A Fixed-Component Point Theorem and Applications

Abstract

We prove a topologically based characterization of the existence of fixed-component points for an arbitrary family of set-valued maps defined on a product set by using topologically based structures, without linear or convexity structures. Then, applying this general result, we derive sufficient conditions for the existence of coincidence-component points of families of set-valued maps and intersection points of families of sets, as examples for many other important points in nonlinear analysis. Applications to systems of variational relations and abstract economies are provided as examples for other optimization-related problems.

1 Introduction

Numerous fixed point theorems have been developed and used for centuries as the most useful tool in dealing with the existence of solutions in various fields of science. Relatively recently, motivated by problems involving functions or sets defined on product spaces, e.g., in game theory, problems on sets with convex sections, systems of variational inequalities, etc., several authors (see [1, 21]) established fixed point theorems for a family of mappings defined in product spaces. These points are called fixed-component points or collectively fixed points. Such results were developed, e.g., in [1, 6, 7, 9, 19, 24, 28, 31, 34], with successful applications in existence studies for optimization-related problems.

For considerations of the existence of fixed points in particular and of many other important points in nonlinear analysis in general, it was believed for a long period that one needed both topological and algebraic machineries. Wu [33] and Horvath [12] started two directions of dealing with existence issues in pure topological settings. Wu’s approach is based on replacing convexity assumptions by connectedness conditions, and Horvath’s one on replacing a convex hull by an image of a simplex through a continuous map. Recently, an attempt to unify these two directions in obtaining topological full (two-way) characterizations of the existence of various important points like intersection points, maximal elements, coincidence points, sectional points, etc., was carried out in [18, 20], based on the so-called KKM-structures and connectedness structures. Realizing the basic role of fixed points in existence studies, in [19] we developed a new characterization of the existence of such points in topologically based settings. An extension to fixed-component points was also included, but only for a family of finite number of mappings. We cannot employ the same proof technique to extend this result for infinitely many mappings. It is worth noting that among the above-encountered references, only [18,19,20] dealt with necessary and sufficient conditions for existence, the others included only sufficient ones.

The above discussions inspire us to find another way in this paper to consider full characterizations of the existence of fixed-component points of general infinite families of mappings. Namely, based on our KKM-structures and using continuous partitions of unity and the classical Tikhonov fixed point theorem, in this paper we extend Theorem 2.5 of [19], which is a necessary and sufficient condition for the existence of fixed-component points, to the case of an arbitrary family of mappings defined on product sets. Applications of this result to aforementioned important points in nonlinear analysis and to optimization-related problems are also included.

The layout of the paper is simple. In the rest of this section, we recall some needed definitions. Section 2 contains full characterizations of the existence of fixed-component points together with discussions on imposed assumptions as well as consequences and relations to previous results in the literature. Section 3 is devoted to applications, including studies of the existence of some important points in nonlinear analysis and for solutions of optimization-related problems.

Throughout this paper, for a nonempty set X, $\langle X \rangle$ stands for the set of all finite subsets of X. For $N=\{x_0, x_1,\ldots , x_n\}\in \langle X \rangle$ and $M = \{x_{i_0}, x_{i_1},\ldots , x_{i_k}\}\subset N$, $\Delta _{|N|}:=\Delta _n$ stands for the standard n-simplex of Euclidean space $\mathbb {R}^{n+1}$ with vertices being unit vectors $e_0=(1, 0,\ldots ,0)$, $e_1=(0, 1, 0,\ldots ,0)$,..., $e_n=(0,0,\ldots ,1)$, $\Delta _M$ denotes the face of $\Delta _{|N|}$ with vertices $e_{i_0}$, $e_{i_1}$,..., $e_{i_k}$. Let $H : X \rightrightarrows Y$ be a set-valued map between nonempty sets X and Y. For $x\in X$ and $y\in Y$, an image and a fiber (or inverse image) of H is the set H(x) and $H^{-1}(y)= \{x\in X\mid y\in H(x)\}$, respectively (resp).

Definition 1

([19, 20]) For nonempty sets X and Y, a pair $\mathscr {F}:=({\varPhi }_X, \mathfrak {I}_Y)$ is called a KKM-structure of the pair (X, Y) if $\mathfrak {I}_Y$ is a topology on Y and ${\varPhi }_X =\{\varphi _N: \Delta _{|N|}\rightarrow Y|\quad N\in \langle X \rangle \}$ is a family of maps with all $\varphi _N\in {\varPhi }_X$ being $\mathfrak {I}_Y$-continuous. In the special case where $X=Y$, such a $\mathscr {F}$ is termed a KKM-structure of X. If $\mathfrak {I}_Y$ is compact, i.e., Y is $\mathfrak {I}_Y$-compact, $({\varPhi }_X, \mathfrak {I}_Y)$ is called a compact KKM-structure.

If $X=Y$ is a convex subset of a topological vector space E, $\mathfrak {I}_X$ is the topology on X induced by that of E, and

$$\begin{aligned} {\varPhi }_X =\Big \{\varphi _N: \Delta _{|N|}\rightarrow X|\quad \varphi _N(e)=\sum _{x_i\in N}\lambda _ix_i\;\,\mathrm{for}\;\, e=\sum \lambda _ie_i\in \Delta _{|N|},\, N\in \langle X \rangle \Big \}, \end{aligned}$$

then $\mathscr {F}:=({\varPhi }_X, \mathfrak {I}_X)$ is called the natural KKM-structure of X.

Let I be any index set, $X_i$ a nonempty set and $\mathscr {F}_i := ({\varPhi }_{X_i}, \mathfrak {I}_{X_i})$ a KKM-structure on $X_i$ for each $i\in I$. We define a KKM-structure of $X:=\prod _{i\in I}X_i$ as follows. Let $\mathfrak {I}_X$ be the Tikhonov product topology on X of the topologies $\mathfrak {I}_{X_i}$. For $N =\{(x_i^1)_{i\in I}, (x_i^2)_{i\in I}, \ldots (x_i^n)_{i\in I}\}\in \langle X\rangle$, we define $N_i=\{x_i^1, x_i^2,\ldots ,x_i^n\}\in \langle X_i\rangle$ and denote $N:=\otimes _{i\in I} N_i$. Each $N_i$ is called the “ith component” of N. We also denote $x:=\otimes _{i\in I}x_i$ for each element $x=(x_i)_{i\in I}\in X$. Let

$$\begin{aligned} {\varPhi }_{X}=\Big \{\varphi _N: \Delta _{|N|}\rightarrow X\mid \varphi _N(e)= \otimes _{i\in I}\varphi _{N_i}(e)\,\mathrm{for}\, e\in \Delta _{|N|},\, N=\otimes _{i\in I}N_i\in \langle X \rangle \Big \}. \end{aligned}$$

Then, $\mathscr {F}:=({\varPhi }_X,\mathfrak {I}_X)$ is a KKM-structure on X, called the product KKM-structure of the KKM-structures $\mathscr {F}_i := ({\varPhi }_{X_i}, \mathfrak {I}_{X_i})$, and denoted by $\mathscr {F}=\prod _{i\in I}\mathscr {F}_i$.

Definition 2

([19, 20]) Let X be a nonempty set and $({\varPhi }_X, \mathfrak {I}_X)$ a KKM-structure of X. We say that a subset B of X is ${\varPhi }_X$-convex if, for all $\varphi _N\in {\varPhi }_X$ and $M\subset N\cap B$, $\varphi _N(\Delta _M)\subset B$. For $C\subset X$, the smallest ${\varPhi }_X$-convex set containing C, denoted by ${\varPhi }_X$-$\mathrm{co}C$, is called ${\varPhi }_X$-convex hull of C. It is not hard to check that ${\varPhi }_X$-$\mathrm{co}C$ = $\bigcup _{N\in \langle C\rangle }{\varPhi }_X$-$\mathrm{co}N$.

When $\mathscr {F}:= ({\varPhi }_X, \mathfrak {I}_X)$ is the natural KKM-structure of X, the notions of a ${\varPhi }_X$-convex set and a ${\varPhi }_X$-convex hull collapse to the usual notions of a convex set and a convex hull, resp. Note that notions of generalized convex sets and convex hulls in some spaces, previously introduced by many authors, such as a convex space [22], H-space [12], G-convex space [26], FC-space [8], GFC-space [17], etc., are particular cases of the notions in Definition 2 because in each of these spaces there is a KKM-structure implicitly. However, there are several convex structures for which notions of convex sets and convex hulls do seemingly not naturally match with the notions in Definition 2. We discuss first the Takahashi-convex structure. Recall that a Takahashi-convex structure on a metric space (X, d) is a function $h: X\times X\times [0,1]\longrightarrow X$ satisfying $d(s, h(u,v,t))\le td(s, u) +(1-t)d(s,v)$ for all $(s, u,v,t)\in X\times X\times X\times [0,1]$ (see [29]). A subset B of X is said to be convex if $h(u,v,t)\in B$ for any $u,v\in B$ and $t\in [0,1]$. We construct a KKM-structure on X as follows. For each $N\in \langle X\rangle$ and $e=\sum _{i=0}^{|N|}\lambda _ie_i\in \Delta _{|N|}$, let $i_e=\min \{i\,\mid \, \lambda _i\ne 0\}$ and $i^e=\max \{i\,\mid \, \lambda _i\ne 0\}$. Let ${\varPhi }_X$ includes maps $\varphi _N:\Delta _{|N|}\longrightarrow X$ defined by $\varphi _N(e)=h(x_{i_e},x_{i^e}, \lambda _{i_e}+\lambda _{i^e})$ for all $e=\sum _{i=0}^{|N|}\lambda _ie_i\in \Delta _{|N|}$. Let

$$\begin{aligned} \mathfrak {I}_X=\bigcap _{N\in \langle X\rangle }\left\{ U\subset X\,|\,\varphi _N^{-1}(U)\,\,\mathrm{is\,\, open\,\, in}\,\, \Delta _{|N|}\right\} . \end{aligned}$$

Then, $({\varPhi }_X, \mathfrak {I}_X)$ is a KKM-structure of X, and any convex subset in Takahashi-convex metric space (X, d, h) is also ${\varPhi }_X$-convex. However, if we fix the topology $\mathfrak {I}$ induced by the metric d of X (and do not consider the above topology $\mathfrak {I}_X$, then we still do not know if there is or not a family ${\varPhi }_X$ such that $({\varPhi }_X, \mathfrak {I})$ is a KKM-structure of X with each convex subset in (X, d, h) being also ${\varPhi }_X$-convex. Another approach to obtaining a notion of convex hull was proposed in [14], which was interesting, without any convex structure. But the notions in Section 4 of [14] and in Definition 2 are not comparable.

