Some fixed point theorems in locally p-convex spaces

Abstract

In this paper we investigate the existence of a fixed point of multivalued mapson almost p-convex and p-convex subsets of topological vectorspaces. Our results extend and generalize some fixed point theorems on the topicin the literature, such as the results of Himmelberg, Fan and Glicksberg.

MSC: 46T99, 47H10, 54H25, 54E50, 55M20, 37C25.

1 Introduction and preliminaries

In nonlinear analysis, one of the dynamic research areas is investigation ofexistence of a fixed point of maps on convex sets and p-convex sets.Recently, a number of fixed point theorems have appeared on the setting ofp-convex sets. For instance, Alimohammady et al.[1] extended the Markov-Kakutani fixed point theorem for compactp-star shaped subsets in topological vector spaces by usingp-convex sets instead of convex sets, see also [2, 3]. Further, in [4] authors achieved a fixed point theorem due to Park for a compact mappingon a p-star shaped subset of a topological vector space via Fan-KKMprinciple in a generalized convex space. In [5, 6], generalized versions of Brouwer and Kakutani fixed point theorems werecharacterized in the context of locally p-convex space.

On the other hand, in 1993 Park and Kim introduced the concept of generalized convexspace, which extends many generalized convex structures on topological vector spaces [7]. This new concept, developed in connection with fixed point theory andKKM theory, generalizes topological vector spaces.

Maki [8] introduced the notion of minimal spaces which is a generalization of theconcept of topological spaces (see also [9]). After these initial papers, many authors have paid attention to thesubject and have published several results in this direction; see, e.g., [10–13]. Very recently, Darzi et al.[14] introduced the notion of minimal generalized convex space as to extendthe construction of the generalized convex space.

For the sake of completeness, we recall some basic definitions and fundamentalresults in the literature. All we need regarding topological vector spaces can befound in [15–18].

Let U be a subset of a vector space V and $x,y\in U$ and $0<p\le 1$. Bayoumi [5] introduced the notion of arc segment joining x and y, as follows:

$$A_x^y=\{s^{\frac{1}{p}}x+t^{\frac{1}{p}}y:s+t=1\}=\{ux+vy:u^p+v^p=1\}.$$

A set X in a vector space V is said to bep-convex if $A_x^y\subseteq X$ for every $x,y\in X$. The p-convex hull of Xdenoted by $C_p(X)$ is the smallest p-convex set containingX[5]. Further, the closed p-convex hull of X denoted by ${\overline{C}}_p(X)$ is the smallest closed p-convex setcontaining $X\subseteq E$, where E is a topological vector space.Notice that if $p=1$ and $s+t=1$, then $A_x^y$ turns out to be the line segment joining xand y. In this case, $C_p(X)$ and ${\overline{C}}_p(X)$ become the convex hull and the closed convex hull ofX, respectively. For more details, we refer to, e.g., [5, 6, 19–23] and references therein.

Let X be a nonempty set. Then a family $\mathcal{M}\subseteq \mathcal{P}(X)$ is said to be a minimal structure onX if $\mathrm{\varnothing },X\in \mathcal{M}$. Moreover, the pair $(X,\mathcal{M})$ is called a minimal space. The naturalexamples of minimal spaces can be listed as follows [8]: τ, the collection of all semi-open sets$\mathit{SO}(X)$, the collection of all pre-open sets$\mathit{PO}(X)$, the collection of all α-open sets$\alpha O(X)$ and the collection of all β-open sets$\beta O(X)$, where $(X,\tau )$ is a topological space. In a minimal space$(X,\mathcal{M})$, a set $A\in \mathcal{P}(X)$ is said to be an m-open set if$A\in \mathcal{M}$. Similarly, a set $B\in \mathcal{P}(X)$ is an m-closed set if$B^c\in \mathcal{M}$. Furthermore, m-interior andm-closure of a set A are defined as follows:

$$m\text{-}Int(A)=\bigcup \{U:U\subseteq A,U\in \mathcal{M}\}\text{and}m\text{-}Cl(A)=\bigcap \{F:A\subseteq F,F^c\in \mathcal{M}\}.$$

For more details on minimal structure and minimal space, we refer the reader to,e.g., [8, 9, 12–14, 24, 25].

