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Some fixed point theorems in locally pconvex spaces
Fixed Point Theory and Applications volume 2013, Article number: 312 (2013)
Abstract
In this paper we investigate the existence of a fixed point of multivalued mapson almost pconvex and pconvex 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 pconvex sets.Recently, a number of fixed point theorems have appeared on the setting ofpconvex sets. For instance, Alimohammady et al.[1] extended the MarkovKakutani fixed point theorem for compactpstar shaped subsets in topological vector spaces by usingpconvex 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 pstar shaped subset of a topological vector space via FanKKMprinciple in a generalized convex space. In [5, 6], generalized versions of Brouwer and Kakutani fixed point theorems werecharacterized in the context of locally pconvex 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 set X in a vector space V is said to bepconvex if ${A}_{x}^{y}\subseteq X$ for every $x,y\in X$. The pconvex hull of Xdenoted by ${C}_{p}(X)$ is the smallest pconvex set containingX[5]. Further, the closed pconvex hull of X denoted by ${\overline{C}}_{p}(X)$ is the smallest closed pconvex 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 semiopen sets$\mathit{SO}(X)$, the collection of all preopen sets$\mathit{PO}(X)$, the collection of all αopen sets$\alpha O(X)$ and the collection of all βopen 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 mopen set if$A\in \mathcal{M}$. Similarly, a set $B\in \mathcal{P}(X)$ is an mclosed set if${B}^{c}\in \mathcal{M}$. Furthermore, minterior andmclosure of a set A are defined as follows:
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$f:(X,\tau )\u27f6(Y,\mathcal{N})$ 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. Asetvalued map or a setvalued 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$x\mid \u22b8T(x)$. The lower inverse of a point$y\in Y$ of a setvalued map T is the setvalued map${T}^{l}$ of Y into X defined by
Analogously, lower inverse of a subset of$B\subset Y$ is defined as
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 setvalued map $T:X\u22b8Y$ 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 setvalued map$T:A\u22b8A$ 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 $\u3008D\u3009$ denote the set of all nonempty finite subsets of aset D, and let ${\mathrm{\Delta}}_{n}$ be the nsimplex with vertices${e}_{0},{e}_{1},\dots ,{e}_{n}$, ${\mathrm{\Delta}}_{J}$ be the face of ${\mathrm{\Delta}}_{n}$ corresponding to $J\in \u3008A\u3009$, where $A\in \u3008D\u3009$. 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 (brieflyMGconvex space) $(X,D,\mathrm{\Gamma})$ consists of a minimal space $(X,\mathcal{M})$, a nonempty set D and a setvalued map$\mathrm{\Gamma}:\u3008D\u3009\u22b8X$ in which for $A\in \u3008D\u3009$ with $n+1$ elements, there exists a $(\tau ,m)$continuous function ${\varphi}_{A}:{\mathrm{\Delta}}_{n}\u27f6{\mathrm{\Gamma}}_{A}:=\mathrm{\Gamma}(A)$ for which $J\in \u3008A\u3009$ implies that ${\varphi}_{A}({\mathrm{\Delta}}_{J})\subseteq {\mathrm{\Gamma}}_{J}$. If $\mathcal{M}=\tau $, then the notion of MGconvex space turnsinto Gconvex 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 setvalued map $\mathrm{\Gamma}:\u3008X\u3009\u22b8X$ 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 MGconvex space. A setvalued map$F:D\u22b8X$ is called a KKM setvalued map if${\mathrm{\Gamma}}_{A}\subseteq F(A)$ for any $A\in \u3008D\u3009$.
We state two useful theorems of Alimohammady et al.[25] as follows.
Theorem 1.3[25]
Suppose that$(X,D,\mathrm{\Gamma})$is an MGconvex space and$F:D\u22b8X$is a setvalued 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 mcompact for some $N\in \u3008D\u3009$,
then${\bigcap}_{z\in D}F(z)\ne \mathrm{\varnothing}$.
Theorem 1.4[25]
Suppose that$(X,D,\mathrm{\Gamma})$is an MGconvex space and$F:D\u22b8X$is a setvalued 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 locallypconvex spaces. In particular, we establish a generalized version ofAlexandroffPasynkoff theorem. Furthermore, we present a generalization of theHimmelberg fixed point theorem. We also prove FanGlicksberg result forpconvex 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$F:B\u22b8A$is a setvalued 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 $N\in \u3008B\u3009$.
Then$\{F(b):b\in B\}$has the finite intersection property.
Proof Consider the setvalued map $\mathrm{\Gamma}:\u3008B\u3009\u22b8A$ defined by
Since ${C}_{p}(B)\subseteq A$, the setvalued 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
Now, one can verify that $(A,B,\mathrm{\Gamma})$ is a Gconvex 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 MGconvex space$(X,D,\mathrm{\Gamma})$, $\{{A}_{0},{A}_{1},\dots ,{A}_{n}\}$is a family of mclosure valued (resp. minteriorvalued) 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 ${C}_{i}=\mathrm{\Gamma}(N\setminus \{{z}_{i1}\})$. Consider the setvalued map $F:D\u22b8X$ defined by $F({z}_{0})={A}_{n}$, $F({z}_{i})={A}_{i1}$ 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
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 AlexandroffPasynkoff theorem [31].
Definition 2.4 A nonempty subset B of a topological vector spaceX is said to be almost pconvex 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 pconvex subset of atopological vector space X is almost pconvex. If we delete acertain subset of the boundary of a closed pconvex set, then we have analmost pconvex set.
Definition 2.6 Let A be a subset of a topological vector spaceX. A setvalued map $T:A\u22b8A$ is said to have the pconvexly almostfixed point property if for every pconvex 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 pconvex dense subset of A. Suppose that$T:A\u22b8X$is a lower (resp. upper) semicontinuoussetvalued map such that$T(b)$is pconvex 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 pconvexly almost fixed point property.
Proof Suppose that U is a pconvex 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 almostpconvex 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$N=\{{b}_{{i}_{0}},{b}_{{i}_{1}},\dots ,{b}_{{i}_{k}}\}\in \u3008D\u3009$ 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
C, $T({x}_{U})$ and U are pconvex and henceM is pconvex. Consequently, ${x}_{U}\in M$, which implies that $T({x}_{U})\cap ({x}_{U}+U)\ne \mathrm{\varnothing}$; i.e., T has thepconvexly 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 pconvex subset of a topological vector space X, and let$T:A\u22b8X$be a lower (resp. upper) semicontinuoussetvalued map such that$T(a)$is pconvex 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 pconvexly 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 pconvex dense subset of A. Suppose that$T:A\u22b8X$is a setvalued map satisfying