2 Fixed-Component Point Theorems

Definition 3

Let I be any index set. For each $i\in I$, let $X_i$ be a nonempty set, $\mathscr {F}_i:=({\varPhi }_{X_i}, \mathfrak {I}_{X_i})$ be a KKM-structure of $X_i$, and $P_i, Q_i: X:=\prod _{i\in I}X_i\rightrightarrows X_i$ be set-valued maps. $\{Q_i\}_{i\in I}$ is called $\{{\varPhi }_{X_i}\}$-weak-convex with respect to (w.r.t.) $\{P_i\}_{i\in I}$ if whenever $x=(x_i)_{i\in I}\in X$, $\varphi _{N_i}\in {\varPhi }_{X_i}$, and $M_i\subset N_i\cap P_i(x)$ satisfying $x_i\in \varphi _{N_i}(\Delta _{M_i})$ simultaneously for all $i\in I$, one has $\varphi _{N_i}(\Delta _{M_i})\subset Q_i(x)$ for each $i\in I$.

If Definition 3 holds with the natural KKM-structures $\mathscr {F}_i$, we say that $\{Q_i\}_{i\in I}$ is weak-convex w.r.t. $\{P_i\}_{i\in I}$.

Remark 1

For each $i\in I$, let us consider the following conditions for the pair $(P_i, Q_i)$.

• (h1) Whenever $x=(x_i)_{i\in I}\in X$, $\varphi _{N_i}\in {\varPhi }_{X_i}$, and $M_i\subset N_i\cap P_i(x)$ satisfying $x_i\in \varphi _{N_i}(\Delta _{M_i})$, one has $\varphi _{N_i}(\Delta _{M_i})\subset Q_i(x)$. In this case, we say that $Q_i$ is ${\varPhi }_{X_i}$-weak-convex w.r.t. $P_i$.

• (h2) For all $x\in X$, $\varphi _{N_i}\in {\varPhi }_{X_i}$ and $M_i\subset N_i\cap P_i(x)$, one has $\varphi _{N_i}(\Delta _{M_i})\subset Q_i(x)$. If this condition holds, $Q_i$ is called ${\varPhi }_{X_i}$-convex w.r.t. $P_i$.

• (h3) ${\varPhi }_{X_i}$-$\mathrm{co}P_i(x) \subset Q_i(x)$ for each $x\in X$.

• (h4) $P_i(x) \subset Q_i(x)$ and $Q_i(x)$ is ${\varPhi }_{X_i}$-convex for each $x\in X$.

It is not hard to see that, if the KKM-structures $\mathscr {F}_i:=({\varPhi }_{X_i}, \mathfrak {I}_{X_i})$ are given, then (h4) $\Rightarrow$ (h3) $\Rightarrow$ (h2) $\Rightarrow$ (h1). Example 1 below shows that, in general, the reverse implications are not true. However, in the case $\mathscr {F}_i$ is the natural KKM-structure of $X_i$, (h2) coincides with (h3). From Definition 3, we see that “if for each $i\in I$, one of conditions (h1)-(h4) holds for$(P_i, Q_i)$, then$\{Q_i\}_{i\in I}$ is $\{{\varPhi }_{X_i}\}$-weak-convex w.r.t.$\{P_i\}_{i\in I}$”. The converse does not hold as shown by Example 2 below.

Example 1

Let $X_1 =X_2= X_3 = [0, 2]$, $\mathscr {F}_1:=({\varPhi }_{X_1}, \mathfrak {I}_{X_1})$ be the natural KKM-structure of $X_1$, $\mathscr {F}_2:=({\varPhi }_{X_2}, \mathfrak {I}_{X_2})$ and $\mathscr {F}_3:=({\varPhi }_{X_3}, \mathfrak {I}_{X_3})$ be KKM-structures of $X_2$ and $X_3$, resp., defined by: $\mathfrak {I}_{X_2}= \mathfrak {I}_{X_3}$ being the usual topology on [0, 2], ${\varPhi }_{X_2}=\{\varphi _{N_2}: \Delta _{|N_2|}\rightarrow X_2 \,|\, \varphi _{N_2}(e) =0 \,\mathrm{for \,all}\, e\in \Delta _{|N_2|},\, N_2\in \langle X_2\rangle \}$, and ${\varPhi }_{X_3}=\{\varphi _{N_3}: \Delta _{|N_3|}\rightarrow X_3 \,|\, \varphi _{N_3}(e) =\frac{\min N_3 +\, \max N_3}{2}\,\mathrm{for \,all}\, e\in \Delta _{|N_3|},\, N_3\in \langle X_3\rangle \}$. Let, for all $x=(x_1, x_2, x_3) \in X:= X_1\times X_2\times X_3$,

$$\begin{aligned} P_1(x)= & {} \left\{ \begin{array}{ll}{} [0, \frac{x_1}{3})&{} \quad \mathrm{if}\quad x_1\in (0, 2],\\ {}0&{} \quad \mathrm{if}\quad x_1=0,\\ \end{array} \right. \quad Q_1(x)=\{0, 1\}, \\ P_2(x)= & {} \left\{ \begin{array}{ll}{} 0&{} \quad \mathrm{if}\quad x_2=0,\\ {}[0, x_2)&{} \quad \mathrm{if}\quad x_2\in (0, 2],\\ \end{array} \right. \quad Q_2(x)=\{0, x_2\}, \\ P_3(x)= & {} \left\{ \begin{array}{ll}{} 0&{} \quad \mathrm{if}\quad x_3=0,\\ {}[0, \frac{x_3}{3})&{} \quad \mathrm{if}\quad x_3\in (0, 2],\\ \end{array} \right. \quad Q_3(x)=\left[ 0, \frac{x_3}{3}\right] \cup \left[ \frac{2x_3}{3}, 2\right] . \end{aligned}$$

For $x=(x_1, x_2, x_3)\in X$, $\varphi _{N_1}\in {\varPhi }_{X_1}$ and $M_1\subset N_1$, we see the following string of equivalent statements

$$\begin{aligned} \left\{ \begin{array}{ll}{}M_1\subset P_1(x),\\ {} x_1\in \varphi _{N_1}(\Delta _{M_1})\\ \end{array} \right.\Longleftrightarrow & {} \left\{ \begin{array}{ll}{}M_1\subset P_1(x),\\ {}x_1\in \mathrm{co}M_1 \\ \end{array} \right. \\\Longleftrightarrow & {} \left\{ \begin{array}{ll}{}M_1\subset P_1(x)=\{0\},\\ {}x_1=0, \\ {}x_1\in [\min M_1, \max M_1]\\ \end{array} \right. \mathrm{or}\,\left\{ \begin{array}{ll}{}x_1\in (0, 2],\\ {} x_1\in [\min M_1, \max M_1],\\ {}M_1\subset P_1(x)=[0, \frac{x_1}{3})\\ \end{array} \right. \\\Longleftrightarrow & {} \left\{ \begin{array}{ll}{}x_1=0,\\ {}M_1=\{0\}.\\ \end{array} \right. \end{aligned}$$

Then, $\varphi _{N_1}(\Delta _{M_1}) =\mathrm{co}\{0\} \subset Q_1(x)$. Thus, $(P_1, Q_1)$ satisfies condition (h1). $(P_1, Q_1)$ does not fulfill condition (h2) because, for $x=(1, 1, 1)$ and $N_1=M_1=\{0, \frac{1}{4}\}$ satisfying $M_1\subset P_1(x)$, but $\varphi _{N_1}(\Delta _{M_1}) =\mathrm{co}\{0,\frac{1}{4}\} \not \subset Q_1(x)$.

We easily see that condition (h2) holds for $(P_2, Q_2)$, but condition (h3) does not, because ${\varPhi }_{X_2}$-$\mathrm{co}P_2(x) =[0, x_2)\not \subset \{ 0, x_2\} =Q_2(x)$ for all $x=(x_1, x_2, x_3) \in X_1\times (0, 2] \times X_3$.

For $(P_3, Q_3)$, we see that the values of $P_3$ are ${\varPhi }_{X_3}$-convex and $P_3(x)\subset Q_3(x)$ for all $x\in X$, i.e., condition (h3) holds, while condition (h4) does not because the values of $Q_3$ are not ${\varPhi }_{X_3}$-convex.

Finally, since $(P_i, Q_i)$ ($i=1, 2, 3$) satisfy at least one of conditions (h1)-(h4), $\{Q_1, Q_2, Q_3\}$ is $\{{\varPhi }_{X_1}, {\varPhi }_{X_2}, {\varPhi }_{X_3}\}$- weak-convex w.r.t. $\{P_1, P_2, P_3\}$.