The continuity of maps in a minimal space is defined as follows.

Definition 1.1[25]

Suppose that $(X,\tau )$ is a topological space, and also suppose that$(Y,\mathcal{N})$ is a minimal space. A function$f:(X,\tau )⟶(Y,\mathcal{N})$ is called $(\tau ,m)$-continuous if $f^{-1}(U)\in \tau$ for any $U\in \mathcal{N}$.

Let X and Y be two nonempty sets and $\mathcal{P}(Y)$ be the set of all subsets of Y. Aset-valued map or a set-valued function from X intoY is a function from X to $\mathcal{P}(Y)$ that assigns an element x of X to anonempty subset $T(x)$ of Y and is denoted by$x\mid ⊸T(x)$. The lower inverse of a point$y\in Y$ of a set-valued map T is the set-valued map$T^l$ of Y into X defined by

$$T^l(y)=\{x\in X:y\in T(x)\}.$$

Analogously, lower inverse of a subset of$B\subset Y$ is defined as

$$T^l(B)=\{x\in X:T(x)\cap B\ne \mathrm{\varnothing }\}.$$

We note that $T^l(\mathrm{\varnothing })=\mathrm{\varnothing }$. The set $\{x\in X:T(x)\subseteq B\}$ is the upper inverse of B and isdenoted by $T^u(B)$. A map T is lower semicontinuous if$T^l(U)$ is open in X for every open set$U\subseteq Y$. Similarly, a map T is uppersemicontinuous if for every open set $U\subseteq Y$, the set $T^u(U)$ is open in X.

A set-valued map $T:X⊸Y$ is said to be closed if its graph,$Graph(T)=\{(x,y):y\in T(x)\}$, is a closed subset of $X\times Y$. Also, T is called compact if itsrange, $T(X)$, is contained in a compact subset of Y.

The notion of almost convex was introduced by Himmelberg [26]. A nonempty subset B of a topological vector space X issaid to be almost convex if for any neighborhood V of 0 and forany finite subset $\{b_1,\dots ,b_n\}$ of B, there exists a finite subset$\{x_1,\dots ,x_n\}\subseteq B$ such that $x_i-b_i\in V$ for each $i=1,\dots ,n$ and $co(\{x_1,\dots ,x_n\})\subseteq B$. It is clear that any convex subset is almost convex.Moreover, if we delete a certain subset of the boundary of a closed convex set, thenwe have an almost convex set. Another example of an almost convex set is thefollowing: Let $C([0,1])$ be the Banach space of all continuous real functionsdefined on the unit interval $[0,1]$, and let $P([0,1])$ be a dense subset of all polynomials. Then any subsetof $C([0,1])$ containing $P([0,1])$ is almost convex.

Let A be a subset of a topological vector space X. A set-valued map$T:A⊸A$ is said to have the (convexly) almostfixed point property if for every (convex) neighborhood U of 0 inX, there exists a point $a_U\in A$ for which $a_U\in T(a_U)+U$ or $T(a_U)\cap (a_U+U)\ne \mathrm{\varnothing }$.