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

(b)
$T(b)$ is pconvex 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 pconvexly almost fixed point property.
Proof It is clear that (a) implies that T is a lower (resp. upper)semicontinuous setvalued map and hence T has the pconvexlyalmost fixed point property by Theorem 2.7. □
Corollary 2.10 Let A be a pconvex subset of a topological vector space X, and let$T:A\u22b8X$be a compact setvalued 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 pconvex for all $a\in A$.
Then T has the pconvexly 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 setvalued map on a locally convex Hausdorff topological vector space goes back to Ky Fan [32]. Corollary 2.8 for a singlevalued 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 setvalued 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 pconvex space X and B is an almost pconvex dense subset of A. Suppose that$T:A\u22b8A$satisfies the following:

(a)
T is compact upper semicontinuous,

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

(c)
$T(b)$ is nonempty pconvex 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 pconvex 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 ${b}_{U}\u27f6{a}_{0}$. Because X is Hausdorff,${a}_{U}\u27f6{a}_{0}$. Since T is an upper semicontinuoussetvalued 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 pconvex subset of a locally pconvex space X. Suppose that$T:A\u22b8A$satisfies the following:

(a)
T is compact upper semicontinuous,

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

(c)
$T(a)$ is nonempty pconvex for all $a\in A$.
Then T has a fixed point.
Theorem 2.14 Suppose that A is a pconvex subset of a locally pconvex space X. Suppose that$T:A\u22b8A$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 ${b}_{U}\u27f6{b}_{0}$. 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 pconvex subset of a locally pconvex space X and that$T:A\u22b8A$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 pconvex 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 setvalued map $T:X\u22b8Y$, set ${T}_{B}=\{x\in X:x\in T(x)+B\}$ for $B\subseteq Y$.
Lemma 2.17 Suppose that A is a pconvex subset of a topological vector space X, and also suppose thatis a fundamental system of open neighborhoods of 0. Then, fora setvalued map$T:A\u22b8X$, the following are equivalent:

(a)
If $a\in A$ satisfies $a\notin T(a)+U$ for some $U\in \mathcal{U}$, then
$$a\notin Cl\left(\{a\in A:a\in T(a)+{C}_{p}(V)\}\right)\phantom{\rule{1em}{0ex}}\mathit{\text{for some}}V\in \mathcal{U},$$ 
(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 pconvex compact subset of a topological vector space X, and let$T:A\u22b8X$be a mapping satisfying the following conditions:

(a)
T has the pconvexly 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 pconvexly 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
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 pconvex compact subset of a topological vector space X, and let$T:A\u22b8X$be a mapping such that

(a)
T has the pconvexly 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 pconvex compact subset of a topological vector space X, and let$T:A\u22b8A$be a mapping such that

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

(b)
T has pconvex 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 pconvex and compact, by (a) and (b)one can see that all the conditions of Corollary 2.8 hold. Then T hasthe pconvexly almost fixed point property. The fact that$\overline{T}$ has a fixed point follows fromTheorem 2.19. □
Corollary 2.22 Let A be a pconvex compact subset of a topological vector space X, and let$T:A\u22b8A$be a mapping satisfying the following conditions:

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

(b)
T has closed pconvex 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 setvalued maps goes back toKim [35] and Park [36], and also this result for $p=1$ and upper semicontinuous setvalued maps is due toHuang and Jeng [37].
Theorem 2.24 Let A be a compact pconvex subset of a locally pconvex space X, and let the setvalued map$T:A\u22b8A$be a mapping such that

(a)
T has the pconvexly almost fixed point property,

(b)
T is a closed setvalued map.
Then T has a fixed point.
Proof Suppose that is a fundamental system of pconvex 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 ${a}_{i}\u27f6a$. 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 ${b}_{i}\u27f6b$ for some $b\in \overline{T(A)}$, and so $ab\in \overline{V}$. $b\in T(a)$, because T is closed. Therefore,
i.e., $a\in {T}_{\overline{V}}$. Finally, since ${T}_{\overline{V}}$ is closed, and $V\subseteq \overline{V}\subseteq U$, so
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 FanGlicksbergtheorem [38, 39] and its convex version can be found in [34]. Notice also that Theorem 2.24 can be derived fromTheorem 2.14.
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Gholizadeh, L., Karapınar, E. & Roohi, M. Some fixed point theorems in locally pconvex spaces. Fixed Point Theory Appl 2013, 312 (2013). https://doi.org/10.1186/168718122013312
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Keywords
 fixed point theorems
 locally convex space
 pconvex set