Collectively coincidence-type results and applications

Abstract

Based on the well-known Brouwder fixed-point theorem in this paper we will present a variety of collectively coincidence-type results for general classes of maps. Our theory will automatically generate analytic alternatives and minimax inequalities.

Keywords:

• Continuous selections
• coincidence theory
• analytic alternatives
• minimax inequalities

2010 Mathematics Subject Classification:

• 47H10

1. Introduction

In this paper, we present a collection of collectively coincidence-type results for very general classes of maps. These maps are usually multivalued and include the admissible maps of Gorniewicz, the maps of Park and also multivalued maps which may have continuous selections (i.e. the DKT maps). Our theory considers multivalued maps in both the compact and coercive cases. The theory presented here was motivated in part from the papers [1–6] and the references therein. Our collectively coincidence results will then generate analytic alternatives and minimax inequalities of Neumann–Sion type.

Now we describe the maps considered in this paper. Let H be the $\breve{C}$ech homology functor with compact carriers and coefficients in the field of rational numbers K from the category of Hausdorff topological spaces and continuous maps to the category of graded vector spaces and linear maps of degree zero. Thus $H(X)=\{H_q(X)\}$ (here X is a Hausdorff topological space) is a graded vector space, $H_q(X)$ being the q-dimensional $\breve{C}$ech homology group with compact carriers of X. For a continuous map $f:X\to X$, $H(f)$ is the induced linear map $f_{\star }=\{f_{\star q}\}$ where $f_{\star q}:H_q(X)\to H_q(X)$. A space X is acyclic if X is nonempty, $H_q(X)=0$ for every $q\ge 1$, and $H_0(X)\approx K$.

Let X, Y and Γ be Hausdorff topological spaces. A continuous single-valued map $p:\mathrm{\Gamma }\to X$ is called a Vietoris map (written $p:\mathrm{\Gamma }\Rightarrow X$) if the following two conditions are satisfied:

1. for each $x\in X$, the set $p^{-1}(x)$ is acyclic

2. p is a perfect map, i.e. p is closed and for every $x\in X$ the set $p^{-1}(x)$ is nonempty and compact.

Let $\varphi :X\to Y$ be a multivalued map (note for each $x\in X$ we assume $\varphi (x)$ is a nonempty subset of Y). A pair $(p,q)$ of single-valued continuous maps of the form $X\stackrel{p}{\leftarrow }\mathrm{\Gamma }\stackrel{q}{\to }Y$ is called a selected pair of ϕ (written $(p,q)\subset \varphi$) if the following two conditions hold:

1. p is a Vietoris map

   and

2. $q(p^{-1}(x))\subset \varphi (x)$ for any $x\in X$.

Now we define the admissible maps of Gorniewicz [7]. A upper semicontinuous map $\varphi :X\to Y$ with compact values is said to be admissible (and we write $\varphi \in Ad(X,Y)$) provided there exists a selected pair $(p,q)$ of ϕ. An example of an admissible map is a Kakutani map. A upper semicontinuous map $\varphi :X\to K(Y)$ is said to Kakutani (and we write $\varphi \in Kak(X,Y)$); here $K(Y)$ denotes the family of nonempty, convex, compact subsets of Y.

The following class of maps will play a major role in this paper. Let Z and W be subsets of Hausdorff topological vector spaces $Y_1$ and $Y_2$ and G a multifunction. We say $G\in DKT(Z,W)$ [4, 5] if W is convex and there exists a map $S:Z\to W$ with $\mathrm{co}(S(x))\subseteq G(x)$ for $x\in Z$, $S(x)\ne \mathrm{\varnothing }$ for each $x\in Z$ and the fibre $S^{-1}(w)=\{z\in Z:w\in S(z)\}$ is open (in Z) for each $w\in W$.

Now we consider a general class of maps, namely the PK maps of Park. Let X and Y be Hausdorff topological spaces. Given a class $\mathcal{X}$ of maps, $\mathcal{X}(X,Y)$ denotes the set of maps $F:X\to 2^Y$ (nonempty subsets of Y) belonging to $\mathcal{X}$, and ${\mathcal{X}}_c$ the set of finite compositions of maps in $\mathcal{X}$. We let $\mathcal{F}(\mathcal{X})=\{Z:FixF\ne \mathrm{\varnothing }\text{for all}F\in \mathcal{X}(Z,Z)\}$where $FixF$ denotes the set of fixed points of F.

The class $\mathcal{U}$ of maps is defined by the following properties:

1. $\mathcal{U}$ contains the class $\mathbf{C}$ of single-valued continuous functions;

2. each $F\in {\mathcal{U}}_c$ is upper semicontinuous and compact valued; and

3. $B^n\in \mathcal{F}({\mathcal{U}}_c)$ for all $n\in \{1,2,\dots .\}$; here $B^n=\{x\in {\mathbf{R}}^n:\Vert x\Vert \le 1\}$.

We say $F\in PK(X,Y)$ if for any compact subset K of X there is a $G\in {\mathcal{U}}_c(K,Y)$ with $G(x)\subseteq F(x)$ for each $x\in K$. Recall PK is closed under compositions.

For a subset K of a topological space X, we denote by ${\mathrm{Cov}}_X(K)$ the directed set of all coverings of K by open sets of X (usually we write $\mathrm{Cov}(K)={\mathrm{Cov}}_X(K)$). Given two maps $F,G:X\to 2^Y$ and $\alpha \in \mathrm{Cov}(Y)$, F and G are said to be α-close if for any $x\in X$ there exists $U_x\in \alpha$, $y\in F(x)\cap U_x$ and $w\in G(x)\cap U_x$.

Let Q be a class of topological spaces. A space Y is an extension space for Q (written $Y\in \mathrm{ES}(Q)$) if for any pair $(X,K)$ in Q with $K\subseteq X$ closed, any continuous function $f_0:K\to Y$ extends to a continuous function $f:X\to Y$. A space Y is an approximate extension space for Q (written $Y\in \mathrm{AES}(Q)$) if for any $\alpha \in \mathrm{Cov}(Y)$ and any pair $(X,K)$ in Q with $K\subseteq X$ closed, and any continuous function $f_0:K\to Y$ there exists a continuous function $f:X\to Y$ such that $f{|}_K$ is α–close to $f_0$.

Let V be a subset of a Hausdorff topological vector space E. Then we say V is Schauder admissible if for every compact subset K of V and every covering $\alpha \in {\mathrm{Cov}}_V(K)$ there exists a continuous functions ${\pi }_{\alpha }:K\to V$ such that

1. ${\pi }_{\alpha }$ and $i:K\to V$ are α–close;

2. ${\pi }_{\alpha }(K)$ is contained in a subset $C\subseteq V$ with $C\in \mathrm{AES}(\text{compact})$.