Let $〈D〉$ denote the set of all nonempty finite subsets of aset D, and let ${\mathrm{\Delta }}_n$ be the n-simplex with vertices$e_0,e_1,\dots ,e_n$, ${\mathrm{\Delta }}_J$ be the face of ${\mathrm{\Delta }}_n$ corresponding to $J\in 〈A〉$, where $A\in 〈D〉$. For instance, if $A=\{a_0,a_1,\dots ,a_n\}$ and $J=\{a_{i_0},a_{i_1},\dots ,a_{i_k}\}\subseteq A$, then ${\mathrm{\Delta }}_J=co\{e_{i_0},e_{i_1},\dots ,e_{i_k}\}$. A minimal generalized convex space (brieflyMG-convex space) $(X,D,\mathrm{\Gamma })$ consists of a minimal space $(X,\mathcal{M})$, a nonempty set D and a set-valued map$\mathrm{\Gamma }:〈D〉⊸X$ in which for $A\in 〈D〉$ with $n+1$ elements, there exists a $(\tau ,m)$-continuous function ${\varphi }_A:{\mathrm{\Delta }}_n⟶{\mathrm{\Gamma }}_A:=\mathrm{\Gamma }(A)$ for which $J\in 〈A〉$ implies that ${\varphi }_A({\mathrm{\Delta }}_J)\subseteq {\mathrm{\Gamma }}_J$. If $\mathcal{M}=\tau$, then the notion of MG-convex space turnsinto G-convex space (see, e.g., [27]). On the other hand, suppose that $(X,\mathcal{M})$ is a minimal vector space which is not a topologicalvector space. Consider the set-valued map $\mathrm{\Gamma }:〈X〉⊸X$ defined by $\mathrm{\Gamma }(\{a_0,a_1,\dots ,a_n\})=\{{\sum }_{i=0}^n{\lambda }_i{a}_i:0\le {\lambda }_i\le 1,{\sum }_{i=0}^n{\lambda }_i=1\}$. Then $(X,\mathrm{\Gamma })$ is a minimal generalized convex space; of course, weknow that $(X,\mathrm{\Gamma })$ is not a generalized convex space [14].

Definition 1.2 Suppose that $(X,D,\mathrm{\Gamma })$ is an MG-convex space. A set-valued map$F:D⊸X$ is called a KKM set-valued map if${\mathrm{\Gamma }}_A\subseteq F(A)$ for any $A\in 〈D〉$.

We state two useful theorems of Alimohammady et al.[25] as follows.

Theorem 1.3[25]

Suppose that$(X,D,\mathrm{\Gamma })$is an MG-convex space and$F:D⊸X$is a set-valued map satisfying

• (a) for all $x\in D$, $F(x)=m\text{-}Cl(A_x)$ for some $A_x\subseteq X$,

• (b) F is a KKM map.

Then$\{F(z):z\in D\}$has the finite intersection property.

Further, if

• (c) ${\bigcap }_{z\in N}F(z)$ is m-compact for some $N\in 〈D〉$,

then${\bigcap }_{z\in D}F(z)\ne \mathrm{\varnothing }$.

Theorem 1.4[25]

Suppose that$(X,D,\mathrm{\Gamma })$is an MG-convex space and$F:D⊸X$is a set-valued map satisfying

• (a) for all $x\in D$, $F(x)=m\text{-}Int(A_x)$ for some $A_x\subseteq X$,

• (b) F is a KKM map.

Then$\{F(z):z\in D\}$has the finite intersection property.

In this paper we investigate the existence of a fixed point on the setting of locallyp-convex spaces. In particular, we establish a generalized version ofAlexandroff-Pasynkoff theorem. Furthermore, we present a generalization of theHimmelberg fixed point theorem. We also prove Fan-Glicksberg result forp-convex sets.

2 Main results

We start this section with the following result which is inspired by Theorem 1.3and Theorem 1.4.

Theorem 2.1 Suppose that A is a subset of a topological vector space X and B is a nonempty subset of A with$C_p(B)\subseteq A$. Also suppose that$F:B⊸A$is a set-valued map satisfying

• (a) $F(b)$ is closed (resp. open) in A for all $b\in B$,

• (b) $C_p(N)\subseteq F(N)$ for each $N\in 〈B〉$.

Then$\{F(b):b\in B\}$has the finite intersection property.

