear alterative, Leray–Schauder alternative, fixed point theorems of Rothe, Petryshyn, Altman type for nonself mappings, and related fixed point, 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 other related social science areas.
It is well known that the best approximation is one of very important aspects for the study of nonlinear problems related to the problems on their solvability for partial differential equations, dynamic systems, optimization, mathematical program, operation research; and in particular, the one approach well accepted for the study of nonlinear problems in optimization, complementarity problems, of variational inequality problems, and so on, is strongly based on what is today called Fan’s best approximation theorem given by Fan [37] in 1969, which acts as a very powerful tool in nonlinear analysis (see also the book of Singh et al. [114] for the related discussion and 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 (TVS) and local topological vector spaces (LCS), which are comprehensively studied by Agarwal et al. [1], Alghamdi et al. [5], Balaj [8], Chang et al. [24], Chang et al. [25–27], Carbone and Conti [20], Ennassik and Taoudi [34], Ennassik et al. [33], Isac [53], Granas and Dugundji [48], Kirk and Shahzad [60], Liu [72], Park [91], Rothe [104, 105], Shahzad [109–111], Xu [126], Yuan [132, 133], Zeidler [134] (see also the references therein).
On the other hand, the celebrated socalled KKM principle established in 1929 in [62] 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 [91]). Since Fan extended the classical KKM theorem to infinitedimensional spaces in 1961 [36–38], 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 (see [14–19] and the related references therein). Among them, Schauder’s fixed point theorem [107] 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 [75] 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?” Recently, this question has been answered by the work of Ennassik and Taoudi [34] by using the pseminorm methods under locally pconvex spaces; see also related contribution given by Cauty [22], plus the works by Askoura and GodetThobie [6], Cauty [21], Chang [23], Chang et al. [24], Chen [29], Dobrowolski [32], Gholizadeh et al. [41], Isac [53], Li [70], Li et al. [69], Liu [72], Nhu [77], Okon [79], Park [90–92], Reich [100], Smart [115], Weber [121, 122], Xiao and Lu [123], Xiao and Zhu [124, 125], Xu [129], Xu et al. [130], Yuan [132, 133] in both TVS, LCS and related references therein under the general framework of pvector spaces for nonself setvalued or singlevalued mappings \((0 < p \leq 1)\).
The goal of this paper is to establish general new tools of nonlinear analysis under the framework of locally pconvex (seminorm) spaces for 1set contractive mappings (here \(0 < p \leq 1\)), but we do wish these new results such as best approximation, theorems of Birkhoff–Kellogg type, nonlinear alternative, fixed point theorems for nonself setvalued with boundary conditions, Rothe, Petryshyn type, Altman type, Leray–Schedule type, and other related nonlinear problems would play important roles for the nonlinear analysis of pseminorm spaces for \(0 < p \leq 1\). In addition, our results on the fixed point theorem for compact (singlevalued) continuous mappings in TVS also provide solutions for Schauder’s conjecture since 1930s in the affirmative way under the general setting of pvector spaces for pconvex sets (which may not be locally convex when \(p \in (0, 1)\), see Kalton [55, 56], Kalton et al. [57], Jarchow [54], Roloewicz [103], and the related references for the study on the development of related nonlinear analysis). In addition, based on the framework for some key results in nonlinear analysis obtained for setvalued mappings with closed pconvex values in this paper, we also conclude that the development of nonlinear analysis for singevalued mappings in locally pconvex spaces for \(0 < p \leq 1\) seem very importnat, too, and can be developed by the approach and method established in this paper.
The paper has seven sections. Section 1 is the introduction. Section 2 describes general concepts for the pconvex subsets of topological vector spaces (\(0 < p \leq 1\)). In Sect. 3, some basic results of the KKM principle related to abstract convex spaces are given by including locally pconvex spaces as a special class. In Sect. 4, as the application of the KKM principle in abstract convex spaces, which include locally pconvex spaces as a special class (\(0< p \leq 1\)), and plus by combining the embedding lemma for compact pconvex subsets from topological vector spaces into locally pconvex spaces, we provide general fixed point theorems for compact (singlevalued) continuous selfmappings defined on pconvex compact in TVS, and condensing upper semicontinuous setvalued mappings defined on noncompact pconvex subsets in locally pconvex vector 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, and then it is used as a tool to establish general existence theorems for solutions of Birkhoff–Kellogg (problem) alternative, general principle of nonlinear alterative, including Leray–Schauder alternative, Rothe type, Altman type associated with different boundary conditions in locally pconvex spaces. In Sect. 6, we give a number of new results based on the general principles of Birkhoff–Kellogg theorems and Leray–Schauder alternative established in Sect. 5 for 1set contractive mappings with different boundary conditions in locally pconvex spaces. In Sect. 7, we focus on the study of fixed point theorems for classes of 1set contractive setvalued mappings under various boundary conditions by including nonexpansive setvalued mappings under pnorm spaces, uniformly convex Banach spaces, or with Opial condition.
For the convenience of our discussion, throughout this paper, all pconvex topological vector spaces and the compact pconvex sets are always assumed to be Hausdorff, and p satisfies the condition for \(0 < p \leq 1\) unless specified otherwise, and also we denote by \(\mathbb{N}\) the set of all positive integers, i.e., \(\mathbb{N}:=\{1,2, \ldots , \}\).
2 The basic results of pvector spaces
We now recall some notions and definitions for pconvex topological vector spaces which will be used in what follows (see Balachandran [7], Bayoumi [9], Jarchow [54], Kalton [55], Rolewicz [103], Gholizadeh et al. [41], Ennassik et al. [33], Ennassik and Taoudi [34], Xiao and Lu [123], Xiao and Zhu [124], and the references therein).
Definition 2.1
A set A in a vector space X is said to be pconvex for \(0 < p \leq 1\) if, for any \(x, y\in A\), \(0 \leq s, t \leq 1\) with \(s^{p} + t^{p}=1\), we have \(s^{1/p}x+t^{1/p}y\in A\); and if A is 1convex, it is simply called convex (for \(p = 1\)) in general vector spaces; the set A is said to be absolutely pconvex if \(s^{1/p}x+t^{1/p}y\in A\) for \(0 \leq s, t \leq 1\) with \(s^{p} + t^{p} \leq 1\).
Definition 2.2
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 are the smallest pconvex set containing A and the smallest closed pconvex set containing A, respectively.
Definition 2.3
Let 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 circled (or balanced) 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 circled subsets of X is easily seen to be closed under the formation of linear combinations, arbitrary unions, and arbitrary intersections. 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 \(\hat{A} =\cup _{\lambda \leq 1} \lambda A\) 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, we use \(\operatorname{int}(A)\) 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 unless specified otherwise.
Definition 2.5
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 p. 52 of Bayoumi [9], Jarchow [54], or Rolewicz [103]).
Definition 2.6
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 \(x, y \in X\).
A pseminorm P is called a pnorm if \(x=0\) whenever \(P(x)=0\), so a vector space with a specific pnorm is called a pnormed space, and of course if \(p=1\), X is a normed space as discussed before (e.g., see Jarchow [54]).
By Lemma 3.2.5 of Balachandran [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 a 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 \(\epsilon > 0\).
Definition 2.7
A topological vector space X is said to be locally pconvex if it has a 0basis consisting of pconvex neighborhoods for \((0 < p \leq 1)\). If \(p=1\), X is a usual locally convex space.
We also need the following notion for the socalled pgauge (see Balachandran [7]).
Definition 2.8
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 the pgauge (gauge if \(p=1\)). The pgauge of A is also known as the Minkowski pfunctional for set A.
By Proposition 4.1.10 of Balachandran [7], we have the following proposition.
Proposition 2.2
Let A be an absorbing subset of X. Then 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).
As mentioned above, a given pseminorm is said to be a pnorm if \(x = 0\) whenever \(P(x) = 0\). A vector space with a specific pnorm is called a pnormed space. The pnorm of an element \(x \in E\) will usually be denoted by \(\x\_{p}\). If \(p = 1\), X is a usual normed space. If X is a pnormed space, then \((X, d_{p})\) is a metric linear space with a translation invariant metric \(d_{p}\) such that \(d_{p}=d_{p}(x, y)=\x y\_{p}\) for \(x, y \in X\). We point out that pnormed spaces are very important in the theory of topological vector spaces. Specifically, a Hausdorff topological vector space is locally bounded if and only if it is a pnormed space for some pnorm \(\ \cdot \_{p}\), where \(0 < p \leq 1\) (see p. 114 of Jarchow [54]). We also note that examples of pnormed spaces include \(L^{p}(\mu )\)  spaces and Hardy spaces \(H_{p}\), \(0 < p < 1\), endowed with their usual pnorms.
Remark 2.1
We would like to make the following important two points.
(1) First, by the fact that (e.g., see Kalton et al. [57] or Ding [31]) there is no open convex nonvoid subset in \(L^{p}[0, 1]\) (for \(0< p < 1\)) except \(L^{p}[0,1]\) itself. This means that pnormed paces with \(0< p <1\) are not necessarily locally convex. Moreover, we know that every pnormed space is locally pconvex; and incorporating Lemma 2.3, it seems that pvector spaces (for \(0 < p \leq 1\)) are nicer spaces as we can use a pvector space to approximate (Hausdorff) topological vector spaces (TVS) in terms of Lemma 2.1(ii) for the convex subsets in TVS by using bigger pconvex subsets in pvector spaces for \(p\in (0,1)\) by also considering Lemma 2.3. In this way, it seems that Pvector spaces have better properties in terms of pconvexity than the usually (1−) convex subsets used in TVS with \(p=1\).
(2) Second, it is worthwhile noting that a 0neighborhood in a topological vector space is always absorbing by Lemma 2.1.16 of Balachandran [7] or Proposition 2.2.3 of Jarchow [54].
Now, by Proposition 4.1.12 of Balachandran [7], we also have the following Proposition 2.3 and Remark 2.2 (which is Remark 2.3 of Ennassik and Taoudi [34]).
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 \(\overline{B_{1}}\) depending on 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. For the reader’s convenience, we provide a sketch of proof below (see also Lemma 2.1 of Ennassik and Taoudi [33], Remark 2.1 of Qiu and Rolewicz [99]).