X is said to be q-Schauder admissible if any nonempty compact convex subset Ω of X is Schauder admissible.

Theorem 1.1

[8, 9]

Let X be a Schauder admissible subset of a Hausdorff topological vector space and $\mathrm{\Psi }\in PK(X,X)$ a compact upper semicontinuous map with closed values. Then there exists a $x\in X$ with $x\in \mathrm{\Psi }(x)$.

Remark 1.2

Other variations of Theorem 1.1 can be found in [10].

2. Coincidence results

In this section, we present a variety of collectively coincidence-type results in both the compact and coercive case. We begin with the compact case.

Theorem 2.1

Let $\{X_i{\}}_{i=1}^N,$ $\{Y_i{\}}_{i=1}^{N_0}$ be families of convex sets each in a Hausdorff topological vector space $E_i$. For each $i\in \{1,\dots ,N_0\}$ suppose $F_i:X\equiv \prod _{i=1}^N{X}_i\to Y_i$ and $F_i\in DKT(X,Y_i)$ and for each $j\in \{1,\dots ,N\}$ suppose $G_j:Y\equiv \prod _{i=1}^{N_0}{Y}_i\to X_j$ and $G_j\in DKT(Y,X_j)$. In addition assume for each $i\in \{1,\dots ,N_0\}$ there exists a compact set $K_i$ with $F_i(X)\subseteq K_i\subseteq Y_i$. Then there exists a $x\in X$ and a $y\in Y$ with $x_i\in G_i(y)$ for $i\in \{1,\dots ,N\}$ and $y_j\in F_j(x)$ for $j\in \{1,\dots ,N_0\}$ $($here $x_i$$($respectively, $y_j)$ is the projection of x $($respectively, $y)$ on $X_i$ $($respectively, $Y_j))$.

Proof.

For $i\in \{1,\dots ,N_0\}$ let $T_i:X\to Y_i$ with $T_i(x)\ne \mathrm{\varnothing }$ for $x\in X$, $\mathrm{co}(T_i(x))\subseteq F_i(x)$ for $x\in X$ and $T_i^{-1}(w)$ is open (in X) for each $w\in Y_i$. For $i\in \{1,\dots ,N\}$ let $S_i:Y\to X_i$ with $S_i(y)\ne \mathrm{\varnothing }$ for $y\in Y$, $\mathrm{co}(S_i(y))\subseteq G_i(y)$ for $y\in Y$ and $S_i^{-1}(w)$ is open (in Y) for each $w\in X_i$. Let $K=\prod _{i=1}^{N_0}{K}_i(\subseteq Y)$ and note K is compact. Let $G_i^{\star }$ (respectively, $S_i^{\star }$) denote the restriction of $G_i$ (respectively, $S_i$) to K. Note for $i\in \{1,\dots ,N\}$ that $G_i^{\star }\in DKT(K,X_i)$ since for $x\in X_i$ we have $(S_i^{\star }{)}^{-1}(x)=\{z\in K:x\in S_i^{\star }(z)\}=\{z\in K:x\in S_i(z)\}=K\cap \{z\in Y:x\in S_i(z)\}=K\cap S_i^{-1}(x)$which is open in $K\cap Y=K$. Now for each $i\in \{1,\dots ,N\}$ from [4] there exists a continuous (single-valued) selection $g_i:K\to X_i$ of $G_i^{\star }$ with $g_i(y)\in \mathrm{co}(S_i^{\star }(y))\subseteq G_i^{\star }(y)$ for $y\in K$ and also there exists a finite set $R_i$ of $X_i$ with $g_i(K)\subseteq \mathrm{co}(R_i)\equiv Q_i$. Let $Q=\prod _{i=1}^N{Q}_i(\subseteq X)$ and note Q is compact. Now let $F_i^{\star }$ (respectively, $T_i^{\star }$) denote the restriction of $F_i$ (respectively, $T_i$) to Q. Note for $i\in \{1,\dots ,N_0\}$ that $F_i^{\star }\in DKT(Q,Y_i)$ since for $y\in Y_i$ we have $(T_i^{\star }{)}^{-1}(y)\{z\in Q:y\in T_i^{\star }(z)\}=\{z\in Q:y\in T_i(z)\}=Q\cap \{z\in X:y\in T_i(z)\}=Q\cap T_i^{-1}(y)$which is open in $Q\cap X=Q$. Now for each $i\in \{1,\dots ,N_0\}$ from [4] there exists a continuous (single-valued) selection $f_i:Q\to Y_i$ of $F_i^{\star }$ and note $f_i(Q)\subseteq F_i^{\star }(Q)\subseteq F_i(X)\subseteq K_i$. Let $f(x)=\prod _{i=1}^{N_0}{f}_i(x)\text{and}g(y)=\prod _{i=1}^N{g}_i(y)$and note $f:Q\to Y,g:K\to X$ are continuous and since $f_i(Q)\subseteq K_i$ for $i\in \{1,\dots .,N_0\}$ and $g_i(K)\subseteq Q_i$ for $i\in \{1,\dots .,N\}$ we have $f(Q)\subseteq K$ and $g(K)\subseteq Q$. Consider the continuous map $h:Q\to Q$ given by $h(x)=g(f(x))$ for $x\in Q$. Note Q is a compact convex subset in a finite-dimensional subspace of $E=\prod _{i=1}^N{E}_i$ so Brouwer's fixed-point theorem guarantees that there exists a $x\in Q$ with $x=h(x)=g(f(x))$. Let $y=f(x)$ so $x=g(y)$. Then since $x\in Q$ and $y=f(x)\in f(Q)\subseteq K$ we have $y_j=f_j(x)\in F_j^{\star }(x)=F_j(x)$ for $j\in \{1,\dots ,N_0\}$ and $x_i=g_i(y)\in G_i^{\star }(y)=G_i(y)$ for $i\in \{1,\dots ,N\}$.

We now consider Theorem 2.1 in a more general setting.

Theorem 2.2

Let I and J be index sets and let $\{X_i{\}}_{i\in I},$ $\{Y_i{\}}_{i\in J}$ be families of convex sets each in a Hausdorff topological vector space $E_i$. For each $i\in J$ suppose $F_i:X\equiv \prod _{i\in I}{X}_i\to Y_i$ and $F_i\in DKT(X,Y_i)$ and for each $j\in I$ suppose $G_j:Y\equiv \prod _{i\in J}{Y}_i\to X_j$ and $G_j\in DKT(Y,X_j)$. In addition assume for each $i\in J$ there exists a compact set $K_i$ with $F_i(X)\subseteq K_i\subseteq Y_i$. Also suppose X is a q-Schauder admissible subset of the Hausdorff topological vector space $E=\prod _{i\in I}{E}_i$. Then there exists a $x\in X$ and a $y\in Y$ with $x_i\in G_i(y)$ for $i\in I$ and $y_j\in F_j(x)$ for $j\in J$.