Example 2

Let $X_1 =X_2= [0, 1]$, $\mathscr {F}_1:=({\varPhi }_{X_1}, \mathfrak {I}_{X_1})=\mathscr {F}_2:=({\varPhi }_{X_2}, \mathfrak {I}_{X_2})$ be the natural KKM-structure of [0, 1]. Let $P_1, Q_1 : X:=X_1\times X_2\rightrightarrows X_1$ and $P_2, Q_2 : X\rightrightarrows X_2$ defined by, for all $x=(x_1, x_2) \in X$,

$$\begin{aligned} P_1(x)=Q_2(x) =[0, x_2], \quad Q_1(x)=P_2(x) = [0, x_1]. \end{aligned}$$

For all $x=(x_1, x_2)\in X$, $\varphi _{N_1}\in {\varPhi }_{X_1}$, $\varphi _{N_2}\in {\varPhi }_{X_2}$, $M_1\subset N_1$ and $M_2\subset N_2$, we have the equivalent assertions:

$$\begin{aligned} \left\{ \begin{array}{ll}{}M_1\subset P_1(x),\\ {} x_1\in \varphi _{N_1}(\Delta _{M_1}),\\ {} M_2\subset P_2(x),\\ {} x_2\in \varphi _{N_2}(\Delta _{M_2})\\ \end{array} \right.\Longleftrightarrow & {} \left\{ \begin{array}{ll}{}M_1\subset [0, x_2],\\ {} x_1\in [\min M_1, \max M_1],\\ {} M_2\subset [0, x_1],\\ {} x_2\in [\min M_2, \max M_2]\\ \end{array} \right. \\\Longleftrightarrow & {} \left\{ \begin{array}{ll}{}\max M_1\le x_2,\\ {} x_1\in [\min M_1, \max M_1],\\ {} \max M_2 \le x_1,\\ {} x_2\in [\min M_2, \max M_2]\\ \end{array} \right. \\\Longleftrightarrow & {} x_1=x_2 =\max M_1 =\max M_2. \end{aligned}$$

Then, $\varphi _{N_1}(\Delta _{M_1})=[\min M_1, \max M_1] \subset [0, x_1] = Q_1(x)$ and $\varphi _{N_2}(\Delta _{M_2})=[\min M_2, \max M_2] \subset [0, x_2] = Q_2(x)$. Thus, $\{Q_1, Q_2\}$ is $\{{\varPhi }_{X_1}, {\varPhi }_{X_2}\}$-weak-convex w.r.t. $\{P_1, P_2\}$. For $N_1=\{0, \frac{1}{2}, 1\}$, $M_1 =\{0, 1\}$ and $x=(0, 1)$, we see that $x_1=0\in [0, 1]=\varphi _{N_1}(\Delta _{M_1})$ and $M_1\subset [0, 1]=P_1(x)$, but $\varphi _{N_1}(\Delta _{M_1})=[0, 1]\not \subset \{0\}=Q_1(x)$. Thus, condition (h1) does not hold for $(P_1, Q_1)$ (hence conditions (h2)-(h4) not either). We also check easily that condition (h1) does not hold for $(P_2, Q_2)$.

From now on, for an index set I, nonempty sets $X_i$ ($i\in I$), $X:=\prod _{i\in I}X_i$, and KKM-structures $\mathscr {F}_i:= ({\varPhi }_{X_i}, \mathfrak {I}_{X_i})$ of $X_i$, if not otherwise stated, $\mathfrak {I}_X$ denotes the Tikhonov product topology of topologies $\mathfrak {I}_{X_i}$ on X.

Coercivity condition Let $\{X_i\}_{i\in I}$ be a family of nonempty sets, $X:=\prod _{i\in I}X_i$, $\mathscr {F}_i= ({\varPhi }_{X_i}, \mathfrak {I}_{X_i})$ a KKM-structure of $X_i$, and $P_i: X\rightrightarrows X_i$. The following condition is a coercivity condition for the family $\{P_i\}_{i\in I}$

• (C) there exists a nonempty $\mathfrak {I}_{X}$-compact subset K of X and, for each $N_i\in \langle X_i\rangle$, there is a $\mathfrak {I}_{X_i}$-compact and ${\varPhi }_{X_i}$-convex subset $L_{N_i}\subset X_i$ containing $N_i$ such that, for each $x\in (\prod _{i\in I}L_{N_i})\backslash K$ and $i\in I$, there exists $x'_i\in L_{N_i}$ with $x\in \mathrm{int}_{ \mathfrak {I}_{X}}P_i^{-1}(x'_i)$.

Let I, $X_i$, X and $Q_i$ be as in Definition 3. A point $\bar{x} := (\bar{x}_i)_{i\in I}\in X$ is called a fixed-component point of the family $\{Q_i\}_{i\in I}$ if $\bar{x}_i \in Q_i(\bar{x})$ for all $i \in I$.

Theorem 1

For an arbitrary index set I and $i\in I$, let $X_i$ be a nonempty set and $Q_i: X:=\prod _{i\in I}X_i\rightrightarrows X_i$. Then, the family $\{Q_i\}_{i\in I}$ has a fixed-component point if and only if, for each $i\in I$, there exist a KKM-structure $\mathscr {F}_i= ({\varPhi }_{X_i}, \mathfrak {I}_{X_i})$ of $X_i$ and a set-valued mapping $P_i: X\rightrightarrows X_i$ such that, for each $i\in I$, $X=\bigcup _{x_i\in X_i}\mathrm{int}_{ \mathfrak {I}_{X}}P_i^{-1}(x_i)$, $\{Q_i\}_{i\in I}$ is $\{{\varPhi }_{X_i}\}_{i\in I}$-weak-convex w.r.t. $\{P_i\}_{i\in I}$, and condition $\mathrm{(C)}$ holds for $\{P_i\}_{i\in I}$.

Proof

Necessity. Suppose that $\bar{x}:=(\bar{x}_i)_{i\in I}\in X$ is a fixed-component point of $\{Q_i\}_{i\in I}$. For $i\in I$, let $P_i: X\rightrightarrows X_i$ be defined by $P_i(x)=\{\bar{x}_i\}$ for all $x\in X$, and a KKM-structure $\mathscr {F}_i$ of $X_i$ defined as follows: $\mathfrak {I}_{X_i} = \{U\subset X_i\mid \bar{x}_i\notin U\}\cup \{X_i\}$ and

$$\begin{aligned} {\varPhi }_{X_i} = \big \{\varphi _{N_i}:\Delta _{|N_i|} \rightarrow X_i|\quad \varphi _{N_i}(e)= \bar{x}_i\quad \mathrm{for\quad all}\quad e\in \Delta _{|{N_i}|}, N_i\in \langle X_i\rangle \big \}. \end{aligned}$$

For each $i\in I$, the condition $X=\bigcup _{x_i\in X_i}\mathrm{int}_{ \mathfrak {I}_{X}}P_i^{-1}(x_i)$ is obviously satisfied. Since each $\mathfrak {I}_{X_i}$ is clearly a compact topology, $\mathfrak {I}_X$ is compact too. Therefore, condition (C) is satisfied with $K=X$ and $L_{N_i}=X_i$ for all $N_i\in \langle X_i\rangle$. For all $x:=(x_i)_{i\in I}\in X$, $\varphi _{N_i}\in {\varPhi }_{X_i}$ and $M_i\subset N_i$, we obtain the equivalent statements:

$$\begin{aligned} \left\{ \begin{array}{ll}{}M_i\subset P_i(x),\\ {}x_i\in \varphi _{N_i}(\Delta _{M_i}),\\ {}i\in I\\ \end{array} \right.\Longleftrightarrow & {} \left\{ \begin{array}{ll}{}M_i\subset \{\bar{x}_i\},\\ {}x_i\in \{\bar{x}_i\},\\ {}i\in I \\ \end{array} \right. \\\Longleftrightarrow & {} x=\bar{x}=(\bar{x}_i)_{i\in I}\quad \mathrm{and}\quad M_i=\{\bar{x}_i\}\quad \mathrm{for\quad all}\quad i\in I. \end{aligned}$$

Then, $\varphi _{N_i}(\Delta _{M_i}) =\{\bar{x}_i\}\subset Q_i(x)$ for all $i\in I$. Hence, $\{Q_i\}_{i\in I}$ is $\{{\varPhi }_{X_i}\}_{i\in I}$-convex w.r.t. $\{P_i\}_{i\in I}$. Thus, the proof of the necessity is complete.

Sufficiency. Assume that, for each $i\in I$, there exist a KKM-structure $\mathscr {F}_i$ of $X_i$ and a set-valued mapping $P_i: X\rightrightarrows X_i$ such that the conditions mentioned in Theorem 1 hold. For each $i\in I$, since $X=\bigcup _{x_i\in X_i}\mathrm{int}_{ \mathfrak {I}_{X}}P_i^{-1}(x_i)$, there exists $N_i\in \langle X_i\rangle$ such that $K\subset \bigcup _{x_i\in N_i}\mathrm{int}_{ \mathfrak {I}_{X}}P_i^{-1}(x_i)$ (K is given by condition (C)). For each $i\in I$, by (C) there is a $\mathfrak {I}_{X_i}$-compact and ${\varPhi }_{X_i}$-convex subset $L_{N_i}\subset X_i$ containing $N_i$ such that $L_N\backslash K\subset \bigcup _{x_i\in L_{N_i}}\mathrm{int}_{ \mathfrak {I}_{X}}P_i^{-1}(x_i)$, where $L_N:= \prod _{i\in I}L_{N_i}$. Hence, $L_N \subset (L_N\backslash K)\cup K\subset$$\bigcup _{x_i\in L_{N_i}}\mathrm{int}_{ \mathfrak {I}_{X}}P_i^{-1}(x_i)$. Since $L_N$ is compact, there exists $\overline{N}_i:=\{x_i^0, x_i^1,\ldots , x_i^{n_i}\}\in \langle L_{N_i}\rangle$ such that $L_N = \bigcup _{j=0}^{n_i}\mathrm{int}_{ \mathfrak {I}_{X}}P_i^{-1}(x_i^j)$. For each $i\in I$, let $\{\psi _j\}_{j=0}^{n_i}$ be a continuous partition of unity of $L_N$ associated with the finite open covering $\{\mathrm{int}_{ \mathfrak {I}_{X}}P_i^{-1}(x_i^j)\}_{j=0}^{n_i}$. Then, for each $x\in L_N$ and $k\in J_i(x) := \{j\in \{0, 1,\ldots , n_i\}| \psi _j(x)\ne 0\}$, one has $x\in \mathrm{int}_{ \mathfrak {I}_{X}}P_i^{-1}(x_i^k)\subset P_i^{-1}(x_i^k)$. Therefore,

$$\begin{aligned} \overline{M}_i(x): = \left\{ x_i^k|\, k\in J_i(x)\right\} \subset \overline{N}_i\cap P_i(x)\,\mathrm{for\, all}\, x\in L_N. \end{aligned}$$