Lemma 2.1
Let A be a subset of a vector space X, then we have:
(i) If A is pconvex with \(0 < p < 1\), 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 pconvex for any \(p \in (0, 1]\).
(iii) If A is pconvex for some \(p \in (0, 1)\), then A is sconvex for any \(s \in (0, p]\).
Proof
(i) As \(r \le 1\), by the fact that “for all \(x \in A\) and all \(\alpha \in [2^{(n+1)(1\frac{1}{p})}, 2^{n(1\frac{1}{p})}]\), we have \(\alpha x \in A\)” is true for all integer \(n\geq 0\), taking into account the fact that \((0, 1]=\cup _{n\geq 0} [2^{(n+1)(1\frac{1}{p})}, 2^{n(1\frac{1}{p})}]\), the result is obtained.
(ii) Assume that A is a convex subset of X with \(0 \in A\) and take a real number \(s \in (0, 1]\). We show that A is sconvex. Indeed, let \(x, y \in A\) and \(\alpha , \beta >0\) with \(\alpha ^{p} + \beta ^{p} = 1\). Since A is convex, then \(\frac{\alpha}{\alpha + \beta} x + \frac{\beta}{\alpha + \beta}y \in A\). Keeping in mind that \(0 < \alpha + \beta < \alpha ^{p} + \beta ^{p}=1\), it follows that \(\alpha x + \beta y=(\alpha + \beta )(\frac{\alpha}{\alpha + \beta}x + \frac{\beta}{\alpha +\beta}y ) + (1\alpha \beta ) 0 \in A\).
(iii) Now, assume that A is rconvex for some \(p \in (0,1)\) and pick up any real \(s \in (0, p]\). We show that A is sconvex. To see this, let \(x, y \in A\) and \(\alpha , \beta > 0\) such that \(\alpha ^{s} + \beta ^{s}=1\). First notice that \(0 < \alpha ^{\frac{p  s}{p}} \leq 1\) and \(0 < \beta ^{\frac{p  s}{p}} \leq 1\), which imply that \(\alpha ^{\frac{p  s}{p}} x \in A\) and \(\beta ^{\frac{p  s}{p}} y \in A\). By the pconvexity of A and the equality \((\alpha ^{\frac{s}{p}})^{p} + (\beta ^{\frac{s}{p}})^{p} =1\), it follows that \(\alpha x + \beta y = \alpha ^{\frac{s}{p}}(\alpha ^{\frac{ps}{p}}x) + \beta ^{\frac{s}{p}}(\beta ^{\frac{ps}{p}} y) \in A\). This competes the sketch of the proof. □
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 \(p \in (0, 1)\) (see also Lemma 2.3 below).
We also need the following proposition, which is Proposition 6.7.2 of Jarchow [54].
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 [124].
Lemma 2.2
Let C be a bounded closed pconvex subset of a pseminorm X with \(0 \in \operatorname{int} C\), where \((0< p\leq 1)\). For every \(x\in X\) 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 [124], Proposition 2.3, and Remark 2.2, the proof is compete. □
Remark 2.4
As discussed by 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”.
Before we close this section, we would like to point out that the structure of pconvexity when \(p \in (0, 1)\) is really different from what we normally have for the concept of “convexity” used in topological vector spaces (TVS). In particular, maybe the following fact is one of the reasons for us to use better (pconvex) structures in pvector spaces to approximate the corresponding structure of the convexity used in TVS (i.e., the pvector space when \(p=1\)). Based on the discussion in p. 1740 of Xiao and Zhu [124](see also Bernués and Pena [12] and Sezer et al. [108]), we have the following fact, which indicates that each pconvex subset is “bigger” than the convex subset in topological vector spaces for \(0 < p < 1\).
Lemma 2.3
Let x be a point of a pvector space E, where assume \(0 < p < 1\), then the pconvex hull and the closure of \(\{x\}\) are given by
and
But note that if x is a given one point in pvector space E, when \(p=1\), we have that \(\overline{C_{1}(\{x\})} =C_{1}(\{x\})=\{ x\}\). This shows to be significantly different for the structure of pconvexity between \(p=1\) and \(p\neq 1\)!
As an application of Lemma 2.3, we have the following fact for (setvalued) mappings with nonempty closed pconvex values in pvector spaces for \(p \in (0, 1)\), which are truly different from any (setvalued) mappings defined in topological vector spaces (i.e., for a pvector space with \(p =1\)).
Lemma 2.4
Let U be a nonempty subset of a pvector space E (where \(0 < p < 1\)) with zero \(0 \in U\), and assume that a (setvalued) mapping \(T: U \rightarrow 2^{E}\) is with nonempty closed pconvex values. Then T has at least one fixed point in U, which is the element zero, i.e., \(0 \in \cap _{x \in U} T(x) \ne \emptyset \).
Proof
For each \(x \in U\), as \(T(x)\) is nonempty closed pconvex, by Lemma 2.3, we have at least \(0 \in T(x)\). It implies that \(0 \in \cap _{x \in U} T(x)\), and thus zero of E is a fixed point of T. This completes the proof. □
Remark 2.5
By following Definitions 2.5 and 2.6, the discussion given by Proposition 2.3, and remarks thereafter, each given (open) pconvex subset U in a pvector space E with the zero \(0 \in \operatorname{int}(U)\) always corresponds to a pseminorm \(P_{U}\), which is indeed the Minkowski pfunctional of U in E, and \(P_{U}\) is continuous in E. In particular, a topological vector space is said to be locally pconvex if the origin 0 of E has a fundamental set (denoted by) \(\mathfrak{U}\), which is a family of absolutely pconvex 0neighborhoods (each denoted by U). This topology can be determined by pseminorm \(P_{U}\), which is indeed the family \(\{P_{U}\}_{U \in \mathfrak{U}}\), where \(P_{U}\) is just the Minkowski pfunctional for each \(U \in \mathfrak{U}\) in E (see also p. 52 of Bayoumi [7], Jarchow [49], or Rolewicz [99]).
Throughout this paper, by following Remark 2.5, without loss of generality unless specified otherwise, for a given pvector space E, where \(p \in (0, 1]\), we always denote by \(\mathfrak{U}\) the base of the pvector space E’s topology structure, which is the family of its 0neighborhoods. For each \(U \in \mathfrak{U}\), its corresponding Pseminorm \(P_{U}\) is the Minkowski pfunctional of U in E. For a given point \(x \in E\) and a subset \(C \subset E\), we denote by \(d_{P_{U}}(x, C): =\inf \{P_{U}(xy): y \in C\}\) (in short, denoted by \(d_{P}(x, C)\) if no confusion below) the distance of x and C by the seminorm \(P_{U}\), where \(P_{U}\) is the Minkowski pfunctional for each \(U \in \mathfrak{U}\) in E.
3 The KKM principle in abstract convex spaces
As mentioned in the introduction, Knaster, Kuratowski, and Mazurkiewicz (in short, KKM) [62] 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 [36] (see also Fan [38]) 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 [85]. Then the KKM theory was extended to general abstract convex spaces by Park [89] (see also Park [90] and [91]), which actually include locally pconvex spaces (\(0 < p \leq 1\)) as a special class.
Here we first give some notion and a brief introduction to 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], Alghamdi et al. [5], Balaj [8], Mauldin [75], Granas and Dugundji [48], Park [91] and [92], Yuan [133], and 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 by Park [89].
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 \(A \in \langle D\rangle \) such that the 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, multimap) \(F: E \rightarrow 2^{Z}\) 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. A KKM mapping \(G: D\rightarrow 2^{E}\) is a KKM mapping with respect to the identity map \(1_{E}\).
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 [89] and also [90]) 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 \(\phi _{A}: \Delta _{n}\rightarrow 2^{X}\) (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 \(G: D\rightarrow 2^{X}\) 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, it is clear that every \(\phi _{A}\)space is a KKM space, thus we have the following fact (see Lemma 1 of Park [90]).
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 [54] or p. 52 of Bayoumi [9]).
Now we have a new KKM space as follows inducted by the concept of pconvexity (see Lemma 2 of Park [90]).
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 \(N\in \langle D\rangle \). Then \((X, D; \Gamma )\) is 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 \(\sum_{i=0}^{n}t_{i}e_{i}\mapsto \sum_{i=0}^{n}(t_{i})^{ \frac{1}{\mathrm{p}}}x_{i}\). Then, clearly, \((X, D; \Gamma )\) is a \(\phi _{A}\)space. This completes the proof. □
4 Fixed point theorems for condensing setvalued mappings in pvector spaces
In this section, we establish fixed point theorems for upper semicontinuous setvalued mappings in locally pconvex spaces, compact (singlevalued) continuous mappings for pconvex subsest in TVS, and condensing mappings for pconvex subsets under the general framework of locally pconvex spaces, which will be a tool used in Sect. 5, Sect. 6 and Sect. 7 to establish the best approximation, fixed points, the principle of nonlinear alternative, Birkhoff–Kellogg problems, Leray–Schauder alternative, which would be useful tools in nonlinear analysis for the study of nonlinear problems arising from theory to the 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 \(T^{1}(B): = \{ x\in X: T(x) \cap B \neq \emptyset \}\) (resp., the set \(\{x \in X: T(x) \subset B\}\)) is closed (resp., 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 related results given by Theorem 2.7 of Gholizadeh et al. [41], Theorem 5 of Park [90], 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 for T is lower semicontinuous (LSC) and upper semicontinuous (USC).
Case 1, by assuming T is lower semicontinuous: As X is a locally pconvex vector space, suppose that \(\mathfrak{U}\) is the 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 \(A \subset \cup _{i=0}^{n} (x_{i} + V)\). 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 \(\emptyset \neq T(c)\cap A \subset T(c)\cap \cup _{i=0}^{n}(x_{i}+ V)\), it follows that \(\cap _{i=0}^{n} F(b_{i})=\emptyset \). 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 [35]) in locally pconvex vector spaces for \((0 < p \leq 1)\), which also improve or generalize the corresponding results given by Yuan [133], Xiao and Lu [123], Xiao and Zhu [124, 125] into locally pconvex vector spaces.
Theorem 4.2
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 upper semicontinuous with nonempty pconvex closed values. Then T has at least one fixed point.