Proof.

For $i\in J$ (respectively, $i\in I$) let $T_i$ (respectively, $S_i$) be as in Theorem 2.1, $K=\prod _{i\in J}{K}_i(\subseteq Y)$ and $G_i^{\star }$ the restriction of $G_i$ to K. The same reasoning as in Theorem 2.1 guarantees that $G_i^{\star }\in DKT(K,X_i)$ for $i\in I$ so from [4] there exists a continuous (single-valued) selection $g_i:K\to X_i$ of $G_i^{\star }$ and also a finite set $R_i$ of $X_i$ with $g_i(K)\subseteq \mathrm{co}(R_i)\equiv Q_i$ Let $Q=\prod _{i\in I}{Q}_i(\subseteq X)$ and let $F_i^{\star }$ denote the restriction of $F_i$ to Q. The same reasoning as in Theorem 2.1 guarantees that $F_i^{\star }\in DKT(Q,Y_i)$ for $i\in J$ so from [4] there exists a continuous (single-valued) selection $f_i:Q\to Y_i$ of $F_i^{\star }$ and note $f_i(Q)\subseteq F_i^{\star }(Q)\subseteq F_i(X)\subseteq K_i$. Let $f(x)=\prod _{i\in J}{f}_i(x)\text{and}g(y)=\prod _{i\in I}{g}_i(y)$and note $f(Q)\subseteq K$ and $g(K)\subseteq Q$. Let the continuous map $h:Q\to Q$ be given by $h(x)=g(f(x))$ for $x\in Q$. Now Theorem 1.1 guarantees that there exists a $x\in Q$ with $x=h(x)=g(f(x))$ and as in Theorem 2.1 we immediately have the result.

Remark 2.3

Note in the statement of Theorems 2.1 and 2.2 we could replace $F_i\in DKT(X,Y_i)$ with $F_i\in DKT(X,K_i)$; here let $T_i:X\to K_i$ with $T_i(x)\ne \mathrm{\varnothing }$ for $x\in X$, $\mathrm{co}(T_i(x))\subseteq F_i(x)$ for $x\in X$ and $T_i^{-1}(w)$ is open (in X) for each $w\in K_i$.

Other classes of maps could also be considered. We illustrate this in our next results.

Theorem 2.4

Let $\{X_i{\}}_{i=1}^N,$ $\{Y_i{\}}_{i=1}^{N_0}$ be families of convex sets each in a Hausdorff topological vector space $E_i$. For each $i\in \{1,\dots ,N_0\}$ suppose $F_i:X\equiv \prod _{i=1}^N{X}_i\to Y_i$ and $F_i\in Ad(X,Y_i)$ and in addition assume there exists a compact set $K_i$ with $F_i(X)\subseteq K_i\subseteq Y_i$. For each $j\in \{1,\dots ,N\}$ suppose $G_j:Y\equiv \prod _{i=1}^{N_0}{Y}_i\to X_j$ and $G_j\in DKT(Y,X_j)$. Then there exists a $x\in X$ and a $y\in Y$ with $x_i\in G_i(y)$ for $i\in \{1,\dots ,N\}$ and $y_j\in F_j(x)$ for $j\in \{1,\dots ,N_0\}$.

Proof.

Let $S_i,K,G_i^{\star }$ and $S_i^{\star }$ be as in Theorem 2.1 and the same reasoning as in Theorem 2.1 guarantees that $G_i^{\star }\in DKT(K,X_i)$ for $i\in \{1,\dots ,N\}$ and from [4] there exists a continuous (single-valued) selection $g_i:K\to X_i$ of $G_i^{\star }$ and also a finite set $R_i$ of $X_i$ with $g_i(K)\subseteq \mathrm{co}(R_i)\equiv Q_i$ Let $Q=\prod _{i=1}^N{Q}_i(\subseteq X)$. Let $F_i^{\star }$ denote the restriction of $F_i$ to Q and let $F^{\star }(x)=\prod _{i=1}^{N_0}{F}_i^{\star }(x)\text{for}x\in Q.$Since a finite product of admissible maps of Gorniewicz is an admissible map of Gorniewicz [7] then $F^{\star }\in Ad(Q,Y)$. Now $F_i^{\star }(Q)\subseteq F_i(X)\subseteq K_i$ for each $i\in \{1,\dots ,N_0\}$ so $F^{\star }(Q)\subseteq K$. Let $g(y)=\prod _{i=1}^N{g}_i(y)$ for $y\in K$ and note since $g_i(K)\subseteq Q_i$ for $i\in \{1,\dots ,N\}$ that $g(K)\subseteq Q$. Thus $g{F}^{\star }\in Ad(Q,Q)$ and note Q is a compact convex subset in a finite-dimensional subspace of $E=\prod _{i=1}^N{E}_i$. Now Theorem 1.1 guarantees that there exists a $x\in Q$ with $x\in g(F^{\star }(x))$. Now let $y\in F^{\star }(x)$ with $x=g(y)$. Note $y\in F^{\star }(Q)\subseteq K$ and $y_j\in F_j^{\star }(x)=F_j(x)$ for $j\in \{1,\dots ,N_0\}$ and $x_i=g_i(y)\in G_i^{\star }(y)=G_i(y)$ for $i\in \{1,\dots ,N\}$.

Theorem 2.5

Let I and J be index sets and let $\{X_i{\}}_{i\in I},$ $\{Y_i{\}}_{i\in J}$ be families of convex sets each in a Hausdorff topological vector space $E_i$. For each $i\in J$ suppose $F_i:X\equiv \prod _{i\in I}{X}_i\to Y_i$ and there exists a compact set $K_i$ with $F_i(X)\subseteq K_i\subseteq Y_i$. For each $j\in I$ suppose $G_j:Y\equiv \prod _{i\in J}{Y}_i\to X_j$ and $G_j\in DKT(Y,X_j)$. Let $F(x)=\prod _{i\in J}{F}_i(x)$ for $x\in X$ and suppose $F\in PK(X,Y)$. Also suppose X is a q-Schauder admissible subset of the Hausdorff topological vector space $E=\prod _{i\in I}{E}_i$. Then there exists a $x\in X$ and a $y\in Y$ with $x_i\in G_i(y)$ for $i\in I$ and $y_j\in F_j(x)$ for $j\in J$.

Proof.