(1)

Now, for each $i\in I$, we define a map $\gamma _i: L_N\rightarrow \Delta _{|\overline{N}_i|}$ by $\gamma _i(x)=\sum _{j=0}^{n_i}\psi _j(x)e_j$ for all $x\in L_N$. We have, for all $x\in L_N$,

$$\begin{aligned} \varphi _{\overline{N}_i}(\gamma _i(x))=\varphi _{\overline{N}_i}\left( \sum _{j=0}^{n_i}\psi _j(x)e_j\right) =\varphi _{\overline{N}_i}\left( \sum _{k\in J_i(x)}\psi _k(x)e_k\right) \in \varphi _{\overline{N}_i}\left( \Delta _{\overline{M}_i(x)}\right) . \end{aligned}$$

(2)

Let ${\varOmega }= \prod _{i\in I}\Delta _{|\overline{N}_i|}$. Then, ${\varOmega }$ is a compact convex subset of the locally convex space $\mathbb {R}^I = \prod _{i\in I}\mathbb {R}^{n_i +1}$. Let ${\varGamma }: L_N\rightarrow {\varOmega }$ and ${\varPsi }:{\varOmega }\rightarrow L_N$ be defined by

• ${\varGamma }(x)=(\gamma _i(x))_{i\in I}$ for all $x\in L_N$, and

• ${\varPsi }(t)=(\varphi _{\overline{N}_i}(p_i(t)))_{i\in I}$ for all $t\in {\varOmega }$,

where $p_i(t)$ is the projection of t on $\Delta _{|\overline{N}_i|}$. Then, ${\varGamma }$ and ${\varPsi }$ are continuous and so is ${\varGamma }\circ {\varPsi }: {\varOmega }\rightarrow {\varOmega }$. By the Tikhonov fixed point theorem (see [32]), a $\bar{t}\in {\varOmega }$ exists such that $\bar{t}=({\varGamma }\circ {\varPsi })(\bar{t})$. Setting $\bar{x} =(\bar{x}_i)_{i\in I} = {\varPsi }(\bar{t})=(\varphi _{\overline{N}_i}(p_i(\bar{t})))_{i\in I}$, we have $\bar{t}=(p_i(\bar{t}))_{i\in I}={\varGamma }(\bar{x})=(\gamma _i(\bar{x}))_{i\in I}$. Then, $\bar{x}_i=\varphi _{\overline{N}_i}(\gamma _i(\bar{x}))$ for all $i\in I$, and by (2),

$$\begin{aligned} \bar{x}_i=\varphi _{\overline{N}_i}(\gamma _i(\bar{x}))\in \varphi _{\overline{N}_i}\Big (\Delta _{\overline{M}_i(\bar{x})}\Big )\,\mathrm{for\, all}\, i\in I. \end{aligned}$$

(3)

Then, (1) gives

$$\begin{aligned} \overline{M}_i(\bar{x}) \subset \overline{N}_i\cap P_i(\bar{x})\,\mathrm{for\, all}\, i\in I. \end{aligned}$$

(4)

Since $\{Q_i\}_{i\in I}$ is $\{{\varPhi }_{X_i}\}_{i\in I}$-convex w.r.t. $\{P_i\}_{i\in I}$, (3) and (4) imply that $\bar{x}_i\in$$\varphi _{\overline{N}_i}(\Delta _{\overline{M}_i(\bar{x})})$$\subset Q_i(\bar{x})$ for all $i\in I$. The proof is complete. $\square$

Remark 2

• (a) For each $i\in I$, the condition $X=\bigcup _{x_i\in X_i}\mathrm{int}_{ \mathfrak {I}_{X}}P_i^{-1}(x_i)$ in Theorem 1 is satisfied if $P_i$ has the nonempty values and $\mathfrak {I}_X$-open inverse images.

• (b) For I, $X_i$, X and $Q_i$ as in Theorem 1, applying this theorem, condition (h4) and the statement at the end of Remark 1 with $P_i$ and $Q_i$ replaced by $Q_i$ and ${\varPhi }_{X_i}$-co$Q_i(\cdot )$, resp, we obtain

Corollary 1

Let I, $X_i$, X, and $Q_i$ be as in Theorem 1. If, for each $i\in I$, there exists a KKM-structure $\mathscr {F}_i:= ({\varPhi }_{X_i}, \mathfrak {I}_{X_i})$ of $X_i$ such that $X=\bigcup _{x_i\in X_i}\mathrm{int}_{ \mathfrak {I}_{X}}Q_i^{-1}(x_i)$ and condition $\mathrm{(C)}$ holds for $\{Q_i\}_{i\in I}$. Then, the family $\{{\varPhi }_{X_i}$-co$Q_i(\cdot )\}_{i\in I}$ has a fixed-component point.

(c) If each $X_i$ has the natural KKM-structure, we easily check the necessary condition of Theorem 1 with maps $P_i$ taken in the proof of the necessity part of Theorem 1. Therefore, we have a particular case of Theorem 1 as follows.

Theorem 2

For an arbitrary index set I and $i\in I$, let $X_i$ be a nonempty convex set of a topological vector space and $Q_i: X:=\prod _{i\in I}X_i\rightrightarrows X_i$. Then, the family $\{Q_i\}_{i\in I}$ has a fixed-component point if and only if, for each $i\in I$, there exists $P_i: X\rightrightarrows X_i$ such that $X=\bigcup _{x_i\in X_i}\mathrm{int}P_i^{-1}(x_i)$, $\{Q_i\}_{i\in I}$ is weak-convex w.r.t. $\{P_i\}_{i\in I}$, and the following condition holds for $\{P_i\}_{i\in I}$:

(C$'$):

there exists a nonempty compact subset K of X and, for each $N_i\in \langle X_i\rangle$, there is a compact and convex subset $L_{N_i}\subset X_i$ containing $N_i$ such that, for each $x\in \prod _{i\in I}L_{N_i}\backslash K$ and $i\in I$, there exists $x'_i\in L_{N_i}$ with $x\in \mathrm{int}P_i^{-1}(x'_i)$.

(d) Let I, $X_i$, X be as in Theorem 1, $X^i:=\prod _{j\in I, j\ne i}X_j$, and $x^i$ be the canonical projection of $x\in X$ on $X^i$. For each $i\in I$, let $\widetilde{P}_i, \widetilde{Q}_i : X^i \rightrightarrows X_i$ be set-valued maps. We can state Definition 3 for $\{\widetilde{P}_i\}_{i\in I}$ and $\{\widetilde{Q}_i\}_{i\in I}$ and change conditions (h1)-(h4) for the pair $(\widetilde{P}_i, \widetilde{Q}_i)$ in the manner that the phrases “$P_i(x)$,” “$Q_i(x)$,” “for all $x\in X$” and “for each $x\in X$” in Definition 3 are replaced by “$\widetilde{P}_i(x^i)$,” “$\widetilde{Q}_i(x^i)$,” “for all $x^i\in X^i$” and “for each $x^i\in X^i$,” resp. A point $\bar{x}: = (\bar{x}_i)_{i\in I}\in X$ satisfying $\bar{x}_i \in \widetilde{Q}_i(\bar{x}^i)$ for all $i \in I$ is also called a fixed-component point of the family $\{\widetilde{Q}_i\}_{i\in I}$.

The following consequence of Theorem 1 is formulated in terms of the family $\{\widetilde{Q}_i\}_{i\in I}$.

Theorem 3

Let I, $X_i$, X be as in Theorem 1, $X^i:=\prod _{j\in I, j\ne i}X_j$ and $\widetilde{Q}_i : X^i \rightrightarrows X_i$. Then, the family $\{\widetilde{Q}_i\}_{i\in I}$ has a fixed-component point if and only if, for each $i\in I$, there exist a KKM-structure $\mathscr {F}_i= ({\varPhi }_{X_i}, \mathfrak {I}_{X_i})$ of $X_i$ and a set-valued mapping $\widetilde{P}_i: X^i\rightrightarrows X_i$ such that $X^i=\bigcup _{x_i\in X_i}\mathrm{int}_{ \mathfrak {I}_{X^i}}\widetilde{P}_i^{-1}(x_i)$ for each $i\in I$, $\{\widetilde{Q}_i\}_{i\in I}$ is $\{{\varPhi }_{X_i}\}_{i\in I}$-weak-convex w.r.t. $\{\widetilde{P}_i\}_{i\in I}$, and the following condition holds:

$(\widetilde{\mathrm{C}})$ :

there exists a nonempty $\mathfrak {I}_X$-compact subset K of X, and, for each $N_i\in \langle X_i\rangle$, there is a $\mathfrak {I}_{X_i}$-compact and ${\varPhi }_{X_i}$-convex subset $L_{N_i}\subset X_i$ containing $N_i$ such that, for each $x\in (\prod _{i\in I}L_{N_i})\backslash K$ and $i\in I$, there exists $x'_i\in L_{N_i}$ with $x^i\in \mathrm{int}_{ \mathfrak {I}_{X^i}}\widetilde{P}_i^{-1}(x'_i)$.