Proof
Assume that \(\mathfrak{U}\) is 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, regarding two nets \(\{a_{U}\}\) and \(\{b_{U}\}\) in 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 also Lemma 1.1 of Yuan [132]), thus the proof is complete. □
For a given set A in a vector space X, we denote by “\(\operatorname{lin}(A)\)” the “linear hull” of A in X.
Definition 4.2
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 embedded Lemma 4.1 is a significant result due to Theorem 1 of Kalton [55], which says that although not every compact convex set can be linearly embedded in a locally convex space (e.g., see Roberts [101] and Kalton et al. [57]), but for pconvex sets when \(0 < p <1\), every compact pconvex set in topological vector spaces can be 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 of a topological vector space containing zero 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 comppact (singlevalued) continuous mappings for compact pconvex subsets in topological vector 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 in pvector spaces.
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 [55] which completes the proof. □
Remark 4.1
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. [57], 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 [101].
Now we give the the following fixed point theorem for (upper semicontinuous continuity, which actually is not needed by applying Lemma 2.4 directly) setvalued mappings with nonempty pconvex closed values defined on closed pconvex subsets of locally pconvex spaces for \(0 < p < 1 \); and also for the singlevalued version of a continuous mapping which is Theorem 3.3 first given by Ennassik and Taoudi [34] by using the pseminorm argument in pvector spaces for \(0 < p \leq 1\). Here we like to point out that though the conclusion for the existence of fixed points for upper semicontinuous setvalued mappings below defined on compact pconvex subsets for \(0 < p < 1 \) is the direct conlusion by Lemma 2.4 (even without assumption for the upper semicontinuity), the following argument shares with readers the way by applying KKM principle how to establish the setvalued versions of fixed points in pvector spaces with combining some special embedded features for pconvex structures for \(0< p <1\).
Theorem 4.3
If K is a nonempty compact pconvex subset of a topological vector space X for \(0 < p < 1\), then any upper semicontinuous setvalued mapping \(T: K \rightarrow 2^{K}\) with nonempty pconvex closed value has at least a fixed point; and secondly, for \(0 < p \leq 1\), any singlevalued continuous mapping \(T: K \rightarrow K\) has at least a fixed point.
Proof
We complete the argument by the following two cases.
First case, for the setvalued mappings by assuming that K is pconvex with \(0 < p < 1\). By Lemma 4.1, it follows that K can be linearly embedded in a locally pconvex space E, which means that there exists a linear map \(L: \operatorname{lin}(K) \rightarrow E\) whose restriction to K is a homeomorphism. Define the mapping \(S: L(K) \rightarrow L(K)\) by \((Sx): = L(Tx)\) for \(x \in X\). This mapping is easily checked to be well defined. The mapping S is upper semicontinuous since L is a (continuous) homeomorphism and T is upper semicontinuous on K. Furthermore, the set \(L(K)\) is compact, being the image of a compact set under a continuous mapping L. It is also pconvex since it is the image of a pconvex set under a linear mapping. Then, by Theorem 4.2, there exists \(x \in K\) such that \(Lx \in S(Lx) =L(Tx)\), thus there exists \(y \in Tx\) such that \(Lx=Ly\), which implies that \(x=y \in T(x)\) since L is a homeomorphism, which is the fixed point of T.
Second case, considering when T is a singlevalued continuous mapping for \(0 < p \leq 1\), this is Theorem 3.3 given by Ennassik and Taoudi [34]. Thus the proof is complete. □
Remark 4.2
As mentoined above, by Lemma 2.4, the conclusion in Theorem 4.3 holds for setvalued mappings with nonempty pconvex closed values without upper semicontinuous assumptions when \(0< p < 1\). Secondly, the the singlevalued version of Theorem 4.3 which was first given by Ennassik and Taoudi (Theorem 3.3 of [34]) for \(0 < p \leq 1\) indeed provides an answer to Schauder’s conjecture under the TVS. Here we also mention a number of related works and discussion by authors in this drection, see Mauldin [75], Granas and Dugundji [48], Park [91, 92] and the references therein.
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 set 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 version of fixed point theorems for compact (singlevalued) continuous mappings defined on a general pconvex subset in topologiocal vector spaces for \(0 < p \leq 1\).
Theorem 4.4
(Schauder’s fixed point theorem for compact mappings)
If C is a nonempty closed pconvex subset of a topological vector space E with (\(0 < p \leq 1\)) and \(T: C \rightarrow C\) is (singlevalued) continuous and compact (i.e., the set \(T(C)\) is contained in a compact subset of C), then T has at least one fixed point.
Proof
As T is compact, there exists a compact subset A in C such that \(T(C) \subset A\). Let \(K: =\overline{C}_{p}(A)\) be the closure of the pconvex hull of set A in C. Then K is compact pconvex by Proposition 2.4, and the mapping \(T: K \rightarrow K\) is continuous. Now, by Theorem 4.3, it follows that T has a fixed point \(x \in K \subset C\) such that \(x \in T(x)\). This completes the proof. □
As an immediate consequence of Theorem 4.4, we have the following result, which gives an affirmative answer to Schauder’s conjecture in in topological vector spaces (TVS).
Corollary 4.1
If K is a nonempty closed convex subset of a topological vector space X, then any (singlevalued) continuous and compact mapping \(T: K \rightarrow K\) has at least a fixed point.
Proof
Apply Theorem 4.4 with \(p = 1\), this completes the proof. □
Theorem 4.4 improves or unifies corresponding results given by Askoura and GodetThobie [6], Cauty [21], Cauty [22], Chen [29], Isac [53], Li [70], Nhu [77], Okon [79], Park [92], Reich [100], Smart [115], Yuan [133], Theorem 3.3 of Ennassik and Taoudi [33], Theorem 3.14 of Gholizadeh et al. [41], Xiao and Lu [123], Xiao and Zhu [124, 125] under the framework of topological vector spaces.
In order to establish fixed point theorems for the classes of 1set contractive and condensing mappings in pvector spaces by using the concept of the measure of noncompactness (or the noncompactness measures), which were introduced and widely accepted in mathematical community by Kuratowski [65], Darbo [30] (see related references therein), 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 [73], Nussbaum [78], Sadovskii [106], Silva et al. [112], Xiao and Lu [123] for the general concepts under the framework of pseminorm or just for locally convex pconvex settings for \(0< p \leq 1\), which will be discussed below, too).
For a given metric space \((X, d)\) (or a pnormed space \((X, \c\ \cdot \_{p})\)), we recall the notions of completeness, boundedness, relative compactness, and compactness as follows. Let \((X, d)\) and \((Y, d)\) be two metric spaces and \(T: X \rightarrow Y\) be a mapping (or an operator). Then: 1) T is said to be bounded if for each bounded set \(A\subset X\), \(T(A)\) is bounded set of Y; 2) T is said to be continuous if for every \(x \in X\), the \(\lim_{n \rightarrow \infty} x_{n} = x\) 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 Kuratowski 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 [28], Ko and Tasi [63], and Kozlov et al. [64] (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 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.
In order to establish the fixed points of setvalued condensing mappings in pvector spaces for \(0 < p \leq 1\), we need to recall some notions introduced by Machrafi and Oubbi [73] 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 [65], Sadovskii [106](see also the related discussion by Alghamdi et al. [5], Nussbaum [78], Silva et al. [112], Xiao and Lu [123] and the references therein). In particular, the measures of noncompactness in locally pconvex 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 [73]) unless stated otherwise. The E is used to denote a Hausdorff topological vector space over the field \(\mathbb{K} \in \{\mathbb{R}, \mathbb{Q}\}\), where \(\mathbb{R}\) denotes all real numbers and \(\mathbb{Q}\) all complex numbers, and \(p \in (0, 1]\). Here, the base set of a 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 [61] or Jarchow [54] 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 pseminormed space. A pnormed space is a Hausdorff pseminormed 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 above.
By Remark 2.5, 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 small of order U by Robertson [102]). Now, we follow the idea of Kaniok [58] in the setting of a topological vector space E to 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
where 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 [58] can be expressed by the Kuratowski (resp., the Hausdorff) 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 following Machrafi and Oubbi [73], in order 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 4.3
(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 will be 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 4.3
By Remark 2.5 again, 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 note that: 1) In the case 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, the collection \((B_{p})_{p \in \mathbb{P}}\) is an 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 an SCZN consisting of \(\mathfrak{W}\) members; and 3) By following Oubbi [84], 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 4.4
Let \(\mathfrak{B}\) be an 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 the related discussion by Machraf and Oubbi [84]).
As we know, under the normed spaces (and even seminormed spaces), Kuratowski [65], Darbo [30], and Sadovskii [106] introduced the notions of ksetcontractions for \(k \in (0, 1)\) and the 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 4.5
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 the \(\beta _{K}\) and \(\beta _{H}\), which are the Kuratowski measure and Hausdorff measure of noncompactness, respectively, in normed spaces (see Kuratowski [65]).
From now on, we denote by \(\mathfrak{V}_{0}\) the set of all shrinkable zero neighborhoods in E. We have the following result which is Theorem 1 of Machrafi and Oubbi [73], saying that in the general setting of locally pconvex spaces, the measure of noncompactness α for U given by Definition 4.4 is stable from U to its pconvex hull \(C_{p}(A)\) of the subset A in E, which is key for us to establish the fixed points for condensing mappings in locally pconvex spaces for \(0< p \leq 1\). This also means that the key property for the Kuratowski and Hausdorff measures of noncompactness in normed (or pseminorm) spaces also holds for the measure of noncompactness by Definition 4.4 in the setting of locally pconvex spaces with (\(0 < p \leq 1\)) (see more similar and related discussion in detail by Alghamdi et al. [5] and Silva et al. [112]).
Lemma 4.2
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 [73]. The proof is complete. □
Now, based on the definition for the measure of noncompactness given by Definition 4.4 (originally from Machrafi and Oubbi [73]), 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 4.5
(Schauder’s fixed point theorem for condensing mappings)
Let \(C \subset E\) be a complete pconvex subset of a Hausdorff locally pconvex space E with \(0 < p \leq 1\). If \(T: C \rightarrow 2^{C}\) is an upper semicontinuous and (α) condensing setvalued mapping with nonempty pconvex closed values, then T has a fixed point in C and the set of fixed points of T is compact.