Proof Consider the set-valued map $\mathrm{\Gamma }:〈B〉⊸A$ defined by

$$\mathrm{\Gamma }(\{b_0,b_1,\dots ,b_n\})=\{\sum _{i=0}^n{\lambda }_i{b}_i:0\le {\lambda }_i\le 1,\sum _{i=0}^n{\lambda }_i^p=1\}.$$

Since $C_p(B)\subseteq A$, the set-valued map Γ is well defined. Condition(b) implies that F is a KKM map. For each $N=\{b_0,b_1,\dots ,b_n\}\subseteq B$, let us define

$$\begin{array}{c}{\varphi }_N:{\mathrm{\Delta }}_n⟶{\mathrm{\Gamma }}_N,\hfill \\ \sum _{i=0}^n{t}_i{e}_i\mapsto \sum _{i=0}^n{(t_i)}^{\frac{1}{p}}{b}_i.\hfill \end{array}$$

Now, one can verify that $(A,B,\mathrm{\Gamma })$ is a G-convex space. The fact that$\{F(b):b\in B\}$ has the finite intersection property follows fromTheorem 1.3 (resp. Theorem 1.4). □

Theorem 2.2 Suppose that A is a subset of an MG-convex space$(X,D,\mathrm{\Gamma })$, $\{A_0,A_1,\dots ,A_n\}$is a family of m-closure valued (resp. m-interiorvalued) subsets of X such that$A\subseteq {\bigcup }_{i=0}^n{A}_i$, and also suppose that$N=\{z_0,z_1,\dots ,z_n\}$is a family of points in D in which$\mathrm{\Gamma }(N)\subseteq A$. If$\mathrm{\Gamma }(N\setminus \{z_i\})\subseteq A_i$for each$i=0,1,\dots ,n$, then${\bigcap }_{i=0}^n{A}_i\ne \mathrm{\varnothing }$.

Proof Set $C_0=\mathrm{\Gamma }(N\setminus z_n)$ and for $i=1,2,\dots ,n$, let $C_i=\mathrm{\Gamma }(N\setminus \{z_{i-1}\})$. Consider the set-valued map $F:D⊸X$ defined by $F(z_0)=A_n$, $F(z_i)=A_{i-1}$ for $i=1,2,\dots ,n$ and $F(z)=X$ for all $z\in D\setminus N$. We claim that F is a KKM map. To see this,we note that $\mathrm{\Gamma }(N)\subseteq A\subseteq {\bigcup }_{i=0}^n{A}_i=F(N)$ and for any choice of a proper subset$\{z_{i_0},z_{i_1},\dots ,z_{i_k}\}$ of N with $0\le k<n$ and $0\le i_0<\cdots <i_k\le n$, one can see that

$$\mathrm{\Gamma }(\{z_{i_0},z_{i_1},\dots ,z_{i_k}\})\subseteq C_{i_j}\subseteq A_{i_j-1}=F(z_{i_j})$$

for some $j\in \{0,1,\dots ,k\}$. Notice that $i_j=0$ if and only if $i_j-1=n$, and so $\mathrm{\Gamma }(\{z_{i_0},z_{i_1},\dots ,z_{i_k}\})\subseteq {\bigcup }_{j=0}^{k}F(z_{i_j})$. The fact that ${\bigcap }_{i=0}^n{A}_i\ne \mathrm{\varnothing }$ follows from Theorem 1.3 (resp.Theorem 1.4). □

Remark 2.3 It should be noted that

• (a) Theorem 1.3 and Theorem 1.4 are extended versions of the corresponding results in [14, 24], and hence they are generalizations of Theorem 1 in [27, 28] and Ky Fan’s lemma [29],

• (b) Theorem 2.2 for closed (open) subsets of a topological vector space goes back to Park [30] and it is an extended version of Alexandroff-Pasynkoff theorem [31].

Definition 2.4 A nonempty subset B of a topological vector spaceX is said to be almost p-convex if for any neighborhood V of 0 and for anyfinite subset $\{b_1,\dots ,b_n\}$ of B, there exists a finite subset$\{x_1,\dots ,x_n\}\subseteq B$ such that $x_i-b_i\in V$ for each $i=1,\dots ,n$ and $C_p(\{x_1,\dots ,x_n\})\subseteq B$.