Let $S_i,K,G_i^{\star }$ and $S_i^{\star }$ be as in Theorem 2.2 and the same reasoning as in Theorem 2.2 guarantees that $G_i^{\star }\in DKT(K,X_i)$ for $i\in I$ and from [4] there exists a continuous (single-valued) selection $g_i:K\to X_i$ of $G_i^{\star }$ and also a finite set $R_i$ of $X_i$ with $g_i(K)\subseteq \mathrm{co}(R_i)\equiv Q_i$. Let $Q=\prod _{i\in I}{Q}_i(\subseteq X)$ and let $F^{\star }$ denote the restriction of F to Q. Now since the composition of PK maps is a PK map then $F^{\star }\in PK(Q,Y)$. Also note since $F_i(X)\subseteq K_i$ for $i\in J$ that $F^{\star }(Q)\subseteq F(X)\subseteq K$. Let $g(y)=\prod _{i\in I}{g}_i(y)$ for $y\in K$ and note since $g_i(K)\subseteq Q_i$ for $i\in I$ that $g(K)\subseteq Q$. Thus $g{F}^{\star }\in PK(Q,Q)$ (note $F(X)\subseteq K$ and $g(K)\subseteq Q$). Now Theorem 1.1 guarantees that there exists a $x\in Q$ with $x\in g(F^{\star }(x))=g(F(x))$ and the conclusion follows as before.

Remark 2.6

1. If I and J are finite sets then the assumption that X is a q-Schauder admissible subset of E is satisfied.

2. Note in the statement of Theorems 2.4 and 2.5 we do not need to have the sets $Y_i$ convex.

Now we will replace the compactness condition on $F_i$ with a coercive type condition [2, 3]. We will consider a subclass of the DKT maps (see [2, 4]). Let G be a multifunction and we say $G\in {\mathrm{\Phi }}^{\star }(Z,W)$ [2] if W is convex and there exists a map $S:Z\to W$ with $S(x)\subseteq G(x)$ for $x\in Z$, $S(x)\ne \mathrm{\varnothing }$ and has convex values for each $x\in Z$ and $S^{-1}(w)$ is open (in Z) for each $w\in W$.

Theorem 2.7

Let $\{X_i{\}}_{i=1}^N,$ $\{Y_i{\}}_{i=1}^{N_0}$ be families of convex sets each in a Hausdorff topological vector space $E_i$. For each $i\in \{1,\dots ,N_0\}$ suppose $F_i:X\equiv \prod _{i=1}^N{X}_i\to Y_i$ and $F_i\in {\mathrm{\Phi }}^{\star }(X,Y_i)$ and for each $j\in \{1,\dots .N\}$ suppose $G_j:Y\equiv \prod _{i=1}^{N_0}{Y}_i\to X_j$ and in addition assume there exists a map $S_j:Y\to X_j$ with $S_j(y)\ne \mathrm{\varnothing }$ and has convex values for each $y\in Y,$ $S_j(y)\subseteq G_j(y)$ for $y\in Y$ and $S_j^{-1}(w)$ is open $($in $Y)$ for each $w\in X_j$. Also suppose there is a compact subset K of Y and for each $i\in \{1,\dots ,N\}$ a convex compact subset $Z_i$ of $X_i$ with $S_i(y)\cap Z_i\ne \mathrm{\varnothing }$ for $y\in Y\mathrm{\setminus }K$. Then there exists a $x\in X$ and a $y\in Y$ with $x_i\in G_i(y)$ for $i\in \{1,\dots ,N\}$ and $y_j\in F_j(x)$ for $j\in \{1,\dots ,N_0\}$.

Proof.