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 \(A_{0}=\cap _{A \in \mathfrak{F}} A\). 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 [73] and Lemma 4.2 (i.e., Theorem 1 and Theorem 2 in [73]), 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 [73]). Now, the conclusion follows by Theorem 4.2. Secondly, let \(C_{0}\) be the set of fixed points of T in C. Then it follows that \(C_{0} \subset T(C_{0})\), and the upper semicontinuity of T implies that its graph is closed, so is the set \(C_{0}\). As T is condensing, we have \(\alpha _{U}(T(C_{0})) \leq \alpha _{U}(C_{0})\), which implies that \(\alpha _{U}(C_{0})=0\). As U is arbitrary from the family \(\mathfrak{B}\), it implies that \(C_{0}\) is compact (by Proposition 4 in [73] again). The proof is complete. □
As applications of Theorem 4.5, we have the following few of fixed points for condensing mappings in locally pconvex spaces for \(0 < p \leq 1\).
Corollary 4.2
(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 4.3
(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 continuous and condensing mapping \(T: C \rightarrow 2^{C}\) with closed and pconvex values has a fixed point.
Proof
In Theorem 4.5, let \(\mathfrak{B}: =\{B_{p}(0, 1) \}\), where \(B_{p}(0,1)\) stands for the closed unit ball of E. By the fact that it is clear that \(\alpha (A)=(\alpha _{\mathfrak{B}}(A))^{p}\) for each \(A \subset E\), T satisfies all the conditions of Theorem 4.5. This completes the proof. □
Corollary 4.4
(Darbo type)
Let \((E, \ \cdot \)\) be a complete pnormed space and C be a bounded, closed pconvex subset of E, where \(0 < p \leq 1\). Then each mapping \(T: C \rightarrow C\) which is a continuous setcontraction with closed pconvex values has a fixed point.
Theorem 4.5 improves Theorem 5 of Machrafi and Oubbi [73] for general condensing mappings, which are general upper semicontinuous mappings with closed pconvex values, and also unifies the corresponding results in the existing literature, e.g., see Alghamdi et al. [5], Górniewicz [46], Górniewicz et al. [47], Nussbaum [78], Silva et al. [112], Xiao and Lu [123], Xiao and Zhu [124, 125], and the references therein.
Secondly, as an application of the KKM principle for abstract convex spaces with Kalton’s remarkable embedded lemma [55] for compact pconvex sets in topological vector spaces, we also establish general fixed point theorems for compact (singlevalued) continuous mappings, which allows us to answer Schauder’s conjecture in the affirmative way under the general framework of closed pconvex subsets in TVS for \(0 < p \leq 1\).
Before ending this section, we would also like to remark that compared with the topological method or related arguments used by Askoura et al. [6], Cauty [21, 22], Nhu [77], Reich [100], the fixed points given in this section improve or unify the corresponding ones given by Alghamdi et al. [5], Darbo [30], Liu [72], Machrafi and Oubbi [73], Sadovskii [106], Silva et al. [112], Xiao and Lu [123], and those from the references therein.
5 Best approximation for the classes 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 [13]) and fixed points for nonself 1set contractive setvalued mappings in locally pconvex spaces for \(0 < p \leq 1\).
Here, we recall that since the Birkhoff–Kellogg theorem was first introduced and proved by Birkhoff and Kellogg [13] 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 the 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 Birkhoff and Kellogg theorem was given by Birkhoff and Kellogg in 1920s, the study on the Birkhoff–Kellogg problem has received a lot of scholars’ attention. For example, one of the fundamental results in nonlinear functional analysis, called the Leray–Schauder alternative by Leray and Schauder [67] in 1934, was established via topological degree. Thereafter, certain other types of Leray–Schauder alternatives were proved using techniques other than topological degree, see the work given by Granas and Dugundji [48], Furi and Pera [39] 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 a general class of mappings in TVS or LCS by Agarwal et al. [2], Agarwal and O’Regan [3, 4], Park [88]; in particular, recently O’Regan [81] used the Leray–Schauder type coincidence theory to establish some Birkhoff–Kellogg problem, 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 and is then used to the general principle for solutions of Birkhoff–Kellogg problems and related nonlinear alternatives, which allows us to give general existence results for the Leray–Schauder type and related fixed point theorems of nonself mappings in pseminorm spaces for \((0< p \leq 1)\). 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 the Leray–Schauder alternative under the framework of pseminorm spaces for \((0< p \leq 1)\) given in this section would be a useful tool for the study of nonlinear problems. In addition, we also note that the corresponding results in the existing literature for Birkhoff–Kellogg problems and the Leray–Schauder alternative have been studied comprehensively by Granas and Dugundji [48], Isac [53], Park [89–91], Carbone and Conti [20], Chang and Yen [27], Chang et al. [25, 26], Kim et al. [59], Shahzad [110]–[111], Singh [114]; and in particular, many general forms have been recently obtained by O’Regan [82] (see also the references therein).
In order to study the existence of fixed points for nonself mappings in pvector spaces for \(0 < p \leq 1\), 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 definitions of the inward set \(I_{C}(x)\) and the outward set \(O_{C}(x)\), respectively, in topological vector spaces introduced by Halpern and Bergman [49] 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 \(I_{C}^{p}(x)\) 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 for \(0 < p \leq 1\). 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 compete. □
By following the original idea by Tan and Yuan [118] 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 below 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 \(\lim_{n \rightarrow \infty} d_{P_{U}}P(x_{n}, F(x_{n})) =0\) for each \(U \in \mathfrak{U}\), where \(d_{P_{U}}P(x, C):= \inf \{P_{U}(xy): y \in C\}\) 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 [118]). 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 converge 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 a locally pconvex space E for \(0< p \leq 1\).
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’s fixed point theorem for 1set contractive mappings)
Let U be a nonempty bounded open 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 an upper semicontinuous and 1set contractive setvalued mapping with nonempty pconvex closed values and satisfies one of the following conditions:
(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 the 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 the 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 subsets containing the zero in E and 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 \(0 < \lambda _{n} < 1\). By Theorem 4.5 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 \(x_{n}=\lambda _{n} y_{n}\). 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 which 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 (\(0 \leq p \leq 1\)) the zero \(0 \in U\), and 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 1set contractive and upper semicontinuous mapping with nonempty closed pconvex values, 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 one of the following conditions:
(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 the 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 the 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 \(x_{0} \in \overline{U} \cap C\) such that \(x_{0} \in F(x_{0})\) (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
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, taking 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 follows by Theorem 4.5 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 one of 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 the (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, if follows that the consequence \(\{x_{n}\}_{n \in \mathbb{N}}\) has a convergent subsequence which 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, if 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 exists \(y \in \overline{U}\) 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 [37, 38]) 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. [69], Liu [72], Xu [129], Xu et al. [130], 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 [87] and the related discussion by 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 \(F: \overline{U}\cap C \rightarrow C\) is a 1setcontractive continuous mapping with closed pconvex values satisfying 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 \(x_{0} \in \overline{U} \cap C\), 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
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 a 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 that either (I) or (II) holds:
(I) \(F_{1}\) has a fixed point \(x_{0} \in \overline{U} \cap C\) (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 and 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 (\(0 \leq p \leq 1\)) with zero \(0 \in \operatorname{int}U=(U)\) (the interior intU as U is open), and C be a closed pconvex subset of E with also zero \(0\in C\). Assume that \(F: \overline{U}\cap C \rightarrow C\) is a 1setcontractive continuous mapping with closed pconvex values satisfying the (H) or (H1) condition above. Then F has at least one of the following two properties:
(I) F has a fixed point \(x_{0} \in \overline{U} \cap C\) 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
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 additional 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 can also have the following general principle of Birkhoff–Kellogg alternative in locally convex spaces by applying Theorem 5.4 for \(p=1\).
Theorem 5.5
(Principle of Birkhoff–Kellogg alternative in TVS)
Let U be a bounded open convex subset of the locally convex space E with the 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 upper semicontinuous mapping with nonempty closed convex values satisfying the (H) or (H1) condition above. Then it has at least one of the following two properties:
(I) F has a fixed point \(x_{0} \in \overline{U} \cap C\) 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 \(P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, \overline{I_{\overline{U}}( x_{0})} \cap C) > 0\).
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 locally pconvex 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 a pseminorm space E with \(0 \leq p \leq 1\) and the zero \(0 \in C\). Assume that \(F: C \rightarrow 2^{C}\) is a 1set contractive and upper semicontinuous mapping with nonempty closed pconvex values 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 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 the 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 the general conditions of Theorem 5.4, and we have that for 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 locally pconvex 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 problem, variational inequality, complementarity problems (see Isac [53] or Yuan [133] 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 upper semicontinuous mapping with nonempty closed pconvex values 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 spaces, we have the following classical Fan’s best approximation (see [37]) as a powerful tool for the study in the optimization, mathematical programming, games theory, 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 the zero \(0 \in U\) and C be a closed convex subset of E with also zero \(0\in C\), and assume \(F: \overline{U}\cap C \rightarrow 2^{C}\) is a 1set contractive and upper semicontinuous mapping with nonempty closed convex values satisfying the (H) or (H1) condition above. Assume \(P_{U}\) to be 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 an inward set \(I_{\overline{U}}(x_{0})\) or an outward set \(O_{\overline{U}}(x_{0})\):
(I) F has a fixed point \(x_{0} \in \overline{U} \cap C\), \(0= P_{U} (y_{0}  x_{0}) = P_{U} (y_{0}  x_{0}) = d_{P}(y_{0}, \overline{U}\cap C) = d_{p}(y_{0}, \overline{W_{\overline{U}}( x_{0})} \cap C))\),
(II) There exist \(x_{0} \in \partial _{C}(U)\) and \(y_{0} \in F(x_{0}) \diagdown \overline{U}\) with
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 similar results on Rothe and Leray–Schauder alternative developed by Isac [53], Park [86], Potter [98], Shahzad [109–111], Xiao and Zhu [124] (see also 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 locally convex spaces, and thus we omit its statement in detail.
6 Principle of nonlinear alternatives for classes of 1set contractive mappings in locally pconvex spaces
As applications of results in Sect. 5, we now establish general results for the existence of solutions for Birkhoff–Kellogg problem and the principle of Leray–Schauder alternatives in locally pconvex spaces for \(0 < p \leq 1\).