Example 2.5 It is easy to see that any p-convex subset of atopological vector space X is almost p-convex. If we delete acertain subset of the boundary of a closed p-convex set, then we have analmost p-convex set.

Definition 2.6 Let A be a subset of a topological vector spaceX. A set-valued map $T:A⊸A$ is said to have the p-convexly almostfixed point property if for every p-convex neighborhood Uof 0 in X, there exists a point $a_U\in A$ for which $a_U\in T(a_U)+U$ or $T(a_U)\cap (a_U+U)\ne \mathrm{\varnothing }$.

Theorem 2.7 Let A be a subset of a topological vector space X and B be an almost p-convex dense subset of A. Suppose that$T:A⊸X$is a lower (resp. upper) semicontinuousset-valued map such that$T(b)$is p-convex for all$b\in B$, and also suppose that there is a precompactsubset K of A such that$T(b)\cap K\ne \mathrm{\varnothing }$for all$b\in B$. Then T has the p-convexly almost fixed point property.

Proof Suppose that U is a p-convex neighborhood of 0 andsuppose that T is lower semicontinuous. There is a symmetric openneighborhood V of 0 for which $\overline{V}+\overline{V}\subseteq U$. Since K is precompact, so there are$x_0,x_1,\dots ,x_n$ in K for which $K\subseteq {\bigcup }_{i=0}^n(x_i+V)$. By using the fact that B is almostp-convex and dense in A, we find $D=\{b_0,b_1,\dots ,b_n\}\subseteq B$ for which $b_i-x_i\in V$ for all $i\in \{0,1,\dots ,n\}$ and also $C=C_p(D)\subseteq B$. Since T is lower semicontinuous, the set$F(b_i):=\{c\in C:T(c)\cap (x_i+V)=\mathrm{\varnothing }\}$ is closed in C for each$i\in \{0,\dots ,n\}$. Regarding $\mathrm{\varnothing }\ne T(c)\cap K\subseteq T(c)\cap {\bigcup }_{i=0}^n(x_i+V)$, we have ${\bigcap }_{i=0}^{n}F(b_i)=\mathrm{\varnothing }$. Now, Theorem 2.1 implies that there is$N=\{b_{i_0},b_{i_1},\dots ,b_{i_k}\}\in 〈D〉$ and $x_U\in C_p(N)\subseteq B$ for which $x_U\notin F(N)$, and so $T(x_u)\cap (x_{i_j}+\overline{V})\ne \mathrm{\varnothing }$ for all $j\in \{0,1,\dots ,k\}$. Both $b_i-x_i\in V$ and $\overline{V}+\overline{V}\subseteq U$ imply that $x_{i_j}+\overline{V}\subseteq b_{i_j}+U$, which implies that $T(x_U)\cap (b_{i_j}+U)\ne \mathrm{\varnothing }$. Therefore

$$N\subseteq M:=\{c\in C:T(x_U)\cap (c+U)\ne \mathrm{\varnothing }\}.$$

C, $T(x_U)$ and U are p-convex and henceM is p-convex. Consequently, $x_U\in M$, which implies that $T(x_U)\cap (x_U+U)\ne \mathrm{\varnothing }$; i.e., T has thep-convexly almost fixed point property. Finally, for the case thatT is upper semicontinuous, we note that $F(b_i):=\{c\in C:T(c)\cap (x_i+\overline{V})=\mathrm{\varnothing }\}$ is open in C for each$i\in \{0,\dots ,n\}$. The rest of the proof is similar to the proof of thecase that T is l.s.c. Regarding the analogy, we skip theproof. □

Corollary 2.8 Let A be a p-convex subset of a topological vector space X, and let$T:A⊸X$be a lower (resp. upper) semicontinuousset-valued map such that$T(a)$is p-convex for all$a\in A$. Suppose that there is a precompact subset K of A such that$T(a)\cap K\ne \mathrm{\varnothing }$for all$a\in A$. Then T has the p-convexly almost fixed point property.