For $i\in \{1,\dots ,N_0\}$ let $T_i:X\to Y_i$ with $T_i(x)\ne \mathrm{\varnothing }$ and has convex values for each $x\in X$, $T_i(x)\subseteq F_i(x)$ for $x\in X$ and $T_i^{-1}(w)$ is open (in X) for each $w\in Y_i$. For $i\in \{1,\dots ,N\}$ let $G_i^{\star }$ (respectively, $S_i^{\star }$) denote the restriction of $G_i$ (respectively, $S_i$) to K and note $G_i^{\star }\in {\mathrm{\Phi }}^{\star }(K,X_i)$ since for $x\in X_i$ then (note $K\subseteq Y$) $(S_i^{\star }{)}^{-1}(x)=\{z\in K:x\in S_i^{\star }(z)\}=\{z\in K:x\in S_i(z)\}=K\cap \{z\in Y:x\in S_i(z)\}=K\cap S_i^{-1}(x)$which is open in $K\cap Y=K$. For each $i\in \{1,\dots .,N\}$ then from [2, 4] there exists a continuous (single-valued) selection $g_i:K\to X_i$ of $G_i^{\star }$ with $g_i(y)\in S_i^{\star }(y)\subseteq G_i^{\star }(y)$ for $y\in K$ and also there exists a finite set $R_i$ of $X_i$ with $g_i(K)\subseteq \mathrm{co}(R_i)$. Let ${\mathrm{\Omega }}_i=\mathrm{co}(\mathrm{co}(R_i)\cup Z_i)\text{for}i\in \{1,\dots ,N\}$which is a convex compact subset of $X_i$. For $i\in \{1,\dots ,N\}$ let $G_i^{\star \star }(y)=G_i(y)\cap {\mathrm{\Omega }}_i$ for $y\in Y$ and we claim $G_i^{\star \star }\in {\mathrm{\Phi }}^{\star }(Y,{\mathrm{\Omega }}_i)$. To see this let $S_i^{\star \star }(y)=S_i(y)\cap {\mathrm{\Omega }}_i$ for $y\in Y$ and $i\in \{1,\dots ,N\}$. If $y\in Y\mathrm{\setminus }K$ then $S_i^{\star \star }(y)\ne \mathrm{\varnothing }$ since $S_i(y)\cap Z_i\ne \mathrm{\varnothing }$ and $Z_i\subseteq {\mathrm{\Omega }}_i$ whereas if $y\in K$ then since $g_i(K)\subseteq S_i(K)$ and $g_i(K)\subseteq \mathrm{co}(R_i)\subseteq {\mathrm{\Omega }}_i$ we have $S_i^{\star \star }(y)=S_i(y)\cap {\mathrm{\Omega }}_i\ne \mathrm{\varnothing }$. Next if $y\in K$ then $S_i^{\star \star }(y)=S_i(y)\cap {\mathrm{\Omega }}_i\subseteq G_i(y)\cap {\mathrm{\Omega }}_i=G_i^{\star \star }(x)$ and also note if $x\in {\mathrm{\Omega }}_i$ then $(S_i^{\star \star }{)}^{-1}(x)=\{z\in Y:x\in S_i^{\star \star }(z)\}=\{z\in Y:x\in S_i(z)\cap {\mathrm{\Omega }}_i\}=\{z\in Y:x\in S_i(z)\}=S_i^{-1}(x)$which is open in Y. Thus $G_i^{\star \star }\in {\mathrm{\Phi }}^{\star }(Y,{\mathrm{\Omega }}_i)$ for $i\in \{1,\dots ,N\}$. Let $\mathrm{\Omega }=\prod _{i=1}^N{\mathrm{\Omega }}_i(\subseteq X)$ and let $F_i^{\star }$ (respectively, $T_i^{\star }$) denote the restriction of $F_i$ (respectively, $T_i$) to Ω. Now for $i\in \{1,\dots ,N_0\}$ note $F_i^{\star }\in {\mathrm{\Phi }}^{\star }(\mathrm{\Omega },Y_i)$ since if $y\in Y_i$ we have $(T_i^{\star }{)}^{-1}(y)=\{z\in \mathrm{\Omega }:y\in T_i^{\star }(z)\}=\{z\in \mathrm{\Omega }:y\in T_i(z)\}=\mathrm{\Omega }\cap \{z\in X:y\in T_i(z)\}=\mathrm{\Omega }\cap T_i^{-1}(y)$which is open in $\mathrm{\Omega }\cap X=\mathrm{\Omega }$. For each $i\in \{1,\dots .,N_0\}$ then from [2, 4] there exists a continuous (single-valued) selection $f_i:\mathrm{\Omega }\to Y_i$ of $F_i^{\star }$ with $f_i(x)\in T_i^{\star }(x)\subseteq F_i^{\star }(x)$ for $x\in \mathrm{\Omega }$ and also there exists a finite set $V_i$ of $Y_i$ with $f_i(\mathrm{\Omega })\subseteq \mathrm{co}(V_i)\equiv W_i$ (note $W_i$ is a convex compact subset of $Y_i$). For $i\in \{1,\dots .,N_0\}$ let $F_i^{\star \star }(x)=F_i^{\star }(x)\cap W_i\text{and}{T}_i^{\star \star }(x)=T_i^{\star }(x)\cap W_i\text{for}x\in \mathrm{\Omega }.$We claim $F_i^{\star \star }\in {\mathrm{\Phi }}^{\star }(\mathrm{\Omega },W_i)$ for $i\in \{1,\dots ,N_0\}$. Note if $x\in \mathrm{\Omega }$ then $T_i^{\star \star }(x)=T_i^{\star }(x)\cap W_i\ne \mathrm{\varnothing }$ since $f_i(\mathrm{\Omega })\subseteq T_i^{\star }(\mathrm{\Omega })$ and $f_i(\mathrm{\Omega })\subseteq W_i$ and also if $x\in \mathrm{\Omega }$ we have $T_i^{\star \star }(x)=T_i^{\star }(x)\cap W_i\subseteq F_i^{\star }(x)\cap W_i=F_i^{\star \star }(x)$. Finally if $y\in W_i$ we have $\begin{array}{rl}(T_i^{\star \star }{)}^{-1}(y)& =\{z\in \mathrm{\Omega }:y\in T_i^{\star \star }(z)\}=\{z\in \mathrm{\Omega }:y\in T_i^{\star }(z)\cap W_i\}=\{z\in \mathrm{\Omega }:y\in T_i^{\star }(z)\}\\ & =\mathrm{\Omega }\cap \{z\in X:y\in T_i(z)\}=\mathrm{\Omega }\cap T_i^{-1}(y)\end{array}$which is open in $\mathrm{\Omega }\cap X=\mathrm{\Omega }$. For each $i\in \{1,\dots .,N_0\}$ we have $F_i^{\star \star }\in {\mathrm{\Phi }}^{\star }(\mathrm{\Omega },W_i)$. Now Theorem 2.1 (with $X_i$ replaced by ${\mathrm{\Omega }}_i$, X replaced by Ω, $F_i$ replaced by $F_i^{\star \star }$, $K_i$ replaced by $W_i$ and $G_i$ replaced by $G_i^{\star \star }$) and Remark 2.3 (note $F_i^{\star \star }\in {\mathrm{\Phi }}^{\star }(\mathrm{\Omega },W_i)$ and $G_i^{\star \star }\in {\mathrm{\Phi }}^{\star }(Y,{\mathrm{\Omega }}_i$)) guarantees there exists a $x\in \mathrm{\Omega }$ and a $y\in Y$ with $x_i\in G_i^{\star \star }(y)=G_i(y)\cap {\mathrm{\Omega }}_i$ for $i\in \{1,\dots ,N\}$ and $y_j\in F_j^{\star \star }(x)=F_j(x)\cap W_j$ for $j\in \{1,\dots ,N_0\}$.

Essentially the same reasoning as in Theorem 2.7 (except Theorem 2.1 is replaced by Theorem 2.2) guarantees the following result.

Theorem 2.8

Let I and J be index sets and let $\{X_i{\}}_{i\in I},$ $\{Y_i{\}}_{i\in J}$ be families of convex sets each in a Hausdorff topological vector space $E_i$. For each $i\in J$ suppose $F_i:X\equiv \prod _{i\in I}{X}_i\to Y_i$ and $F_i\in {\mathrm{\Phi }}^{\star }(X,Y_i)$ and for each $j\in I$ suppose $G_j:Y\equiv \prod _{i\in J}{Y}_i\to X_j$ and in addition assume there exists a map $S_j:Y\to X_j$ with $S_j(y)\ne \mathrm{\varnothing }$ and has convex values for each $y\in Y,$ $S_j(y)\subseteq G_j(y)$ for $y\in Y$ and $S_j^{-1}(w)$ is open $($in $Y)$ for each $w\in X_j$. Also suppose there is a compact subset K of Y and for each $i\in I$ a convex compact subset $Z_i$ of $X_i$ with $S_i(y)\cap Z_i\ne \mathrm{\varnothing }$ for $y\in Y\mathrm{\setminus }K$. Finally assume X is a q-Schauder admissible subset of the Hausdorff topological vector space $E=\prod _{i\in I}{E}_i$. Then there exists a $x\in X$ and a $y\in Y$ with $x_i\in G_i(y)$ for $i\in I$ and $y_j\in F_j(x)$ for $j\in J$.

In Theorem 2.1, we assumed for $i\in \{1,\dots .,N_0\}$ that $T_i(x)\ne \mathrm{\varnothing }$ for $x\in X$ and for $i\in \{1,\dots .,N\}$ that $S_i(y)\ne \mathrm{\varnothing }$ for $y\in Y$. In our next few results we will relax this condition.