We derive Theorem 3 from Theorem 1. The necessary condition is easily checked with the KKM-structures $\mathscr {F}_i:= ({\varPhi }_{X_i}, \mathfrak {I}_{X_i})$ taken from the proof of the necessity part of Theorem 1 and maps $\widetilde{P}_i: X^i \rightrightarrows X_i$ defined by $\widetilde{P}_i(x^i) =\{\bar{x}_i\}$ for all $x^i\in X^i$, where $\bar{x}:=(\bar{x}_i)_{i\in I}$ is the fixed-component point of $\{\widetilde{Q}_i\}_{i\in I}$. For the sufficiency, we define maps $P_i, Q_i: X\rightrightarrows X_i$ by $P_i(x) =\widetilde{P}_i(x^i)$ and $Q_i(x) =\widetilde{Q}_i(x^i)$ for all $x\in X$. Then, for each $i\in I$ and $x_i\in X_i$, $P_i^{-1}(x_i)=X_i\times \widetilde{P}_i^{-1}(x_i)$. Hence,

$$\begin{aligned} \bigcup _{x_i\in X_i}\mathrm{int}_{\mathfrak {I}_X}P_i^{-1}(x_i)&= \bigcup _{x_i\in X_i}\mathrm{int}_{\mathfrak {I}_X}\left( X_i\times \widetilde{P}_i^{-1}(x_i)\right) \supset \bigcup _{x_i\in X_i}\left( X_i\times \mathrm{int}_{\mathfrak {I}_{X^i}}\widetilde{P}_i^{-1}(x_i)\right) \\&\supset X_i\times \left( \bigcup _{x_i\in X_i}\mathrm{int}_{\mathfrak {I}_{X^i}}\widetilde{P}_i^{-1}(x_i)\right) = X_i \times X^i = X. \end{aligned}$$

Moreover, it is clear that the $\{{\varPhi }_{X_i}\}_{i\in I}$-weak-convexity w.r.t. $\{\widetilde{P}_i\}_{i\in I}$ of $\{\widetilde{Q}_i\}_{i\in I}$ implies the $\{{\varPhi }_{X_i}\}_{i\in I}$-weak-convexity w.r.t. $\{P_i\}_{i\in I}$ of $\{Q_i\}_{i\in I}$ and the condition ($\widetilde{\mathrm{C}}$) for $\{\widetilde{P}_i\}_{i\in I}$ implies the condition (C) for $\{P_i\}_{i\in I}$. Thus, by Theorem 1, $\{Q_i\}_{i\in I}$ has a fixed-component point $\bar{x}$. This $\bar{x}$ is also a fixed-component point of $\{\widetilde{Q}_i\}_{i\in I}$.

Conversely, Theorem 3 implies Theorem 1 or not is still an open question for us, though this looks likely.

We can suitably modify the proof of Theorem 1 to obtain Theorem 3 with the condition ($\widetilde{\mathrm{C}}$) replaced by the following weaker condition:

($\widehat{\mathrm{C}}$):

for each $i\in I$, there exists a nonempty $\mathfrak {I}_{X^i}$-compact subset $K^i$ of $X^i$ and, for each $N_i\in \langle X_i\rangle$, there is a $\mathfrak {I}_{X_i}$-compact and ${\varPhi }_{X_i}$-convex subset $L_{N_i}\subset X_i$ containing $N_i$ such that there exists $x'_i\in L_{N_i}$ with $x^i\in \mathrm{int}_{\mathfrak {I}_{X^i}}\widetilde{P}_i^{-1}(x'_i)$ for each $x^i\in (\prod _{j\in I, j\ne i}L_{N_i})\backslash K^i$.

(e) The sufficiency part of Theorem 1 implies Theorems 3.1–3.4 of [9]. Hence, it also implies Theorem 3.2 of [7], Theorem 7 of [34], Theorems 2.3 and 3.1 of [31], Theorem 2.1 of [28], and Theorem 2.2 of [4]. We also deduce Theorem 3.1 of [17] and thus Theorem 1 of [1] and Theorem 2.1 of [21] from this part of Theorem 1. When I is a singleton, Theorem 1 becomes Theorem 2.5 of [19] and hence its sufficiency part generalizes many fixed point theorems, including the classical Browder fixed point theorem in [4] (which implies the seminal Kakutani fixed point theorem in [13]) and Tarafdar’s fixed point theorem in [30], etc. (cf. [19]).

(f) To illustrate Theorem 1, we revisit Examples 1 and 2.

For Example 1, we see that

$$\begin{aligned} P_1^{-1}(x_1)= & {} \left\{ \begin{array}{ll}{} X_1\times X_2\times X_3&{} \quad \mathrm{if}\quad x_1=0,\\ {}(3x_1, 2]\times X_2\times X_3&{} \quad \mathrm{if}\quad 0<x_1\le \frac{2}{3}, \\ {}\emptyset &{} \quad \mathrm{if}\quad \frac{2}{3}<x_1\le 2,\\ \end{array} \right. \quad \mathrm{for\quad all}\quad x_1 \in X_1, \\ P_2^{-1}(x_2)= & {} \left\{ \begin{array}{ll}{} X_1\times X_2\times X_3&{} \quad \mathrm{if}\quad x_2=0,\\ {}X_1\times (x_2, 2]\times X_3&{} \quad \mathrm{if}\quad x_2\in (0, 2],\\ \end{array} \right. \quad \quad \mathrm{for\quad all}\quad x_2 \in X_2, \\ P_3^{-1}(x_3)= & {} \left\{ \begin{array}{ll}{} X_1\times X_2\times X_3&{} \quad \mathrm{if}\quad x_3=0,\\ {}X_1\times X_2\times (3x_3, 2] &{} \quad \mathrm{if}\quad 0<x_3<\frac{2}{3}, \\ {}\emptyset &{} \quad \mathrm{if}\quad \frac{2}{3}\le x_3\le 2,\\ \end{array} \right. \quad \mathrm{for\quad all}\quad x_3 \in X_3. \end{aligned}$$

Clearly, the maps $P_1$, $P_2$, $P_3$ have the nonempty values and open fibers. Moreover, $X= X_1\times X_2\times X_3$ is compact. Thus, by Theorem 1, $\{Q_1, Q_2, Q_3\}$ has a fixed-component point.

For Example 2, we have $P_1^{-1}(x_1) =X_1\times [x_1, 1]$ for all $x_1\in X_1$, and $P_2^{-1}(x_2) = [x_2, 1]\times X_2$ for all $x_2\in X_2$. It is clear that X is compact, and $\bigcup _{x_i\in X_i}\mathrm{int}_{\mathfrak {I}_X}P_i^{-1}(x_i)=X$ for $i\in \{1, 2\}$. Thus, $\{Q_1, Q_2\}$ has a fixed-component point.

Remark 3

Using Theorem 1, we deduce a result on the existence of common fixed points. Common fixed point theorems were studied by many authors (e.g., [2, 3, 5]). Let I be any index set, A be a nonempty set and $\{T_i: A\rightrightarrows A,\,\, i\in I\}$ be a family of set-valued maps. A point $\bar{a} \in A$ is called a common fixed point of the family $\{T_i\}_{i\in I}$ if $\bar{a} \in T_i(\bar{a})$ for all $i \in I$.

Let $\mathscr {F}=({\varPhi }_{A}, \mathfrak {I}_{A})$ be a KKM-structure of A and $P: A\rightrightarrows A$ be a set-valued map. We say that the family $\{T_i\}_{i\in I}$ is ${\varPhi }_{A}$-weak-convex with respect to (w.r.t.) P if whenever $a\in A$, $\varphi _{N}\in {\varPhi }_{A}$, and $M\subset N\cap P(a)$ satisfying $a\in \varphi _{N}(\Delta _{M})$, one has $\varphi _{N}(\Delta _{M})\subset T_i(a)$ for each $i\in I$. When this notion holds with the natural KKM-structure, we say that $\{T_i\}_{i\in I}$ is weak-convex w.r.t. P.

Theorem 4

Let A be a nonempty set and I be an arbitrary index set. For each $i\in I$, let $T_i: A\rightrightarrows A$. Then, the family $\{T_i\}_{i\in I}$ has a common fixed point if and only if there exist a KKM-structure $\mathscr {F}= ({\varPhi }_{A}, \mathfrak {I}_{A})$ of A and a set-valued mapping $P: A\rightrightarrows A$ such that $A=\bigcup _{a\in A}\mathrm{int}_{ \mathfrak {I}_{A}}P^{-1}(a)$, $\{T_i\}_{i\in I}$ is ${\varPhi }_{A}$-weak-convex w.r.t. P, and the following condition holds:

• (D) there exists a nonempty $\mathfrak {I}_{A}$-compact subset K of A and, for each $N\in \langle A\rangle$, there is a $\mathfrak {I}_{A}$-compact and ${\varPhi }_{A}$-convex subset $L_{N}\subset A$ containing N such that, for each $a\in L_{N}\backslash K$, there exists $a'\in L_{N}$ with $a\in \mathrm{int}_{ \mathfrak {I}_{A}}P^{-1}(a')$.

This theorem is deduced from Theorem 1 by setting $X_i = A$, $X=\prod _{i\in I}X_i=A^I$, $P_i(x)=P(x_i)$, and $Q_i(x)=\bigcap _{j\in I}T_j(x_i)$.