Proof It is sufficient to take $A=B$ in Theorem 2.7. □

Corollary 2.9 Let A be a subset of a topological vector space X, and let B be an almost p-convex dense subset of A. Suppose that$T:A⊸X$is a set-valued map satisfying

• (a) $T^l(x)$ (resp. $T^u(x)$) is open for all $x\in X$,

• (b) $T(b)$ is p-convex for all $b\in B$,

• (c) there is a precompact subset K of A such that $T(b)\cap K\ne \mathrm{\varnothing }$ for all $b\in B$.

Then T has the p-convexly almost fixed point property.

Proof It is clear that (a) implies that T is a lower (resp. upper)semicontinuous set-valued map and hence T has the p-convexlyalmost fixed point property by Theorem 2.7. □

Corollary 2.10 Let A be a p-convex subset of a topological vector space X, and let$T:A⊸X$be a compact set-valued map satisfying the following conditions:

• (a) $T^l(x)$ (resp. $T^u(x)$) is open for all $x\in X$,

• (b) $T(a)$ is nonempty and p-convex for all $a\in A$.

Then T has the p-convexly almost fixed point property.

Proof Consider $A=B$, it is easy to see that all the conditions ofCorollary 2.9 are satisfied. □

Remark 2.11 It should be noted that

• (a) Corollary 2.8 for a lower semicontinuous set-valued map on a locally convex Hausdorff topological vector space goes back to Ky Fan [32]. Corollary 2.8 for a single-valued map might be regarded as a generalization of the Thychonoff fixed point theorem to a noncompact (or precompact) convex set [32]. Also, Lassonde obtained Corollary 2.8 for a compact upper semicontinuous set-valued map with nonempty convex values [33].

• (b) Convex versions of Theorem 2.7, Corollary 2.9 and Corollary 2.10 are due to Park [30].

Theorem 2.12 Suppose that A is a subset of a locally p-convex space X and B is an almost p-convex dense subset of A. Suppose that$T:A⊸A$satisfies the following:

• (a) T is compact upper semicontinuous,

• (b) $T(a)$ is closed for all $a\in A$,

• (c) $T(b)$ is nonempty p-convex for all $b\in B$.

Then T has a fixed point.

Proof Since all the conditions of Theorem 2.7 are satisfied and sinceX is a locally p-convex space, T has the almost fixedpoint property. Then, for an arbitrary neighborhood U of 0, there exist$a_U$ and $b_U$ in A for which $b_U\in T(a_U)\cap (a_U+U)$. Since T is compact, we conclude that thereis $a_0\in \overline{T(A)}\subseteq A$ in which the net $b_U⟶a_0$. Because X is Hausdorff,$a_U⟶a_0$. Since T is an upper semicontinuousset-valued map with closed values, $Graph(T)$ is closed. Consequently, $a_0$ is a fixed point of T. □

Corollary 2.13 Suppose that A is a p-convex subset of a locally p-convex space X. Suppose that$T:A⊸A$satisfies the following:

• (a) T is compact upper semicontinuous,

• (b) $T(a)$ is closed for all $a\in A$,

• (c) $T(a)$ is nonempty p-convex for all $a\in A$.

Then T has a fixed point.

Theorem 2.14 Suppose that A is a p-convex subset of a locally p-convex space X. Suppose that$T:A⊸A$satisfies the following:

• (a) T is compact and closed,

• (b) T has the almost fixed point property.

Then T has a fixed point.