Theorem 6.1
(Birkhoff–Kellogg alternative in locally pconvex spaces)
Let U be a bounded open pconvex subset of a locally pconvex space E (where \(0 \leq p \leq 1\)) with the zero \(0 \in U\), and let C be a closed pconvex subset of E with also zero \(0\in C\), and assume \(F: \overline{U}\cap C \rightarrow 2^{C}\) is a 1set contractive and upper semicontinuous mapping with nonempty closed pconvex values satisfying condition (H) or (H1) above. 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 symbols used in the proof of Theorem 5.2, we have that either
(1) F has a fixed point \(x_{0} \in \overline{U} \cap C\); 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 above (2) holds and \(x_{0} \notin F(x_{0})\). As given by the proof of Theorem 5.2, 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 6.2
(Birkhoff–Kellogg alternative in TVS)
Let U be a bounded open convex subset of a locally convex space E with the zero \(0 \in U\), and let C be a closed convex subset of E with also zero \(0\in C\), and assume \(F: \overline{U}\cap C \rightarrow 2^{C}\) is a 1set contractive and upper semicontinuous mapping with nonempty closed convex values satisfying condition (H) or (H1) above. Then we have either (I) or (II) of the following holds, where \(W_{\overline{U}}(x_{0})\) is either an inward set \(I_{\overline{U}}(x_{0})\) or an 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 \(p=1\), then it automatically satisfies the inequality \(P^{\frac{1}{p}}_{U}(y) 1 \leq P^{\frac{1}{p}}_{U} (yx)\), and indeed, 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 5.3 and 5.4. The proof is complete. □
Indeed, we have the following fixed points for nonself mappings in pvector spaces for \(0 < p \leq 1\) under different boundary conditions.
Theorem 6.3
(Fixed points of nonself mappings)
Let U be a bounded open pconvex subset of a locally pconvex space E (where \(0 \leq p \leq 1\)) with the zero \(0 \in U\), and let C be a closed pconvex subset of E with also zero \(0\in C\), and assume \(F: \overline{U}\cap C \rightarrow 2^{C}\) is a 1set contractive and upper semicontinuous mapping with nonempty closed pconvex values satisfying condition (H) or (H1) above. 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 5.2 (see also Theorem 5.4), we have that either
(1) F has a fixed point \(x_{0} \in \overline{U} \cap C\); 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 above (2) 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 must have a fixed point.
Second, suppose that F satisfies condition (ii), if F has no fixed point, then above (2) 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 by (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 above (2) holds and \(x_{0} \notin F(x_{0})\). As given by the proof of Theorem 5.2, 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 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, like 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 above (2) 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 the 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 6.1 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 6.3, we have the following fixed points for nonself upper semicontinuous setvalued mappings associated with inward or outward sets in locally convex spaces.
Theorem 6.4
(Fixed points of nonself mappings with boundary conditions)
Let U be a bounded open convex subset of a locally convex space E with the zero \(0 \in U\), and assume that \(F: \overline{U} \rightarrow 2^{E}\) is a 1set contractive and upper semicontinuous mapping with nonempty closed convex values satisfying condition (H) or (H1) above. 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 [17], Petryshyn [94, 95], Reich [100]), which would play roles in nonlinear analysis for pseminorm space as shown below.
First, as discussed by Remark 5.2, the proof of Theorem 5.2 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 locally pconvex spaces.
Theorem 6.5
(Rothe type)
Let U be a bounded open pconvex subset of a locally pconvex space E (where \(0 \leq p \leq 1\)) with the zero \(0 \in U\). Assume that \(F: \overline{U}\rightarrow 2^{E}\) is a 1set contractive and upper semicontinuous mapping with nonempty closed pconvex values satisfying condition (H) or (H1) above, and such that \(F(\partial (U)) \subset \overline{U}\), then F must have a fixed point.
Now, as applications of Theorem 6.5, 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 [53] and the references therein for the study of complementary problems and related topics in optimization).
Theorem 6.6
(Leray–Schauder alternative in locally pconvex spaces)
Let E be a locally pconvex space E, where \(0 < p \leq 1\), let \(B \subset E\) be bounded closed pconvex such that \(0 \in \operatorname{int} B\). Let \(F: [0, 1] \times B \rightarrow E\) be 1set contractive and upper semicontinuous setvalued with nonempty closed pconvex values satisfying condition (H) or (H1) above, 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 6.5 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 the 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 we 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 have 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 6.6, we have the following principle for the implicit form of Leray–Schauder type alternative for setvalued mappings in locally pspaces for \(0< p \leq 1\).
Corollary 6.1
(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 \(F: [0, 1] \times B \rightarrow E\) be 1set contractive and continuous with nonempty closed pconvex values satisfying condition (H) or (H1) above, and 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 6.6, 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 [39], Granas and Dugundji [48], Górniewicz [46], Górniewicz et al. [47], Isac [53], Li et al. [69], Liu [72], Park [86], Potter [98], Shahzad [109–111], Xu [129], Xu et al. [130] (see also 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, a general class of mappings for nonlinear alternative of Leray–Schauder type in normal topological spaces, and some Birkhoff–Kellogg type theorems for general class mappings in topological vector spaces have also been established by Agarwal et al. [2], Agarwal and O’Regan [3, 4], Park [88] (see the references therein for more details); and in particular, recently O’Regan [81] used the Leray–Schauder type coincidence theory to establish some Birkhoff–Kellogg problem, 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 seem to play important roles for the nonlinear analysis under the framework of pseminorm spaces, as the achievement of nonlinear analysis for locally convex spaces, normed spaces, or in Banach spaces.
7 Fixed points for classes of 1set contractive mappings
In this section, based on the best approximation Theorem 5.2 for classes of 1set contractive mappings developed in Sect. 5, we will show how it can be used as a useful tool to establish fixed point theorems for nonself upper semicontinuous mappings in pseminorm spaces (for \(p \in (0, 1]\), and including norm spaces, uniformly convex Banach spaces as special classes).
By following Browder [17], Li [68], Goebel and Kirk [43], Petryshyn [94, 95], Tan and Yuan [118], Xu [129] (see also the references therein), we recall some definitions as follows for pseminorm spaces, where \(p \in (0, 1]\).
Definition 7.1
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 Browder [17]) 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 [118]) 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 subsets containing the zero in E; (4) f is said to be demicompact (by Petryshyn [94]) 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 the identity mapping (by Li [68]); and (6) f is said to be semicontractive (see Petryshyn [95] and Browder [17]) 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 in detail here 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 in p. 380 by Tan and Yuan [118]. 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 [118], 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) used by Theorem 5.1 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 \(F: \overline{D} \rightarrow 2^{E}\) 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 the same one as condition (C) used by Theorem 1 of Petryshyn [95]. Secondly, it was shown by Browder [17] that indeed the nonexpansive mapping in a uniformly convex Banach X enjoys condition (H1) as shown below.
Lemma 7.1
Let D be a nonempty bounded 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 the proof of Theorem 2.2 and Corollary 2.1) by Petryshyn [95], 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 7.1
When a pvector space E is with a pnorm, the (H) condition satisfies the (H1) condition. The (H1) condition is mainly supported by the socalled demiclosedness principle after the work by Browder [17].
By applying Theorem 5.2, we have the following result for nonself mappings in pseminorm spaces for \(p \in (0, 1]\).
Theorem 7.1
Let U be a bounded open pconvex subset of a p(semi)norm space E (\(0 < p \leq 1\)) the zero \(0 \in U\). Assume that \(F: \overline{U} \rightarrow 2^{E}\) is a 1set contractive and upper semicontinuous mapping with nonempty closed pconvex values, satisfying condition (H) or (H1) above. 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 Theorem 5.2 with \(C= E\), it follows that we have the following either (I) or (II) holding:
(I) F has a fixed point \(x_{0} \in \overline{U} \), 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 above (II) holds and \(x_{0} \notin F(x_{0})\). By the proof of Theorem 5.2, 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 [17], Li [68], Li et al. [69], Goebel and Kirk [43], Petryshyn [94, 95], Tan and Yuan [118], Xu [129], Xu et al. [130] (see also the references therein), we have a number of existence theorems for the principle of Leray–Schauder type alternatives in pseminorm spaces \((E, \\cdot \_{p})\) for \(p \in (0, 1]\) as follows.
Theorem 7.2
Let U be a bounded open pconvex subset of a p(semi)norm space \((E, {\\cdot \_{p}})\) (\(0 < p \leq 1\)) the zero \(0 \in U\). Assume that \(F: \overline{U} \rightarrow 2^{E}\) is a 1set contractive and upper semicontinuous mapping with nonempty closed pconvex values satisfying condition (H) or (H1) above. 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
We prove the conclusion by showing that the Leray–Schauder boundary condition in Theorem 7.1 does not hold. If we assume that F has no fixed point, by the boundary condition of Theorem 7.1, 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 7.2. Thus, the conclusion follows and the proof is complete. □
Theorem 7.3
Let U be a bounded open pconvex subset of a p(semi)norm space \((E, {\\cdot \_{p}})\) (\(0 < p \leq 1\)) the zero \(0 \in U\). Assume that \(F: \overline{U} \rightarrow 2^{E}\) is a 1set contractive and upper semicontinuous mapping with nonempty closed pconvex values satisfying condition (H) or (H1) above. 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 7.1 does not hold. If we assume F has no fixed point, by the boundary condition of Theorem 7.1, 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 7.3, we have that
which contradicts the boundary condition given by Theorem 7.3. Thus, the conclusion follows and the proof is complete. □
Theorem 7.4
Let U be a bounded open pconvex subset of a p(semi)norm space \((E, {\\cdot \_{p}})\) (\(0 < p \leq 1\)) the zero \(0 \in U\). Assume that \(F: \overline{U} \rightarrow 2^{E}\) is a 1set contractive and upper semicontinuous mapping with nonempty closed pconvex values satisfying condition (H) or (H1) above. 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 7.1 does not hold. If we assume F has no fixed point, by the boundary condition of Theorem 7.1, 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 7.4, we have that
which contradicts the boundary condition given by Theorem 7.4. Thus, the conclusion follows and the proof is complete. □
Theorem 7.5
Let U be a bounded open pconvex subset of a p(semi)norm space \((E, {\\cdot \_{p}})\) (\(0 < p \leq 1\)) the zero \(0 \in U\). Assume that \(F: \overline{U} \rightarrow 2^{E}\) is a 1set contractive and upper semicontinuous mapping with nonempty closed pconvex values satisfying condition (H) or (H1) above. 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 7.1 does not hold. If we assume F has no fixed point, by the boundary condition of Theorem 7.1, 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 7.5, 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 7.5. Thus, the conclusion follows and the proof is complete. □
As an application of Theorems 7.1, 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 7.1
Let U be a bounded open pconvex subset of a p(semi)norm space \((E, {\\cdot \_{p}})\) (\(0 < p \leq 1\)) the zero \(0 \in U\). Assume that \(F: \overline{U} \rightarrow 2^{E}\) is a 1set contractive and upper semicontinuous mapping with nonempty closed pconvex values satisfying condition (H) or (H1) above. Then F has at least one fixed point if one of the following 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 pseminorm 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 actually is true for nonexpansive setvalued mappings).