Example 3

Let $A= [0, 1]$, $\mathscr {F}$ be the natural KKM-structure of [0, 1]. Let $T_1, T_2, T_3 : A\rightrightarrows A$ be defined by, for all $a \in A$,

$$\begin{aligned} T_1(a) =\left[ 0, \frac{a}{3}\right] ,\quad T_2(a) = \left[ \frac{2a}{3}, 1\right] ,\quad T_3(a) = \left[ \frac{a}{3}, \frac{2a}{3}\right] . \end{aligned}$$

For $P: A\rightrightarrows A$ defined by $P(0) =\{0\}$ and $P(a)= [0, \frac{a}{4})$ if $a\ne 0$, we have: for all $a\in A$, $\varphi _{N}\in {\varPhi }_{A}$ and $M\subset N$,

$$\begin{aligned} \left\{ \begin{array}{ll}{}M\subset P(a),\\ {} a\in \varphi _{N}(\Delta _{M})\\ \end{array} \right.\Longleftrightarrow & {} \left\{ \begin{array}{ll}{}M\subset P(0)=\{0\},\\ {}a=0\\ \end{array} \right. \mathrm{or}\,\left\{ \begin{array}{ll}{}M\subset [0, \frac{a}{4}],\\ {} a\in [\min M, \max M]\\ \end{array} \right. \\\Longleftrightarrow & {} \left\{ \begin{array}{ll}{}M=\{0\},\\ {} a=0.\\ \end{array} \right. \end{aligned}$$

Then, $\varphi _{N}(\Delta _{M})=\{0\} \subset T_i(0)$ ($i=1,2,3$). Thus, $\{T_1, T_2, T_3\}$ is weak-convex w.r.t. P. Moreover, P has the nonempty values and fibers, for $a\in A$,

$$\begin{aligned} P^{-1}(a) = \left\{ \begin{array}{ll}{} A&{} \quad \mathrm{if}\quad a=0,\\ {}(4y, 1]&{} \quad \mathrm{if}\quad 0<a\le \frac{1}{4}, \\ {}\emptyset &{} \quad \mathrm{if}\quad \frac{1}{4}<a\le 1\\ \end{array} \right. \end{aligned}$$

are open in A. Hence, $A=\bigcup _{a\in A}\mathrm{int}P^{-1}(a)$. Since A is compact, the condition (D) of Theorem 4 holds. Thus, by Theorem 4, there exists $\bar{a}\in A$ such that $\bar{a}\in T_i(\bar{a})$ ($i=1,2,3$).

3 Applications

Since fixed point theorems have a wide range of applications and fixed-component point results imply corresponding ones for fixed points by considering a family consisting of one element, we can deduce many applications from the results obtained in Sect. 2. Here, we focus on two kinds of applications only. First, we study the existence of some important points which were mentioned in Sect. 1 and seemingly more general than fixed points, to emphasize the generality of our results. Then, we show that from these results the existence of solutions to numerous optimization-related problems can be obtained.

3.1 Coincidence-Component Point Theorems

Theorem 5

For each $i\in I$, let $X_i$, $Y_i$ be nonempty sets, $X:=\prod _{i\in I}X_i$, $Y:=\prod _{i\in I}Y_i$, and $F_i: X\rightrightarrows Y_i$ and $G_i: Y\rightrightarrows X_i$ be nonempty-valued. Assume that, for each $i\in I$, there exist KKM-structures $\mathscr {F}_i:= ({\varPhi }_{X_i}, \mathfrak {I}_{X_i})$ of $X_i$ and $\mathscr {G}_i:= ({\varPhi }_{Y_i}, \mathfrak {I}_{Y_i})$ of $Y_i$ such that

• (i) for each $(x_i, y_i)\in X_i\times Y_i$, $F_i^{-1}(y_i)$ and $G_i^{-1}(x_i)$ are $\mathfrak {I}_X$-open and $\mathfrak {I}_Y$-open, resp;

• (ii) for each $(x, y)\in X\times Y$, $F_i(x)$ and $G_i(y)$ are ${\varPhi }_{Y_i}$-convex and ${\varPhi }_{X_i}$-convex, resp;

• (iii) there exists a nonempty $\mathfrak {I}_{X\times Y}$-compact subset K of $X\times Y$ and, for each $N_i\in \langle X_i\times Y_i\rangle$, there is a $\mathfrak {I}_{X_i\times Y_i}$-compact and ${\varPhi }_{X_i\times Y_i}$-convex subset $L_{N_i}\subset X_i\times Y_i$ containing $N_i$ such that, for each $(x,y)\in (\prod _{i\in I}L_{N_i})\backslash K$ and $i\in I$, there exists $(x'_i, y'_i)\in L_{N_i}$ such that $(x,y)\in F_i^{-1}(y'_i)\times G_i^{-1}(x'_i)$.

Then, there exists $\bar{x}:=(\bar{x}_i)_{i\in I}\in X$ and $\bar{y}:=(\bar{y}_i)_{i\in I}\in Y$ such that $\bar{y}_i \in F_i(\bar{x})$ and $\bar{x}_i \in G_i(\bar{y})$ for all $i\in I$.

Proof

For each $i\in I$, let $\mathscr {H}_i:=({\varPhi }_{X_i\times Y_i}, \mathfrak {I}_{X_i\times Y_i})$ be the product KKM-structure of $\mathscr {F}_i$ and $\mathscr {G}_i$ on $X_i\times Y_i$. Let $D_i: X\times Y\rightrightarrows X_i\times Y_i$ be defined by $D_i(x, y):= G_i(y)\times F_i(x)$ for all $(x, y)\in X\times Y$. Then, $D_i$ has the nonempty values, and $D_i^{-1}(x_i, y_i) =F_i^{-1}(y_i)\times G_i^{-1}(x_i)$ is $\mathfrak {I}_{X\times Y}$-open for all $(x_i, y_i)\in X_i\times Y_i$. Hence, $X\times Y=\bigcup _{(x_i,y_i)\in X_i\times Y_i}\mathrm{int}_{\mathfrak {I}_{X\times Y}}D_i^{-1}(x_i, y_i)$. Assumption (ii) implies that $D_i(x, y)$ is ${\varPhi }_{X_i\times Y_i}$-convex for each $(x, y)\in X \times Y$. Assumption (iii) shows that condition (C) holds for $\{D_i\}_{i\in I}$. Thus, applying Theorem 1 together with condition (h4) and the statement at the end of Remark 1 with $X\times Y$, $X_i\times Y_i$ and $D_i$ replacing X, $X_i$, $P_i\equiv Q_i$, resp, we have a $(\bar{x},\bar{y})\in X\times Y$ such that $(x_i, y_i)\in D_i(x, y)$ for all $i\in I$, i.e., $\bar{y}_i \in F_i(\bar{x})$ and $\bar{x}_i \in G_i(\bar{y})$ for all $i\in I$.$\square$

3.2 Maximal Element Theorems

Theorem 6

For an arbitrary index set I and $i\in I$, let $X_i$ be a nonempty set and $S_i, T_i: X:=\prod _{i\in I}X_i\rightrightarrows X_i$. Assume that, for each $i\in I$, there exists a KKM-structure $\mathscr {F}_i:= ({\varPhi }_{X_i}, \mathfrak {I}_{X_i})$ of $X_i$ such that

• (i) for each $x_i\in X_i$, $S_i^{-1}(x_i)$ is $\mathfrak {I}_X$-open;

• (ii) for each $x\in X$, ${\varPhi }_{X_i}$-co$S_i(x)\subset T_i(x)$;

• (iii) there exists $i\in I$ such that $x_i\notin T_i(x)$ for all $x=(x_i)_{i\in I}\in X$;

• (iv) there exists a nonempty $\mathfrak {I}_{X}$-compact subset K of X, and, for each $N_i\in \langle X_i\rangle$, there is a $\mathfrak {I}_{X_i}$-compact and ${\varPhi }_{X_i}$-convex subset $L_{N_i}\subset X_i$ containing $N_i$ such that, for each $x\in (\prod _{i\in I}L_{N_i})\backslash K$ and $i\in I$, $S_i(x) \cap L_{N_i}\ne \emptyset$.

Then, there exist $\bar{x}\in X$ and $i_0\in I$ such that $S_{i_0}(\bar{x})=\emptyset$.

Proof

Suppose to the contrary that, for all $x\in X$ and $i\in I$, $S_{i}(x)\ne \emptyset$. Then, this and (i) imply that $X=\bigcup _{x_i\in X_i}S_i^{-1}(x_i) = \bigcup _{x_i\in X_i}\mathrm{int}_{\mathfrak {I}_X}S_i^{-1}(x_i)$ for each $i\in I$. (iv) together with (i) ensures condition (C) for $\{S_i\}_{i\in I}$. Applying Theorem 1 for $\{P_i\equiv S_i\}_{i\in I}$ and $\{Q_i\equiv T_i\}_{i\in I}$ via condition (h3) of Remark 1 and the statement at the end of this remark, we have $\bar{x}: =(\bar{x}_i)_{i\in I}\in X$ such that $\bar{x}_i\in T_i(\bar{x})$ for all $i\in I$, contradicting (iii).$\square$

Corollary 2

Let X be a nonempty set and $S, T: X \rightrightarrows X$. Assume that there exists a KKM-structure $\mathscr {F}:= ({\varPhi }_{X}, \mathfrak {I}_{X})$ of X such that

• (i) for each $x\in X$, $S^{-1}(x)$ is $\mathfrak {I}_X$-open, ${\varPhi }_{X}$-co$S(x)\subset T(x)$, and $x\notin T(x)$;

• (ii) there exists a nonempty $\mathfrak {I}_{X}$-compact subset K of X and, for each $N\in \langle X\rangle$, there is a $\mathfrak {I}_{X}$-compact and ${\varPhi }_{X}$-convex subset $L_{N}\subset X$ containing N such that $S(x) \cap L_{N}\ne \emptyset$ for each $x\in L_{N}\backslash K$.

Then, there exists $\bar{x}\in X$ such that $S(\bar{x})=\emptyset$.

Proof

This result is a special case of Theorem 6 with the index set I being a singleton.

$\square$

3.3 Intersection Point Theorems

Theorem 7

Let $\{X_i\}_{i\in I}$ be a family of nonempty sets, $X:=\prod _{i\in I}X_i$, $\{A_i\}_{i\in I}$, $\{B_i\}_{i\in I}$ two families of nonempty subsets of X, and $x^i$ the canonical projection of x on $X^i:=\prod _{j\in I, j\ne i}X_j$. Assume that, for each $i\in I$, there exists a KKM-structure $\mathscr {F}_i:= ({\varPhi }_{X_i}, \mathfrak {I}_{X_i})$ of $X_i$ such that

• (i) for each $x_i\in X_i$, $\{x'\in X| (x_i, x'^i)\in B_i\}$ is $\mathfrak {I}_X$-open;

• (ii) for each $x\in X$, $\{x_i'\in X_i| (x_i', x^i)\in B_i\}$ is nonempty, and ${\varPhi }_{X_i}$-co$\{x_i'\in X_i| (x_i', x^i)\in B_i\}\subset \{x_i'\in X_i| (x_i', x^i)\in A_i\}$;

• (iii) there exists a nonempty $\mathfrak {I}_{X}$-compact subset K of X and, for each $N_i\in \langle X_i\rangle$, there is a $\mathfrak {I}_{X_i}$-compact and ${\varPhi }_{X_i}$-convex subset $L_{N_i}\subset X_i$ containing $N_i$ such that, for each $x\in (\prod _{i\in I}L_{N_i})\backslash K$ and $i\in I$, there exists $x'_i\in L_{N_i}$ such that $(x'_i,x^i)\in B_i$.