Proof Suppose that is the family of neighborhoods of 0 in X. For anyelement U of ,since T has the almost fixed point property, so there exist$a_U,b_U\in A$ for which $b_U\in T(a_U)$ and $b_U\in a_U+U$. Now, consider the nets $\{a_U\}$ and $\{b_U\}$. By (a) we have $\overline{T(A)}$ is compact and hence $\{b_U\}$ has a subnet converging to $b_0$. We may assume that $b_U⟶b_0$. Since X is Hausdorff, there is a subnet of$a_U$ converging to $b_0$. The fact that $b_0\in T(b_0)$ follows from $(a_U,b_U)\in Graph(T)$ and the fact that $Graph(T)$ is closed. □

Corollary 2.15 Suppose that A is a p-convex subset of a locally p-convex space X and that$T:A⊸A$satisfies the following:

• (a) T is compact and closed,

• (b) $T^l(x)$ (resp. $T^u(x)$) is open for all $x\in X$,

• (c) $T(a)$ is nonempty and p-convex for all $a\in A$.

Then T has a fixed point.

Proof It is an immediate consequence of Corollary 2.10 andTheorem 2.14. □

Remark 2.16 Corollary 2.13 is a generalization of the main results ofHimmelberg [26]. Theorem 2.12 for $p=1$ goes back to Park [30]. Further, Theorem 2.14 for $p=1$ is an extension of Himmelberg’s theorem (see,e.g., [34]).

For a set-valued map $T:X⊸Y$, set $T_B=\{x\in X:x\in T(x)+B\}$ for $B\subseteq Y$.

Lemma 2.17 Suppose that A is a p-convex subset of a topological vector space X, and also suppose thatis a fundamental system of open neighborhoods of 0. Then, fora set-valued map$T:A⊸X$, the following are equivalent:

• (a) If $a\in A$ satisfies $a\notin T(a)+U$ for some $U\in \mathcal{U}$, then

  $$a\notin Cl\left(\{a\in A:a\in T(a)+C_p(V)\}\right)\mathit{\text{for some }}V\in \mathcal{U},$$

• (b) ${\bigcap }_{U\in \mathcal{U}}{T}_U={\bigcap }_{U\in \mathcal{U}}\overline{T_{C_p(U)}}$.

Proof It is straightforward. □

Remark 2.18 The conditions (a) and (b) considered in Lemma 2.17 for$p=1$ are due to Kim [35].

Theorem 2.19 Let A be a p-convex compact subset of a topological vector space X, and let$T:A⊸X$be a mapping satisfying the following conditions:

• (a) T has the p-convexly almost fixed point property,

• (b) ${\bigcap }_{U\in \mathcal{U}}{T}_U={\bigcap }_{U\in \mathcal{U}}\overline{T_{C_p(U)}}$.

Then$\overline{T}$has a fixed point.

Proof Suppose that is a fundamental system of open neighborhoods of 0. SinceT has the p-convexly almost fixed point property, for any$U\in \mathcal{U}$, there is an $a_U\in A$ such that $a_U\in T(a_U)+C_p(U)$. Hence, $T_{C_p(U)}\ne \mathrm{\varnothing }$ for each $U\in \mathcal{U}$. Now, since is a fundamental system of open neighborhoods of 0, wededuce that for any $U,V\in \mathcal{U}$, there is $W\in \mathcal{U}$ such that

$$T_{C_p(U)}\cap T_{C_p(V)}\supseteq T_{C_p(U\cap V)}\supseteq T_{C_p(W)}\ne \mathrm{\varnothing }.$$

Therefore $\{T_{C_p(U)}:U\in \mathcal{U}\}$ has the finite intersection property. It follows fromthe compactness of A that ${\bigcap }_{U\in \mathcal{U}}\overline{T_{C_p(U)}}\ne \mathrm{\varnothing }$. Therefore, by the condition (b) there is an$a_0\in A$ for which $a_0\in {\bigcap }_{U\in \mathcal{U}}{T}_U$, that is, $a_0\in T(a_0)+U$ for all $U\in \mathcal{U}$. Regarding ${\bigcap }_{U\in \mathcal{U}}(T(a_0)+U)=\overline{T(a_0)}$, we derive that $\overline{T}$ has a fixed point. □

Corollary 2.20 Let A be a p-convex compact subset of a topological vector space X, and let$T:A⊸X$be a mapping such that

• (a) T has the p-convexly almost fixed point property,

• (b) ${\bigcap }_{U\in \mathcal{U}}{T}_U={\bigcap }_{U\in \mathcal{U}}\overline{T_{C_p(U)}}$,

• (c) T has closed values.