Theorem 2.9

Let $\{X_i{\}}_{i=1}^N,$ $\{Y_i{\}}_{i=1}^{N_0}$ be families of convex sets each in a Hausdorff topological vector space $E_i$ with $\prod _{i=1}^N{X}_i$ and $\prod _{i=1}^{N_0}{Y}_i$ paracompact. For each $i\in \{1,\dots ,N_0\}$ suppose $F_i:X\equiv \prod _{i=1}^N{X}_i\to Y_i$ and there exists a map $T_i:X\to Y_i$ with $T_i(x)\subseteq F_i(x)$ for $x\in X,$ $T_i(x)$ has convex values for each $x\in X$ and $T_i^{-1}(w)$ is open (in X) for each $w\in Y_i$. In addition assume for each $i\in \{1,\dots ,N_0\}$ there exists a convex compact set $K_i$ with $F_i(X)\subseteq K_i\subseteq Y_i$. For each $j\in \{1,\dots .N\}$ suppose $G_j:Y\equiv \prod _{i=1}^{N_0}{Y}_i\to X_j$ and there exists a map $S_j:Y\to X_j$ with $S_j(y)\subseteq G_j(y)$ for $y\in Y,$ $S_j(y)$ has convex values for each $y\in Y$ and $S_j^{-1}(w)$ is open $($in $Y)$ for each $w\in X_j$. Finally suppose for each $x\in X$ there exists a $i\in \{1,\dots ,N_0\}$ with $T_i(x)\ne \mathrm{\varnothing }$ and suppose for each $y\in Y$ there exists a $j\in \{1,\dots ,N\}$ with $S_j(y)\ne \mathrm{\varnothing }$. Then there exists a $x\in X,$ a $y\in Y,$ a $j_0\in \{1,\dots ,N_0\}$ and a $i_0\in \{1,\dots ,N\}$ with $y_{j_0}\in F_{j_0}(x)$ and $x_{i_0}\in G_{i_0}(y)$.

Proof.

Note $A_i=\{x\in X:T_i(x)\ne \mathrm{\varnothing }\},i\in \{1,\dots ,N_0\}$ is an open covering of X (recall the fibres of $T_i$ are open). Now from [11, Lemma 5.1.6, p.301] there exists a covering $\{B_i{\}}_{i=1}^{N_0}$ of X where $B_i$ is closed in X and $B_i\subset A_i$ for all $i\in \{1,\dots ,N_0\}$. Also $C_i=\{y\in Y:S_i(y)\ne \mathrm{\varnothing }\},i\in \{1,\dots ,N\}$ is an open covering of Y and from [11, Lemma 5.1.6, p.301] there exists a covering $\{D_i{\}}_{i=1}^N$ of Y where $D_i$ is closed in Y and $D_i\subset C_i$ for all $i\in \{1,\dots ,N\}$. Now for each $i\in \{1,\dots ,N_0\}$ let $H_i:X\to Y_i$ and $J_i:X\to Y_i$ be given by $H_i(x)=\{\begin{array}{ll}{F}_i(x),& x\in B_i\\ Y_i,& x\in X\mathrm{\setminus }{B}_i\end{array}\text{and}{J}_i(x)=\{\begin{array}{ll}{T}_i(x),& x\in B_i\\ Y_i,& x\in X\mathrm{\setminus }{B}_i.\end{array}$For each $i\in \{1,\dots ,N\}$ let $M_i:Y\to X_i$ and $L_i:Y\to X_i$ be given by $M_i(y)=\{\begin{array}{ll}{G}_i(y),& y\in D_i\\ X_i,& y\in Y\mathrm{\setminus }{D}_i\end{array}\text{and}{L}_i(y)=\{\begin{array}{ll}{S}_i(y),& y\in D_i\\ X_i,& y\in Y\mathrm{\setminus }{D}_i.\end{array}$We claim for each $i\in \{1,\dots ,N_0\}$ that $H_i\in {\mathrm{\Phi }}^{\star }(X,Y_i)$. Note first for $i\in \{1,\dots ,N_0\}$ that $J_i(x)\ne \mathrm{\varnothing }$ for $x\in X$ since if $x\in B_i$ then $J_i(x)=T_i(x)\ne \mathrm{\varnothing }$ since $B_i\subset A_i$ whereas if $x\in X\mathrm{\setminus }{B}_i$ then $J_i(x)=Y_i$. Also for $x\in X$ and $i\in \{1,\dots ,N_0\}$ then if $x\in B_i$ we have $J_i(x)=T_i(x)\subseteq F_i(x)=H_i(x)$ whereas if $x\in X\mathrm{\setminus }{B}_i$ we have $J_i(x)=Y_i=H_i(x)$. Also note if $y\in Y_i$ then $\begin{array}{rl}{J}_i^{-1}(y)& =\{z\in X:y\in J_i(z)\}=\{z\in X\mathrm{\setminus }{B}_i:y\in J_i(z)=Y_i\}\cup \{z\in B_i:y\in J_i(z)\}\\ & =(X\mathrm{\setminus }{B}_i)\cup \{z\in B_i:y\in T_i(z)\}=(X\mathrm{\setminus }{B}_i)\cup [B_i\cap \{z\in X:y\in T_i(z)\}]\\ & =(X\mathrm{\setminus }{B}_i)\cup [B_i\cap T_i^{-1}(y)]=X\cap [(X\mathrm{\setminus }{B}_i)\cup T_i^{-1}(y)]=(X\mathrm{\setminus }{B}_i)\cup T_i^{-1}(y)\end{array}$which is open in X (note $T_i^{-1}(y)$ is open in X and $B_i$ is closed in X). Thus for each $i\in \{1,\dots ,N_0\}$ we have $H_i\in {\mathrm{\Phi }}^{\star }(X,Y_i)$. Similar reasoning yields for each $i\in \{1,\dots ,N\}$ that $M_i\in {\mathrm{\Phi }}^{\star }(Y,X_i)$.

Let $K=\prod _{i=1}^N{K}_i$ (note K is compact) and let $M_i^{\star }$ (respectively, $L_i^{\star }$) denote the restriction of $M_i$ (respectively, $L_i$) to K. We note for $i\in \{1,\dots ,N\}$ that $M_i^{\star }\in {\mathrm{\Phi }}^{\star }(K,X_i)$ since if $x\in X_i$ then $\begin{array}{rl}(L_i^{\star }{)}^{-1}(x)& =\{z\in K:x\in L_i^{\star }(z)\}=\{z\in K:x\in L_i(z)\}\\ & =K\cap \{z\in Y:x\in L_i(z)\}=K\cap L_i^{-1}(x)\end{array}$which is open in $K\cap Y=K$. Thus for $i\in \{1,\dots ,N\}$ from [2, 4] there exists a continuous (single-valued) selection $q_i:K\to X_i$ of $M_i^{\star }$ with $q_i(y)\in L_i^{\star }(y)\subseteq M_i^{\star }(y)$ for $y\in K$ and there exists a finite subset $R_i$ of $X_i$ with $q_i(K)\subseteq \mathrm{co}(R_i)\equiv Q_i$. Let $Q=\prod _{i=1}^N{Q}_i(\subseteq X)$.