Then, $\bigcap _{i\in I}A_i\ne \emptyset$

Proof

For each $i\in I$, we define $P_i, Q_i: X\rightrightarrows X_i$ by $P_i(x):=\{x_i'\in X_i| (x_i', x^i)\in B_i\}$ and $Q_i(x):=\{x_i'\in X_i| (x_i', x^i)\in A_i\}$ for all $x\in X$. It is clear that the assumptions of Theorem 1 (under the condition (h3)) are satisfied for $\{P_i\}_{i\in I}$ and $\{Q_i\}_{i\in I}$. Applying this theorem, we have $\bar{x} := (\bar{x}_i)_{i\in I}\in X$ such that $\bar{x}_i \in \{x_i'\in X_i| (x_i', \bar{x}^i)\in A_i\}$ for each $i \in I$, i.e., $\bar{x}=(\bar{x}_i, \bar{x}^i)\in A_i$ for all $i\in I$.$\square$

Corollary 3

Let $\{X_i\}_{i\in I}$ be a family of nonempty sets, $X:=\prod _{i\in I}X_i$, $\{A_i\}_{i\in I}$ a family of nonempty subsets of X, and $x^i$ the canonical projection of x on $X^i:=\prod _{j\in I, j\ne i}X_j$. Assume that, for each $i\in I$, there exists a KKM-structure $\mathscr {F}_i= ({\varPhi }_{X_i}, \mathfrak {I}_{X_i})$ of $X_i$ such that

• (i) for each $x_i\in X_i$, $\{x'\in X| (x_i, x'^i)\in A_i\}$ is $\mathfrak {I}_X$-open;

• (ii) for each $x\in X$, $\{x_i'\in X_i| (x_i', x^i)\in A_i\}$ is nonempty and ${\varPhi }_{X_i}$-convex;

• (iii) there exists a nonempty $\mathfrak {I}_{X}$-compact subset K of X and, for each $N_i\in \langle X_i\rangle$, there is a $\mathfrak {I}_{X_i}$-compact and ${\varPhi }_{X_i}$-convex subset $L_{N_i}\subset X_i$ containing $N_i$ such that, for each $x\in (\prod _{i\in I}L_{N_i})\backslash K$ and $i\in I$, there exists $x'_i\in L_{N_i}$ such that $(x'_i,x^i)\in A_i$.

Then, $\bigcap _{i\in I}A_i\ne \emptyset$.

Proof

This is a particular case of Theorem 7 with $A_i\equiv B_i$ for all $i\in I$.$\square$

An equivalent formulation of Corollary 3 is the following.

Corollary 4

For $i\in I$, let $X_i$ be a nonempty set, $X:=\prod _{i\in I}X_i$, $X^i:=\prod _{j\in I, j\ne i}X_j$, $x^i$ the canonical projection of x on $X^i$, $f_i : X \rightarrow \mathbb {R}$ and $\delta _i\in \mathbb {R}$. Assume that, for each $i\in I$, there exists a KKM-structure $\mathscr {F}_i:= ({\varPhi }_{X_i}, \mathfrak {I}_{X_i})$ of $X_i$ such that

• (i) for each $x_i\in X_i$, $\{x'\in X|\, f_i(x_i,x'^i) > \delta _i\}$ is $\mathfrak {I}_{X}$-open;

• (ii) for each $x\in X$, $\{x'_i\in X_i|\, f_i(x'_i, x^i) > \delta _i\}$ is nonempty and ${\varPhi }_{X_i}$-convex;

• (iii) there exists a $\mathfrak {I}_{X}$-compact subset K of X, and, for each $N_i\in \langle X_i\rangle$, there is a $\mathfrak {I}_{X_i}$-compact and ${\varPhi }_{X_i}$-convex subset $L_{N_i}\subset X_i$ containing $N_i$ such that $(\prod _{i\in I}L_{N_i})\backslash K \subset \bigcup _{x_i\in L_{N_i}}\{x'\in X|\, f_i(x_i,x'^i) > \delta _i\}$.

Then, there exists $\bar{x}\in X$ such that $f_i(\bar{x}) > \delta _i$ for all $i\in I$.

Proof

Corollary 3 implies Corollary 4. It is clear that the assumptions of Corollary 4 imply the assumptions of Corollary 3 with $A_i:=\{x\in X|\, f_i(x) > \delta _i\}$ for $i\in I$. Applying Corollary 3, we have $\bar{x}\in X$ such that $\bar{x} \in A_i$ for all $i \in I$, which means that $f_i(\bar{x}) > \delta _i$ for all $i\in I$.

Corollary 4 implies Corollary 3. We define real functions $f_i : X \rightarrow \mathbb {R}$ by, for $x\in X$,

$$\begin{aligned} f_i(x)=\left\{ \begin{array}{ll}{} 1&{} \quad \mathrm{if}\quad x\in A_i,\\ {}0&{} \quad \mathrm{if}\quad x\notin A_i. \\ \end{array} \right. \end{aligned}$$

Then, for each $i\in I$, we have $\{x'\in X|\, f_i(x_i,x'^i) > 0\}=\{x'\in X|\, (x_i, x'^i)\in A_i\}$ for each $x_i\in X_i$, and $\{x'_i\in X_i|\, f_i(x'_i, x^i) > 0\}= \{x_i'\in X_i|\, (x_i', x^i)\in A_i\}$ for each $x\in X$. Hence, the assumptions of Corollary 3 imply those of Corollary 4 for $\{f_i\}_{i\in I}$ and $\{\delta _i=0\}_{i\in I}$. Corollary 4 gives $\bar{x}\in X$ satisfying $f_i(\bar{x})= 1$ for all $i\in I$, i.e., $\bar{x}\in A_i$ for all $i\in I$. $\square$

Remark 4

• (a) We can prove that Theorem 6 is still true if we replace x, $x'$, X and $\mathfrak {I}_X$ by $x^i$, $x'^i$, $X^i$ and $\mathfrak {I}_{X^i}$, resp, in assumptions (i) and (ii). A similar replacement can be made for Corollaries 3 and 4. In this case, Theorem 6 and Corollary 3 extend some results on sets with convex sections in [11] (Theorems 14–16) and [21] (Theorems 2.3 and 2.4).

• (b) Assumption (i) of Corollary 4 is satisfied if, for each $x_i\in X_i$, $f_i(x_i,\cdot )$ is $\mathfrak {I}_{X^i}$-lower semicontinuous on $X^i$. Corollary 4 generalizes Theorem 3 of [10] and Theorem 2.5 of [21].

Corollary 5

For $i\in I$, let $X_i$ be a nonempty set, $X:=\prod _{i\in I}X_i$, $X^i:=\prod _{j\in I, j\ne i}X_j$, $x^i$ the canonical projection of x on $X^i$, and $f_i : X \rightarrow \mathbb {R}$. Assume that there exists a compact KKM-structure $\mathscr {F}_i= ({\varPhi }_{X_i}, \mathfrak {I}_{X_i})$ of $X_i$ for each $i\in I$ such that

• (i) $f_i$ is $\mathfrak {I}_{X}$-continuous;

• (ii) for any $\epsilon >0$ sufficiently close to 0 and $x\in X$, the set $\{a_i\in X_i\mid f_i(x^i, a_i) > \max _{a'_i \in X_i}f_i(x^i, a'_i)-\epsilon \}$ is ${\varPhi }_{X_i}$-convex.

Then, there exists $\bar{x}\in X$ such that $f_i(\bar{x})=\max _{a_i\in X_i}f_i(\bar{x}^i, a_i)$ for all $i\in I$.

Proof

Applying Corollary 4 for functions $h_i : X\longrightarrow \mathbb {R}$ defined by $h_i (x) = f_i(x) - \max _{a_i \in X_i}f_i(a_i, x^i)$ and $\delta _i=-\epsilon$, we have $\bar{x}\in X$ such that $h_i(\bar{x}) > -\epsilon$. Since $\epsilon$ is arbitrary, the proof is complete. $\square$

Note that results similar to Corollaries 4 and 5 were proved in [19] for the case with a finite index set I.

3.4 Systems of Variational Relations

In this subsection, we discuss applications to a general model of systems of variational relations since it encompasses most problems related to optimization. Variational relations were first studied in [16, 25] and then extended to a case of a system of relations in [23]. Let I be an index set, $\{X_i\}_{i\in I}$ be a family of nonempty sets and $X:=\prod _{i\in I}X_i$. Let ${\varLambda }_i, {\varOmega }_i: X\rightrightarrows X_i$ be nonempty-valued and $R_i(x, a_i)$ be a relation linking $x\in X$ and $a_i\in X_i$. We consider the following system of variational relations

$$\begin{aligned} \mathrm{(SVR)}\quad \left\{ \begin{array}{ll}\mathrm{find}\quad \bar{x}\in X\quad \mathrm{such\quad that},\quad \mathrm{for\quad all} \quad i\in I,\quad \bar{x}_i\in {\varLambda }_i(\bar{x})\quad \mathrm{and}\quad a_i\in {\varOmega }_i(\bar{x}),\\ {}R_i(\bar{x}, a_i)\quad \mathrm{holds} .\\ \end{array} \right. \end{aligned}$$

Problem (SVR) was studied in [15] for the case where $\{X_i\}_{i\in I}$ is a family of nonempty convex subsets of topological vector spaces. In this section, using Theorem 1 we will establish an existence result for the general case.