Theorem 7.6
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 with nonempty values. 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 the assumption that F is a semicontractive and continuous singlevalued mapping with nonempty values, it follows by Lemma 3.2 in p. 338 of Petryshyn [95] that f is a 1set contractive singlevalued mapping. Moreover, by the assumption that E is uniformly convex Banach, indeed \((IF)\) is closed at zero, i.e., F is semiclosed (see Browder [17] or Goebel and Kirk [43]). Thus all assumptions of Theorem 7.1 are satisfied with the (H1) condition. The conclusion follows by Theorem 7.1, and the proof is complete. □
Like Lemma 7.1 shows, a singlevalued nonexpansive mapping defined in a uniformly convex Banach space (see also Theorem 7.6) satisfies the (H1) condition. Actually, the nonexpansive setvalued mappings defined on a special class of Banach spaces with the socalled Opial 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 condition was first given by Opial [80], and it 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 [66], Goebel and Kirk [44], Xu [127], and the references therein. The following result shows that nonexpansive setvalued mappings in Banach spaces with Opial’s condition (see Lami Dozo [66]) satisfy condition (H1).
Lemma 7.2
Let C be a convex weakly compact subset of a Banach space X which 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, \(\sigma (X, X^{*})\) the weak topology, and \(\\cdot \\) the norm (or strong) topology.
Proof
By following Theorem 3.1 of Lami Dozo [66], it follows that the mapping T is demiclosed, thus T satisfies the (H1) condition. The proof is complete. □
As an application of Lemma 7.2, we have the following results for nonexpansive mappings.
Theorem 7.7
Let C be a nonempty convex weakly compact subset of a Banach space X which 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 7.1, it is then semicontractive and continuous. Then the (H1) condition of Theorem 7.1 is satisfied. The conclusion follows by Theorem 7.1, and the proof is complete. □
Before the end of this section, by considering the pseminorm space \((E, \\cdot \)\) is a seminorm space with \(p=1\), the following result is a special case of corresponding results from Theorem 7.2 to Theorem 7.5, and thus we omit its proof.
Corollary 7.2
Let U be a bounded open convex subset of a norm space \((E, \\cdot \)\). Assume that \(F: \overline{U} \rightarrow 2^{E}\) is a 1set contractive and upper semicontinuous mapping with nonempty closed pconvex values satisfying condition (H) or (H1) above. 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 7.2
As discussed in Lemma 7.1 and the proof of Theorem 7.6, when the pvector space is a uniformly convex Banach space, semicontractive or nonexpansive mappings automatically satisfy condition (H) or (H1). Moreover, our results from Theorem 7.1 to Theorem 7.6, Corollary 7.1, and Corollary 7.2 also improve or unify corresponding results given by Browder [17], Li [68], Li et al. [69], Goebel and Kirk [43], Petryshyn [94, 95], Reich [100], Tan and Yuan [118], Xu [126], Xu [129], Xu et al. [130], and the results from the references therein by extending the nonself mappings to the classes of 1set contractive setvalued mappings in pseminorm spaces with \(p \in (0.1]\) (including the normed space or Banach space when \(p=1\) and for pseminorm spaces).
Before the end of this paper, we would like to share with readers that the main goal of this paper is to develop some new results and tools in the nature way for the category of nonlinear analysis for 1set contractive mappings under the general framework of locally pconvex spaces (where \((0< p \leq 1)\)), and we expect that they become useful tools for the study on optimization, nonlinear programming, variational inequality, complementarity, game theory, mathematical economics, and other related social science areas. In particular, we first establish one best approximation, acting as a tool to establish the principle of nonlinear alternative, which then allows us to give general principle of nonlinear alternative for 1set contractive mappings.
As mentioned at the beginning of this paper, we do expect that nonlinear results and principles of the best approximation theorem established in this paper would play a very important role for the nonlinear analysis under the general framework of locally pconvex spaces for \((0< p \leq 1)\), as shown by those results given in Sects. 4, 5, 6 and 7 above for the fixed point theorems of nonself mappings, the principle of nonlinear alternative, Rothe type, Leray–Schauder alternative which not only include corresponding results in the existing literature as special cases, but also would be important tools for the study of optimization, nonlinear programming, variational inequality, complementarity, game theory, mathematical economics, and related topics and areas forthcoming; and in Sect. 7, by considering pseminorm spaces for \(p \in (0,1]\), as an application of best approximation, we unified and improved the corresponding results in the existing literature under the general framework of locally pconvex spaces.
But one thing we like to point out that the results mainly are estabished for setvalued mappings with nonempty closed pconvex values for \(0 < p \leq 1\), not much attention given to singvalued mappings. As suggested by the title of this paper, in Sects. 4, 5, 6 and 7 of this paper, we focus on the development of results in nonlinear analysis mainly related to fixed points, the best approximation, and the general principle of nonlinear alternative and related boundary conditions under the framework of locally pconvex spaces for \(0 < p \leq 1\), for nonlinear setvalued mappings, which are upper semicontinuous, 1set contractive with nonempty closed pconvex values. On the other hand, as shown by Lemma 2.4 and the discussion based on the conclusion of Theorem 4.3 for setvalued mappings, it seems that the setvalued nonlinear mappings with closed pconvex values are very strong assumptions for \(0 < p < 1\), which maybe the major reason to result in trivial conclusions for the existence of fixed points, alternative principle and related approximation results as given in this paper from Sects. 4, 5, 6 and 7, thus we do expect that the most interesting results in nonlinear analysis for locally pconvex space would be given for singlevalued (nonlinear) mappings instead of setvalued mappings with closed pconvex values.
On the other hand, based on the framework established in this paper (though focus on setvalued mappings), the nonlinear analysis for singlevalued mappings actually can be developed by using Theorem 4.4 (instead of Theorem 4.2) as a starting tool, then we can obtain the similar result of Theorem 4.5 for singlevalued mappings which are continuous condensing in Sect. 4; and then it can help us to establish the corresponding results similar to Theorems 5.1, 5.2, 5.3, 5.4, 5.5 and 5.6 for singlevalued mappings under the locally pconvex spaces for \(0 < p \leq 1\). In addition, the similar results of Theorems 6.1, 6.2, 6.3, 6.4, 6.5 and 6.6; and those results related to Theorems 7.1, 7.2, 7.3, 7.4, 7.5, 7.6 and 7.7 for singlevalued mappings can be obtained, too under the framework of locally pconvex spaces for \(0 < p \leq 1\). Though not much results in details for singlevalued mappings are given here, but we do wish to share with readers that how important they are for the development of nonlinear analysis based on singevalued mappings in pvector spaces for \(0 < p \leq 1\) in general.
Based on the framework for some key results in nonlinear analysis obtained for setvalued mappings with closed pconvex values in this paper, we conclude that the development of nonlinear analysis for singevalued mappings in locally pconvex spaces for \(0 < p \leq 1\) seem even more important, and they can be also developed by the approach ane method established in this paper.