Let $H_i^{\star }$ (respectively, $J_i^{\star }$) denote the restriction of $H_i$ (respectively, $J_i$) to Q. Note for $i\in \{1,\dots ,N_0\}$ that $H_i^{\star }\in {\mathrm{\Phi }}^{\star }(Q,Y_i)$ since for $y\in Y_i$ we have $(J_i^{\star }{)}^{-1}(y)=\{z\in Q:y\in J_i^{\star }(z)\}=\{z\in Q:y\in J_i(z)\}=Q\cap \{z\in X:y\in J_i(z)\}=Q\cap J_i^{-1}(y)$which is open in $Q\cap X=Q$. Now for $i\in \{1,\dots ,N_0\}$ let $H_i^{\star \star }$ be given by $H_i^{\star \star }(x)=H_i^{\star }(x)\cap K_i$ for $x\in Q$ and we claim $H_i^{\star \star }\in {\mathrm{\Phi }}^{\star }(Q,K_i)$. To see this let $J_i^{\star \star }$ be given by $J_i^{\star \star }(x)=J_i^{\star }(x)\cap K_i$ for $x\in Q$. First note for each $i\in \{1,\dots ,N_0\}$ that $J_i^{\star \star }(x)\ne \mathrm{\varnothing }$ for $x\in Q$ since if $x\in B_i\cap Q$ then $J_i^{\star \star }(x)=T_i(x)\cap K_i\ne \mathrm{\varnothing }$ since $B_i\subset A_i$ and $T_i(x)\subseteq F_i(x)\subseteq K_i$ whereas if $x\in Q\mathrm{\setminus }{B}_i$ then $J_i^{\star \star }(x)=Y_i\cap K_i\ne \mathrm{\varnothing }$. Next note if $x\in Q$ then $J_i^{\star \star }(x)=J_i^{\star }(x)\cap K_i\subseteq H_i^{\star }(x)\cap K_i=H_i^{\star \star }(x)$. Also note if $y\in K_i$ then $\begin{array}{rl}(J_i^{\star \star }{)}^{-1}(y)& =\{z\in Q:y\in J_i^{\star \star }(z)\}=\{z\in Q:y\in J_i^{\star }(z)\cap K_i\}\\ & =Q\cap \{z\in X:y\in J_i(z)\cap K_i\}=Q\cap \{z\in X:y\in J_i(z)\}=Q\cap J_i^{-1}(y)\end{array}$which is open in $Q\cap X=Q$. Thus for $i\in \{1,\dots ,N_0\}$ we have $H_i^{\star \star }\in {\mathrm{\Phi }}^{\star }(Q,K_i)$ and recall Q is compact so from [2, 4] there exists a continuous (single-valued) selection $h_i:Q\to K_i$ of $H_i^{\star \star }$ with $h_i(x)\in J_i^{\star \star }(x)\subseteq H_i^{\star \star }(x)$ for $x\in Q$. Let $h(x)=\prod _{i=1}^{N_0}{h}_i(x)\text{for}x\in Q\text{and}q(y)=\prod _{i=1}^N{q}_i(y)\text{for}y\in K;$note $h:Q\to K$ and $q:K\to Q$ are continuous (recall $h_i:Q\to K_i$ and $q_i:K\to Q_i$). Consider the continuous map $\theta :Q\to Q$ given by $\theta (x)=q(h(x))$ for $x\in Q$. Note Q is a compact convex subset in a finite-dimensional subspace of $E=\prod _{i=1}^N{E}_i$ so Brouwer's fixed-point theorem guarantees that there exists a $x\in Q$ with $x=\theta (x)=q(h(x))$. Let $y=h(x)$ so $x=q(y)$. Then since $x\in Q$ we have $y_j=h_j(x)\in J_j^{\star \star }(x)\subseteq H_j^{\star \star }(x)$ i.e. $y_j\in H_j^{\star }(x)\cap K_j$ (i.e. $y_j\in H_j(x)$) for $j\in \{1,\dots ,N_0\}$ and since $y\in K$ we have $x_i=q_i(y)\in L_i^{\star }(y)\subseteq M_i^{\star }(y)$ i.e. $x_i\in M_i^{\star }(y)=M_i(y)$ for $i\in \{1,\dots ,N\}$. Next since $\{B_i{\}}_{i=1}^{N_0}$ is a covering of X there exists a $j_0\in \{1,\dots ,N_0\}$ with $x\in B_{j_0}$ so $y_{j_0}\in H_{j_0}(x)=F_{j_0}(x)$. In addition since $\{D_i{\}}_{i=1}^N$ is a covering of Y there exists a $i_0\in \{1,\dots ,N\}$ with $x\in D_{i_0}$ so $x_{i_0}\in M_{i_0}(y)=G_{i_0}(y)$.