Then T has a fixed point.

Corollary 2.21 Let A be a p-convex compact subset of a topological vector space X, and let$T:A⊸A$be a mapping such that

• (a) T is lower (resp. upper) semicontinuous,

• (b) T has p-convex values,

• (c) ${\bigcap }_{U\in \mathcal{U}}{T}_U={\bigcap }_{U\in \mathcal{U}}\overline{T_{C_p(U)}}$.

Then$\overline{T}$has a fixed point.

Proof Since A is a p-convex and compact, by (a) and (b)one can see that all the conditions of Corollary 2.8 hold. Then T hasthe p-convexly almost fixed point property. The fact that$\overline{T}$ has a fixed point follows fromTheorem 2.19. □

Corollary 2.22 Let A be a p-convex compact subset of a topological vector space X, and let$T:A⊸A$be a mapping satisfying the following conditions:

• (a) T is lower (resp. upper) semicontinuous,

• (b) T has closed p-convex values,

• (c) ${\bigcap }_{U\in \mathcal{U}}{T}_U={\bigcap }_{U\in \mathcal{U}}\overline{T_{C_p(U)}}$.

Then T has a fixed point.

Remark 2.23 Corollary 2.22 for $p=1$ and lower semicontinuous set-valued maps goes back toKim [35] and Park [36], and also this result for $p=1$ and upper semicontinuous set-valued maps is due toHuang and Jeng [37].

Theorem 2.24 Let A be a compact p-convex subset of a locally p-convex space X, and let the set-valued map$T:A⊸A$be a mapping such that

• (a) T has the p-convexly almost fixed point property,

• (b) T is a closed set-valued map.

Then T has a fixed point.

Proof Suppose that is a fundamental system of p-convex openneighborhoods of 0. Then, for any $U\in \mathcal{U}$, there is $V\in \mathcal{U}$ for which $V\subseteq \overline{V}\subseteq U$. Now, we claim that $T_{C_p(\overline{V})}=T_{\overline{V}}$ is closed. To see this, let $a\in \overline{T_{\overline{V}}}$. There is a net $\{a_i:i\in I\}\subseteq T_{\overline{V}}$ for which $a_i⟶a$. Then, for each $i\in I$, there exists $b_i\in T(a_i)$ in which $a_i-b_i\in \overline{V}$. Since T is compact and since$b_i\in T(A)$, so one can assume that $b_i⟶b$ for some $b\in \overline{T(A)}$, and so $a-b\in \overline{V}$. $b\in T(a)$, because T is closed. Therefore,

$$a\in (b+\overline{V})\cap A\subseteq (T(a)+\overline{V})\cap A;$$

i.e., $a\in T_{\overline{V}}$. Finally, since $T_{\overline{V}}$ is closed, and $V\subseteq \overline{V}\subseteq U$, so

$$\bigcap _{U\in \mathcal{U}}{T}_{C_p(U)}=\bigcap _{U\in \mathcal{U}}{T}_U=\bigcap _{V\in \mathcal{U}}{T}_{\overline{V}}=\bigcap _{V\in \mathcal{U}}\overline{T_{\overline{V}}}=\bigcap _{U\in \mathcal{U}}\overline{T_U}=\bigcap _{U\in \mathcal{U}}\overline{T_{C_p(U)}}.$$

Consequently, all the conditions of Corollary 2.20 hold and hence Thas a fixed point. □

Remark 2.25 Theorem 2.24 is a generalization of the Fan-Glicksbergtheorem [38, 39] and its convex version can be found in [34]. Notice also that Theorem 2.24 can be derived fromTheorem 2.14.