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References
Agarwal, R.P., Balaj, M., O’Regan, D.: Intersection theorems for weak KKM setvalued mappings in the finite dimensional setting. Topol. Appl. 262, 64–79 (2019)
Agarwal, R.P., Meehan, M., O’Regan, D.: Fixed Point Theory and Applications. Cambridge Tracts in Mathematics, vol. 141. Cambridge University Press, Cambridge (2001)
Agarwal, R.P., O’Regan, D.: BirkhoffKellogg theorems on invariant directions for multimaps. Abstr. Appl. Anal. 7, 435–448 (2003)
Agarwal, R.P., O’Regan, D.: Essential \(U_{c}^{k}\)type maps and BirkhoffKellogg theorems. J. Appl. Math. Stoch. Anal. 1, 1–8 (2004)
Alghamdi, M.A., O’Regan, D., Shahzad, N.: Krasnosel’skii type fixed point theorems for mappings on nonconvex sets. Abstr. Appl. Anal. 2020, 267531 (2012)
Askoura, Y., GodetThobie, C.: Fixed points in contractible spaces and convex subsets of topological vector spaces. J. Convex Anal. 13(2), 193–205 (2006)
Balachandran, V.K.: Topological Algebras, vol. 185. Elsevier, Amsterdam (2000)
Balaj, M.: Intersection theorems for generalized weak KKM setvalued mappings with applications in optimization. Math. Nachr. 294, 1262–1276 (2021). https://doi.org/10.1002/mana.201900243
Bayoumi, A.: Foundations of Complex Analysis in Nonlocally Convex Spaces. Function Theory Without Convexity Condition. NorthHolland Mathematics Studies, vol. 193. Elsevier Science B.V., Amsterdam (2003)
Bayoumi, A., Faried, N., Mostafa, R.: Regularity properties of pdistance transformations in image analysis. Int. J. Contemp. Math. Sci. 10, 143–157 (2015)
Bernstein, S.: Sur les equations de calcul des variations. Ann. Sci. Éc. Norm. Supér. 29, 431–485 (1912)
Bernuées, J., Pena, A.: On the shape of pconvex hulls \(0 < p < 1\). Acta Math. Hung. 74(4), 345–353 (1997)
Birkhoff, G.D., Kellogg, O.D.: Invariant points in function space. Trans. Am. Math. Soc. 23(1), 96–115 (1922)
Browder, F.E.: Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. USA 54, 1041–1044 (1965)
Browder, F.E.: Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces. Arch. Ration. Mech. Anal. 24(1), 82–90 (1967)
Browder, F.E.: The fixed point theory of multivalued mappings in topological vector spaces. Math. Ann. 177, 283–301 (1968)
Browder, F.E.: Semicontractive and semiaccretive nonlinear mappings in Banach spaces. Bull. Am. Math. Soc. 74, 660–665 (1968)
Browder, F.E.: Nonlinear Functional Analysis. Proc. Sympos. Pure Math., vol. 18. Am. Math. Soc., Providence (1976)
Browder, F.E.: Fixed point theory and nonlinear problems. Bull. Am. Math. Soc. (N.S.) 9(1), 1–39 (1983)
Carbone, A., Conti, G.: Multivalued maps and existence of best approximations. J. Approx. Theory 64, 203–208 (1991)
Cauty, R.: Rétractès absolus de voisinage algébriques. (French) [algebraic absolute neighborhood retracts]. Serdica Math. J. 31(4), 309–354 (2005)
Cauty, R.: Le théorėme de Lefschetz  Hopf pour les applications compactes des espaces ULC. (French) [The LefschetzHopf theorem for compact maps of uniformly locally contractible spaces]. J. Fixed Point Theory Appl. 1(1), 123–134 (2007)
Chang, S.S.: Some problems and results in the study of nonlinear analysis. Proceedings of the Second World Congress of Nonlinear Analysts, Part 7 (Athens, 1996), Nonlinear Anal. 30(7), 4197–4208 (1997)
Chang, S.S., Cho, Y.J., Zhang, Y.: The topological versions of KKM theorem and Fan’s matching theorem with applications. Topol. Methods Nonlinear Anal. 1(2), 231–245 (1993)
Chang, T.H., Huang, Y.Y., Jeng, J.C.: Fixed point theorems for multifunctions in SKKM class. Nonlinear Anal. 44, 1007–1017 (2001)
Chang, T.H., Huang, Y.Y., Jeng, J.C., Kuo, K.H.: On SKKM property and related topics. J. Math. Anal. Appl. 229, 212–227 (1999)
Chang, T.H., Yen, C.L.: KKM property and fixed point theorems. J. Math. Anal. Appl. 203, 224–235 (1996)
Chen, Y.K., Singh, K.L.: Fixed points for nonexpansive multivalued mapping and the Opial’s condition. Jñānābha 22, 107–110 (1992)
Chen, Y.Q.: Fixed points for convex continuous mappings in topological vector spaces. Proc. Am. Math. Soc. 129(7), 2157–2162 (2001)
Darbo, G.: Punti uniti in trasformazioni a condominio non compatto. Rend. Semin. Mat. Univ. Padova 24, 84–92 (1955)
Ding, G.G.: New Theory in Functional Analysis. Academic Press, Beijing (2007)
Dobrowolski, T.: Revisiting Cauty’s proof of the Schauder conjecture. Abstr. Appl. Anal. 7, 407–433 (2003)
Ennassik, M., Maniar, L., Taoudi, M.A.: Fixed point theorems in rnormed and locally rconvex spaces and applications. Fixed Point Theory 22(2), 625–644 (2021)
Ennassik, M., Taoudi, M.A.: On the conjecture of Schauder. J. Fixed Point Theory Appl. 23(4), 52 (2021)
Fan, K.: Fixedpoint and minimax theorems in locally convex topological linear spaces. Proc. Natl. Acad. Sci. USA 38, 121–126 (1952)
Fan, K.: A generalization of Tychonoff’s fixed point theorem. Math. Ann. 142, 305–310 (1960/61)
Fan, K.: Extensions of two fixed point theorems of F.E. Browder. Math. Z. 112, 234–240 (1969)
Fan, K.: A minimax inequality and applications. In: Inequalities, III (Proc. Third Sympos. Univ. California, Los Angeles, Calif., 1969; Dedicated to the Memory of Theodore S. Motzkin), pp. 103–113. Academic Press, New York (1972)
Furi, M., Pera, M.P.: A continuation method on locally convex spaces and applications to ordinary differential equations on noncompact intervals. Ann. Pol. Math. 47(3), 331–346 (1987)
Gal, S.G., Goldstein, J.A.: Semigroups of linear operators on pFréchet spaces \(0 < p <1\). Acta Math. Hung. 114(1–2), 13–36 (2007)
Gholizadeh, L., Karapinar, E., Roohi, M.: Some fixed point theorems in locally pconvex spaces. Fixed Point Theory Appl. 2013, 312 (2013). 10 pp.
Goebel, K.: On a fixed point theorem for multivalued nonexpansive mappings. Ann. Univ. Mariae CurieSkłodowska, Sect. A 29, 69–72 (1975)
Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics, vol. 28. Cambridge University Press, Cambridge (1990)
Goebel, K., Kirk, W.A.: Some problems in metric fixed point theory. J. Fixed Point Theory Appl. 4(1), 13–25 (2008)
Göhde, D.: Zum Prinzip der kontraktiven Abbildung. Math. Nachr. 30, 251–258 (1965)
Górniewicz, L.: Topological Fixed Point Theory of Multivalued Mappings. Mathematics and Its Applications, vol. 495. Kluwer Academic, Dordrecht (1999)
Górniewicz, L., Granas, A., Kryszewski, W.: On the homotopy method in the fixed point index theory of multivalued mappings of compact absolute neighborhood retracts. J. Math. Anal. Appl. 161(2), 457–473 (1991)
Granas, A., Dugundji, J.: Fixed Point Theory. Springer Monographs in Mathematics. Springer, New York (2003)
Halpern, B.R., Bergman, G.H.: A fixedpoint theorem for inward and outward maps. Trans. Am. Math. Soc. 130, 353–358 (1965)
Huang, N.J., Lee, B.S., Kang, M.K.: Fixed point theorems for compatible mappings with applications to the solutions of functional equations arising in dynamic programmings. Int. J. Math. Math. Sci. 20(4), 673–680 (1997)
Husain, T., Latif, A.: Fixed points of multivalued nonexpansive maps. Math. Jpn. 33, 385–391 (1988)
Husain, T., Tarafdar, E.: Fixed point theorems for multivalued mappings of nonexpansive type. Yokohama Math. J. 28(1–2), 1–6 (1980)
Isac, G.: LeraySchauder Type Alternatives, Complementarity Problems and Variational Inequalities. Nonconvex Optimization and Its Applications, vol. 87. Springer, New York (2006)
Jarchow, H.: Locally Convex Spaces. B.G. Teubner, Stuttgart (1981)
Kalton, N.J.: Compact pconvex sets. Q. J. Math. Oxf. Ser. 28(2), 301–308 (1977)
Kalton, N.J.: Universal spaces and universal bases in metric linear spaces. Stud. Math. 61, 161–191 (1977)
Kalton, N.J., Peck, N.T., Roberts, J.W.: An FSpace Sampler. London Mathematical Society Lecture Note Series, vol. 89. Cambridge University Press, Cambridge (1984)
Kaniok, L.: On measures of noncompactness in general topological vector spaces. Comment. Math. Univ. Carol. 31(3), 479–487 (1990)
Kim, I.S., Kim, K., Park, S.: LeraySchauder alternatives for approximable maps in topological vector spaces. Math. Comput. Model. 35, 385–391 (2002)
Kirk, W., Shahzad, N.: Fixed Point Theory in Distance Spaces. Springer, Cham (2014)
Klee, V.: Convexity of Chevyshev sets. Math. Ann. 142, 292–304 (1960/61)
Knaster, H., Kuratowski, C., Mazurkiwiecz, S.: Ein beweis des fixpunktsatzes für ndimensional simplexe. Fundam. Math. 63, 132–137 (1929)
Ko, H.M., Tsai, Y.H.: Fixed point theorems for pointtoset mappings in locally convex spaces and a characterization of complete metric spaces. Bull. Acad. Sin. 7(4), 461–470 (1979)
Kozlov, V., Thim, J., Turesson, B.: A fixed point theorem in locally convex spaces. Collect. Math. 61(2), 223–239 (2010)
Kuratowski, K.: Sur les espaces complets. Fundam. Math. 15, 301–309 (1930)
Lami Dozo, E.: Multivalued nonexpansive mappings and Opial’s condition. Proc. Am. Math. Soc. 38, 286–292 (1973)
Leray, J., Schauder, J.: Topologie et equations fonctionnelles. Ann. Sci. Éc. Norm. Supér. 51, 45–78 (1934)
Li, G.Z.: The fixed point index and the fixed point theorems of 1setcontraction mappings. Proc. Am. Math. Soc. 104, 1163–1170 (1988)
Li, G.Z., Xu, S.Y., Duan, H.G.: Fixed point theorems of 1setcontractive operators in Banach spaces. Appl. Math. Lett. 19(5), 403–412 (2006)
Li, J.L.: An extension of Tychonoff’s fixed point theorem to pseudonorm adjoint topological vector spaces. Optimization 70(5–6), 1217–1229 (2021)
Lim, T.C.: A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach space. Bull. Am. Math. Soc. 80, 1123–1126 (1974)