For $i\in I$, we set ${\varTheta }_i(x):=\{a_i\in X_i\,|\quad R_i(x, a_i)$ does not hold$\}$ for all $x\in X$, and $W_i:=\{x\in X\,|$ for all $a_i\in {\varOmega }_i(x)$, $R_i(x, a_i)$ holds$\}$.

Theorem 8

For problem (SVR), assume that there exists a compact KKM-structure $\mathscr {F}_i:= ({\varPhi }_{X_i}, \mathfrak {I}_{X_i})$ of $X_i$ for each $i\in I$ such that

• (i) $X=\bigcup _{a_i\in X_i}\mathrm{int}_{\mathfrak {I}_X}\big (({\varTheta }_i^{-1}(a_i)\cup W_i)\cap {\varOmega }_i^{-1}(a_i)\big )$;

• (ii) ${\varLambda }_i$ is ${\varPhi }_{X_i}$-weak-convex w.r.t. ${\varOmega }_i$;

• (iii) for each $x:=(x_i)_{i\in I}\in X$, $x_i\notin {\varPhi }_{X_i}$-co${\varTheta }_i(x)$.

Then, $\mathrm{(SVR)}$ has a solution.

Proof

For each $i\in I$, we define set-valued maps $P_i, Q_i : X\rightrightarrows X_i$ by

$$\begin{aligned} P_i(x)= & {} \left\{ \begin{array}{ll}{}{\varTheta }_i(x)\cap {\varOmega }_i(x)&{} \quad \mathrm{if}\quad x \notin W_i,\\ {}{\varOmega }_i(x)&{} \quad \mathrm{if}\quad x\in W_i,\\ \end{array} \right. \\ Q_i(x)= & {} \left\{ \begin{array}{ll}{}{\varPhi }_{X_i}\mathrm{-co}{\varTheta }_i(x)&{} \quad \mathrm{if}\quad x \notin W_i,\\ {}{\varLambda }_i(x)&{} \quad \mathrm{if}\quad x\in W_i.\\ \end{array} \right. \end{aligned}$$

For each $a_i\in X_i$ we have

$$\begin{aligned} P_i^{-1}(a_i)= & {} \Big ({\varTheta }_i^{-1}(a_i)\cap {\varOmega }_i^{-1}(a_i)\cap (X\setminus W_i)\Big )\cup \Big ({\varOmega }_i^{-1}(a_i)\cap W_i \Big ) \nonumber \\= & {} \Big ([{\varTheta }_i^{-1}(a_i)\cap (X\setminus W_i)]\cup W_i\Big )\cap {\varOmega }_i^{-1}(a_i) \nonumber \\= & {} ({\varTheta }_i^{-1}(a_i)\cup W_i)\cap {\varOmega }_i^{-1}(a_i). \end{aligned}$$

(5)

The equality (5) and assumption (i) imply that $X= \bigcup _{a_i\in X_i}\mathrm{int}_{ \mathfrak {I}_{X}}P_i^{-1}(a_i)$.

Take any $\varphi _{N_i}\in {\varPhi }_{X_i}$, $M_i\subset N_i$, $x=(x_i)_{i\in I}\in X$ satisfying $x_i\in \varphi _{N_i}(\Delta _{M_i})$, and $M_i\subset P_i(x)$. As $x\notin W_i$, $M_i\subset {\varTheta }_i(x)\cap {\varOmega }_i(x)\subset {\varTheta }_i(x)$. Then, $\varphi _{N_i}(\Delta _{M_i})\subset {\varPhi }_{X_i}$-co${\varTheta }_i(x)=Q_i(x)$. If $x\in W_i$, then $M_i\subset {\varOmega }_i(x)$. By (ii) we obtain $\varphi _{N_i}(\Delta _{M_i})\subset {\varLambda }_i(x)=Q_i(x)$. Thus, $Q_i$ is ${\varPhi }_{X_i}$-weak-convex w.r.t. $P_i$.

Theorem 1 under condition (h1) of Remark 1 implies that an $\bar{x}:=(\bar{x}_i)_{i\in I}\in X$ exists such that $\bar{x}_i \in Q_i(\bar{x})$ for each $i\in I$. If $\bar{x} \notin W_i$, then $\bar{x}_i \in {\varPhi }_{X_i}$-co${\varTheta }_i(\bar{x})$, contradicting (iii). Hence, $\bar{x} \in W_i$, and therefore $\bar{x}_i \in {\varLambda }_i(\bar{x})$ and $R_i(\bar{x}, a_i)$ holds for all $a_i\in {\varOmega }_i(\bar{x})$. $\square$

Remark 5

(a) Because the map $P_i$ in the proof of Theorem 8 has the nonempty values, assumption (i) can be replaced by the following

(i$'$):

for each $a_i\in X_i$, $({\varTheta }_i^{-1}(a_i)\cup W_i)\cap {\varOmega }_i^{-1}(a_i)$ is$\mathfrak {I}_X$-open.

Furthermore, (i$'$) is satisfied if

(i$''$):

$W_i$ is $\mathfrak {I}_X$-open, and, for each $a_i\in X_i$, ${\varTheta }_i^{-1}(a_i)\cap (X\setminus W_i)$ is relatively$\mathfrak {I}_X$-open in $X\setminus W_i$ and ${\varOmega }_i^{-1}(a_i)$ is$\mathfrak {I}_X$-open.

Indeed, as ${\varTheta }_i^{-1}(a_i)\cap (X\setminus W_i)$ is relatively $\mathfrak {I}_X$-open in $X\setminus W_i$, a $\mathfrak {I}_X$-open set $U_i\subset X$ exists such that ${\varTheta }_i^{-1}(a_i)\cap (X\setminus W_i) = U_i\cap (X\setminus W_i)$. Then, $({\varTheta }_i^{-1}(a_i)\cup W_i)\cap {\varOmega }_i^{-1}(a_i)$ = $([U_i\cap (X\setminus W_i)]\cup W_i)\cap {\varOmega }_i^{-1}(a_i)$ = $(U_i\cup W_i)\cap {\varOmega }_i^{-1}(a_i)$ is $\mathfrak {I}_X$-open.

(b) It is clear that assumption (iii) of Theorem 8 is fulfilled if “for each $x:=(x_i)_{i\in I}\in X$, $R_i(x, x_i)$holds, and ${\varTheta }_i(x)$ is ${\varPhi }_{X_i}$-convex.”

3.5 Abstract Economies

Finally, we discuss a practical model. Let I be any set of agents. For each $i\in I$, let $X_i$ be a nonempty set of actions available for the agent i and $X :=\prod _{i\in I}X_i$. An abstract economy (see [27]) is a family of ordered triples $\mathscr {E}:=(X_i, A_i, B_i)_{i\in I}$, where $A_i: X\rightrightarrows X_i$ is a constraint correspondence such that $A_i(x)$ is the state attainable for the agent i at x, and $B_i:X \rightrightarrows X_i$ is a preference correspondence such that $B_i(x)$ is the state preference by the agent i at x. An equilibrium point of $\mathscr {E}$ is a point $\bar{x}\in X$ such that $\bar{x}_i \in A_i(\bar{x})$ and $B_i(\bar{x})\cap A_i(\bar{x})=\emptyset$ for each $i \in I$.

Of course, we can deduce the following existence result from our results in Sect. 2. But, to explain the generality of variational relations studied in the preceding subsection we will apply Theorem 7.

Theorem 9

For the abstract economy $\mathscr {E}:=(X_i, A_i, B_i)_{i\in I}$, let $W_i:=\{x\in X \,| B_i(x)\cap A_i(x) =\emptyset \}$ for each $i\in I$. Assume that, for each $i\in I$, there exists a compact KKM-structure $\mathscr {F}_i= ({\varPhi }_{X_i}, \mathfrak {I}_{X_i})$ of $X_i$ such that

• (i) $X=\bigcup _{a_i\in X_i}\mathrm{int}_{ \mathfrak {I}_{X}}\big ((B_i^{-1}(a_i)\cup W_i)\cap A_i^{-1}(a_i)\big )$;

• (ii) for each $x\in X$, $A_i(x)$ is ${\varPhi }_{X_i}$-convex;

• (iii) for each $x=(x_i)_{i\in I}\in X$, $x_i\notin {\varPhi }_{X_i}$-$\mathrm{co}B_i(x)$;

Then, $\mathscr {E}$ has an equilibrium point.

Proof

We see that $\bar{x}\in X$ is an equilibrium point of $\mathscr {E}$ if and only if, for each $i \in I$, $\bar{x}_i \in A_i(\bar{x})$, and for all $a_i\in A_i(\bar{x})$, $a_i \notin B_i(\bar{x})$. This means that $\bar{x}$ is a solution of $\mathrm{(SVR)}$ with ${\varLambda }_i =A_i$, ${\varOmega }_i=A_i$ and the relation $R_i$ defined by: $R_i(x, a_i)$ holds if and only if $a_i\notin B_i(x)$. Applying Theorem 7, we complete the proof. $\square$

Remark 6

The observations in Remark 5 are valid also for Theorem 9. Furthermore, if $A_i$ and $B_i$ are $(\mathfrak {I}_X, \mathfrak {I}_{X_i})$-closed (i.e., their graphs are $\mathfrak {I}_X \times \mathfrak {I}_{X_i}$-closed in $X\times X_i$), then $W_i$ is $\mathfrak {I}_X$-closed. Hence, assumption (i) of Theorem 9 is satisfied if

• (i’) $A_i$ and $B_i$ are $(\mathfrak {I}_X, \mathfrak {I}_{X_i})$-closed, and their fibers are $\mathfrak {I}_X$-open.

In some cases condition (i’) is restrictive, e.g., if the topologies $\mathfrak {I}_{X_i}$ are connected (in particular, if KKM-structures $\mathscr {F}_i= ({\varPhi }_{X_i}, \mathfrak {I}_{X_i})$ are the natural KKM-structures), then (i’) is fulfilled if and only if $A_i$ and $B_i$ are constant maps. However, there are instances where this condition is relatively easy to check.