Liu, L.S.: Approximation theorems and fixed point theorems for various classes of 1setcontractive mappings in Banach spaces. Acta Math. Sin. Engl. Ser. 17(1), 103–112 (2001)
Machrafi, N., Oubbi, L.: Realvalued noncompactness measures in topological vector spaces and applications. [Corrected title: realvalued noncompactness measures in topological vector spaces and applications]. Banach J. Math. Anal. 14(4), 1305–1325 (2020)
Mańka, R.: The topological fixed point property  an elementary continuumtheoretic approach. Fixed point theory and its applications. In: Banach Center Publ., vol. 77, pp. 183–200. Polish Acad. Sci. Inst. Math, Warsaw (2007)
Mauldin, R.D.: The Scottish Book, Mathematics from the Scottish Café with Selected Problems from the New Scottish Book, 2nd edn. Birkhäuser, Basel (2015)
Muglia, L., Marino, G.: Some results on the approximation of solutions of variational inequalities for multivalued maps on Banach spaces. Mediterr. J. Math. 18(4), 157 (2021)
Nhu, N.T.: The fixed point property for weakly admissible compact convex sets: searching for a solution to Schauder’s conjecture. Topol. Appl. 68(1), 1–12 (1996)
Nussbaum, R.D.: The fixed point index and asymptotic fixed point theorems for ksetcontractions. Bull. Am. Math. Soc. 75, 490–495 (1969)
Okon, T.: The Kakutani fixed point theorem for Robert spaces. Topol. Appl. 123(3), 461–470 (2002)
Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 595–597 (1967)
O’Regan, D.: Abstract LeraySchauder type alternatives and extensions. An. tiin. Univ. “Ovidius” Constana Ser. Mat. 27(1), 233–243 (2019)
O’Regan, D.: Continuation theorems for Monch countable compactnesstype setvalued maps. Appl. Anal. 100(7), 1432–1439 (2021)
O’Regan, D., Precup, R.: Theorems of LeraySchauder Type and Applications. Gordon & Breach, New York (2001)
Oubbi, L.: Algebras of Gelfandcontinuous functions into ArensMichael algebras. Commun. Korean Math. Soc. 34(2), 585–602 (2019)
Park, S.: Some coincidence theorems on acyclic multifunctions and applications to KKM theory. In: In, Fixed Point Theory and Applications, Halifax, NS, 1991, pp. 248–277. World Sci. Publ., River Edge, NJ (1992)
Park, S.: Generalized LeraySchauder principles for compact admissible multifunctions. Topol. Methods Nonlinear Anal. 5(2), 271–277 (1995)
Park, S.: Acyclic maps, minimax inequalities and fixed points. Nonlinear Anal. 24(11), 1549–1554 (1995)
Park, S.: Generalized LeraySchauder principles for condensing admissible multifunctions. Ann. Mat. Pura Appl. 172(4), 65–85 (1997)
Park, S.: The KKM principle in abstract convex spaces: equivalent formulations and applications. Nonlinear Anal. 73(4), 1028–1042 (2010)
Park, S.: On the KKM Theory of Locally pConvex Spaces (Nonlinear Analysis and Convex Analysis), vol. 2011, pp. 70–77. Institute of Mathematical Research(Kyoto University), Japan (2016). http://hdl.handle.net/2433/231597
Park, S.: One hundred years of the Brouwer fixed point theorem. J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 60(1), 1–77 (2021)
Park, S.: Some new equivalents of the Brouwer fixed point theorem. Adv. Theory Nonlinear Anal. Appl. 6(3), 300–309 (2022). https://doi.org/10.31197/atnaa.1086232
Petrusel, A., Rus, I.A., Serban, M.A.: Basic problems of the metric fixed point theory and the relevance of a metric fixed point theorem for a multivalued operator. J. Nonlinear Convex Anal. 15(3), 493–513 (2014)
Petryshyn, W.V.: Construction of fixed points of demicompact mappings in Hilbert space. J. Math. Anal. Appl. 14, 276–284 (1966)
Petryshyn, W.V.: Fixed point theorems for various classes of 1setcontractive and 1ballcontractive mappings in Banach spaces. Trans. Am. Math. Soc. 182, 323–352 (1973)
Pietramala, P.: Convergence of approximating fixed points sets for multivalued nonexpansive mappings. Comment. Math. Univ. Carol. 32(4), 697–701 (1991)
Poincare, H.: Sur un theoreme de geometric. Rend. Circ. Mat. Palermo 33, 357–407 (1912)
Potter, A.J.B.: An elementary version of the LeraySchauder theorem. J. Lond. Math. Soc. 5(2), 414–416 (1972)
Qiu, J., Rolewicz, S.: Ekeland’s variational principle in locally pconvex spaces and related results. Stud. Math. 186(3), 219–235 (2008)
Reich, S.: Fixed points in locally convex spaces. Math. Z. 125, 17–31 (1972)
Roberts, J.W.: A compact convex set with no extreme points. Stud. Math. 60(3), 255–266 (1977)
Robertson, L.B.: Topological vector spaces. Publ. Inst. Math. 12(26), 19–21 (1971)
Rolewicz, S.: Metric Linear Spaces. PWNPolish Scientific Publishers, Warszawa (1985)
Rothe, E.H.: Some homotopy theorems concerning LeraySchauder maps. In: Dynamical Systems, II, Gainesville, Fla., 1981, pp. 327–348. Academic Press, New York (1982)
Rothe, E.H.: Introduction to Various Aspects of Degree Theory in Banach Spaces. Mathematical Surveys and Monographs, vol. 23. Am. Math. Soc., Providence (1986)
Sadovskii, B.N.: On a fixed point principle [in Russian]. Funkc. Anal. Prilozh. 1(2), 74–76 (1967)
Schauder, J.: Der Fixpunktsatz in Funktionalraumen. Stud. Math., pp. 171–180 (1930)
Sezer, S., Eken, Z., Tinaztepe, G., Adilov, G.: pConvex functions and some of their properties. Numer. Funct. Anal. Optim. 42(4), 443–459 (2021)
Shahzad, N.: Fixed point and approximation results for multimaps in \(SKKM\) class. Nonlinear Anal. 56(6), 905–918 (2004)
Shahzad, N.: Approximation and LeraySchauder type results for \(U_{c}^{k}\) maps. Topol. Methods Nonlinear Anal. 24(2), 337–346 (2004)
Shahzad, N.: Approximation and LeraySchauder type results for multimaps in the SKKM class. Bull. Belg. Math. Soc. 13(1), 113–121 (2006)
Silva, E.B., Fernandez, D.L., Nikolova, L.: Generalized quasiBanach sequence spaces and measures of noncompactness. An. Acad. Bras. Ciênc. 85(2), 443–456 (2013)
Simons, S.: Boundedness in linear topological spaces. Trans. Am. Math. Soc. 113, 169–180 (1964)
Singh, S.P., Watson, B., Srivastava, F.: Fixed Point Theory and Best Approximation: The KKMMap Principle. Mathematics and Its Applications, vol. 424. Kluwer Academic, Dordrecht (1997)
Smart, D.R.: Fixed Point Theorems. Cambridge University Press, Cambridge (1980)
Tabor, J.A., Tabor, J.O., Idak, M.: Stability of isometries in pBanach spaces. Funct. Approx. 38, 109–119 (2008)
Tan, D.N.: On extension of isometries on the unit spheres of \(L^{p}\)  spaces for \(0 < p \leq 1\). Nonlinear Anal. 74, 6981–6987 (2011)
Tan, K.K., Yuan, X.Z.: Random fixedpoint theorems and approximation in cones. J. Math. Anal. Appl. 185, 378–390 (1994)
Tychonoff, A.: Ein Fixpunktsatz. Math. Ann. 111, 767–776 (1935)
Wang, J.Y.: An Introduction to Locally pConvex Spaces pp. 26–64. Academic Press, Beijing (2013)
Weber, H.: Compact convex sets in nonlocally convex linear spaces. Dedicated to the memory of Professor Gottfried Köthe. Note Mat. 12, 271–289 (1992)
Weber, H.: Compact convex sets in nonlocally convex linear spaces, SchauderTychonoff fixed point theorem. In: Topology, Measures, and Fractals, Warnemunde, 1991. Math. Res., vol. 66, pp. 37–40. AkademieVerlag, Berlin (1992)
Xiao, J.Z., Lu, Y.: Some fixed point theorems for sconvex subsets in pnormed spaces based on measures of noncompactness. J. Fixed Point Theory Appl. 20(2), 83 (2018)
Xiao, J.Z., Zhu, X.H.: Some fixed point theorems for sconvex subsets in pnormed spaces. Nonlinear Anal. 74(5), 1738–1748 (2011)
Xiao, J.Z., Zhu, X.H.: The Chebyshev selections and fixed points of setvalued mappings in Banach spaces with some uniform convexity. Math. Comput. Model. 54(5–6), 1576–1583 (2011)
Xu, H.K.: Inequalities in Banach spaces with applications. Nonlinear Anal. 16(12), 1127–1138 (1991)
Xu, H.K.: Metric fixed point theory for multivalued mappings. Dissertationes Math. (Rozprawy Mat.), 389 (2000). 39 pp.
Xu, H.K., Muglia, L.: On solving variational inequalities defined on fixed point sets of multivalued mappings in Banach spaces. J. Fixed Point Theory Appl. 22(4), 79 (2020)
Xu, S.Y.: New fixed point theorems for 1setcontractive operators in Banach spaces. Nonlinear Anal. 67(3), 938–944 (2007)
Xu, S.Y., Jia, B.G., Li, G.Z.: Fixed points for weakly inward mappings in Banach spaces. J. Math. Anal. Appl. 319(2), 863–873 (2006)
Yanagi, K.: On some fixed point theorems for multivalued mappings. Pac. J. Math. 87(1), 233–240 (1980)
Yuan, G.X.Z.: The study of minimax inequalities and applications to economies and variational inequalities. Mem. Am. Math. Soc. 132, 625 (1998)
Yuan, G.X.Z.: KKM Theory and Applications in Nonlinear Analysis. Monographs and Textbooks in Pure and Applied Mathematics, vol. 218. Marcel Dekker, Inc., New York (1999)
Zeidler, E.: Nonlinear Functional Analysis and Its Applications, Vol. I, FixedPoint Theorems. Springer, New York (1986)
Acknowledgements
The author thanks Professor Shihsen Chang (Shisheng Zhang), Professor K.K. Tan, Professor Hong Ma, Professor Y.J. Cho, Professor S. Park for their unceasing encouragements in the past for more than two decades; also my thanks go to Professor HongKun Xu, Professor XiaoLong Qin, Professor Ganshan Yang, Professor Xian Wu, Professor Nanjing Huang, Professor Mohamed Ennassik, Professor Tiexin Guo, Professor Lishan Liu, and my colleagues and friends across China, Australia, Canada, UK, USA, and elsewhere.
Funding
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 Professor Shihsen Chang on his 90th Birthday
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Yuan, G.X. Nonlinear analysis by applying best approximation method in pvector spaces. Fixed Point Theory Algorithms Sci Eng 2022, 20 (2022). https://doi.org/10.1186/s1366302200730x
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DOI: https://doi.org/10.1186/s1366302200730x