Remark 2.10

In the statement of Theorem 2.9, we could replace ‘there exists a map $T_i:X\to Y_i$ with $T_i(x)\subseteq F_i(x)$ for $x\in X$, $T_i(x)$ has convex values for each $x\in X$ and $T_i^{-1}(w)$ is open (in X) for each $w\in Y_i$’ with ‘there exists a map $T_i:X\to K_i$ with $T_i(x)\subseteq F_i(x)$ for $x\in X$, $T_i(x)$ has convex values for each $x\in X$ and $T_i^{-1}(w)$ is open (in X) for each $w\in K_i$’. Here we define $H_i:X\to K_i$ and $J_i:X\to K_i$ by $H_i(x)=\{\begin{array}{ll}{F}_i(x),& x\in B_i\\ K_i,& x\in X\mathrm{\setminus }{B}_i\end{array}\text{and}{J}_i(x)=\{\begin{array}{ll}{T}_i(x),& x\in B_i\\ K_i,& x\in X\mathrm{\setminus }{B}_i.\end{array}$The reasoning in Theorem 2.9 guarantees that for each $i\in \{1,\dots ,N_0\}$ we have $H_i\in {\mathrm{\Phi }}^{\star }(X,K_i)$. Also as in Theorem 2.9 for each $i\in \{1,\dots ,N\}$ we have $M_i\in {\mathrm{\Phi }}^{\star }(Y,X_i)$ and $M_i^{\star }\in {\mathrm{\Phi }}^{\star }(K,X_i)$ (here $M_i$ and $M_i^{\star }$ are as in Theorem 2.9) and so there exists a continuous (single-valued) selection $q_i:K\to X_i$ of $M_i^{\star }$ with $q_i(y)\in L_i^{\star }(y)\subseteq M_i^{\star }(y)$ for $y\in K$ (here $L_i$ and $L_i^{\star }$ are as in Theorem 2.9) and there exists a finite subset $R_i$ of $X_i$ with $q_i(K)\subseteq \mathrm{co}(R_i)\equiv Q_i$. Let $Q=\prod _{i=1}^N{Q}_i(\subseteq X)$. Next let $H_i^{\star }$ (respectively, $J_i^{\star }$) denote the restriction of $H_i$ (respectively, $J_i$) to Q and note for $i\in \{1,\dots ,N_0\}$ that $H_i^{\star }\in {\mathrm{\Phi }}^{\star }(Q,K_i)$ since for $y\in K_i$ we have $(J_i^{\star }{)}^{-1}(y)=Q\cap J_i^{-1}(y)$ which is open in $Q\cap X=Q$ so from [2, 4] there exists a continuous (single-valued) selection $h_i:Q\to K_i$ of $H_i^{\star }$ with $h_i(x)\in J_i^{\star }(x)\subseteq H_i^{\star }(x)$ for $x\in Q$. Let $h(x)=\prod _{i=1}^{N_0}{h}_i(x)\text{for}x\in Q\text{and}q(y)=\prod _{i=1}^N{q}_i(y)\text{for}y\in K$and let $\theta :Q\to Q$ (note $h:Q\to K$ since $h_i:Q\to K_i$ and $q:K\to Q$ since $q_i:K\to Q_i$) be given by $\theta (x)=q(h(x))$ for $x\in Q$ and so there exists a $x\in Q$ with $x=\theta (x)=q(h(x))$ Let $y=h(x)$ so $x=q(y)$ and since $x\in Q$ we have $y_j=h_j(x)\in J_j^{\star }(x)\subseteq H_j^{\star }(x)=H_j(x)$ for $j\in \{1,\dots ,N_0\}$ and since $y\in K$ we have $x_i=q_i(y)\in L_i^{\star }(y)\subseteq M_i^{\star }(y)=M_i(y)$ for $i\in \{1,\dots ,N\}$. Next since $\{B_i{\}}_{i=1}^{N_0}$ is a covering of X and $\{D_i{\}}_{i=1}^N$ is a covering of Y we have our result as in Theorem 2.9.

The same reasoning as in Theorem 2.9 except Theorem 1.1 is used instead of Brouwer's fixed point theorem immediately yields our next result.

Theorem 2.11

Let I and J be index sets and let $\{X_i{\}}_{i\in I},$ $\{Y_i{\}}_{i\in J}$ be families of convex sets each in a Hausdorff topological vector space $E_i$ with $\prod _{i\in I}{X}_i$ and $\prod _{i\in J}{Y}_i$ paracompact. For each $i\in J$ suppose $F_i:X\equiv \prod _{i\in I}{X}_i\to Y_i$ and there exists a map $T_i:X\to Y_i$ with $T_i(x)\subseteq F_i(x)$ for $x\in X,$ $T_i(x)$ has convex values for each $x\in X$ and $T_i^{-1}(w)$ is open $($in $X)$ for each $w\in Y_i$. In addition assume for each $i\in J$ there exists a convex compact set $K_i$ with $F_i(X)\subseteq K_i\subseteq Y_i$. For each $j\in I$ suppose $G_j:Y\equiv \prod _{i\in J}{Y}_i\to X_j$ and there exists a map $S_j:Y\to X_j$ with $S_j(y)\subseteq G_j(y)$ for $y\in Y,$ $S_j(y)$ has convex values for each $y\in Y$ and $S_j^{-1}(w)$ is open $($in $Y)$ for each $w\in X_j$. Also assume X is a q-Schauder admissible subset of the Hausdorff topological vector space $E=\prod _{i\in I}{E}_i$. Finally suppose for each $x\in X$ there exists a $i\in J$ with $T_i(x)\ne \mathrm{\varnothing }$ and suppose for each $y\in Y$ there exists a $j\in I$ with $S_j(y)\ne \mathrm{\varnothing }$. Then there exists a $x\in X,$ a $y\in Y,$ a $j_0\in J$ and a $i_0\in I$ with $y_{j_0}\in F_{j_0}(x)$ and $x_{i_0}\in G_{i_0}(y)$.

Remark 2.12

If in Theorem 2.11 we replace ‘suppose for each $x\in X$ there exists a $i\in J$ with $T_i(x)\ne \mathrm{\varnothing }$ and suppose for each $y\in Y$ there exists a $j\in I$ with $S_j(y)\ne \mathrm{\varnothing }$’ with ‘suppose there exists a finite subset $J_0$ of J such that for each $x\in X$ there exists a $i\in J_0$ with $T_i(x)\ne \mathrm{\varnothing }$ and also suppose there exists a finite subset $I_0$ of I such that for each for each $y\in Y$ there exists a $j\in I_0$ with $S_j(y)\ne \mathrm{\varnothing }$’ then X being a q-Schauder admissible subset of the Hausdorff topological vector space E can be removed. We use the Brouwer fixed-point theorem instead of Theorem 1.1 since $A_i=\{x\in X:T_i(x)\ne \mathrm{\varnothing }\},i\in J$ is an open covering of X (respectively, $C_i=\{y\in Y:S_i(y)\ne \mathrm{\varnothing }\},i\in I$ is an open covering of Y) can be replaced by $A_i=\{x\in X:T_i(x)\ne \mathrm{\varnothing }\},i\in J_0$ is an open covering of X (respectively, $C_i=\{y\in Y:S_i(y)\ne \mathrm{\varnothing }\},i\in I_0$ is an open covering of Y) and proceed as in Theorem 2.9 with $\{1,\dots ,N\}$ replaced by $I_0$ and $\{1,\dots ,N_0\}$ replaced by $J_0$. An example of the above situation is if $X\equiv \prod _{i\in I}{X}_i$ and $Y\equiv \prod _{i\in J}{Y}_i$ are compact (of course no reference to paracompactness is needed in this situation and one could restate Theorem 2.11).

Other classes of maps could also be considered. We illustrate this in our next result.

Theorem 2.13

Let $\{X_i{\}}_{i=1}^N,$ $\{Y_i{\}}_{i=1}^{N_0}$ be families of convex sets each in a Hausdorff topological vector space $E_i$ with $\prod _{i=1}^{N_0}{Y}_i$ paracompact. For each $i\in \{1,\dots ,N_0\}$ suppose $F_i:X\equiv \prod _{i=1}^N{X}_i\to Y_i$ and $F_i\in Ad(X,Y_i)$ and in addition assume there exists a compact set $K_i$ with $F_i(X)\subseteq K_i\subseteq Y_i$. For each $j\in \{1,\dots .N\}$ suppose $G_j:Y\equiv \prod _{i=1}^{N_0}{Y}_i\to X_j$ and there exists a map $S_j:Y\to X_j$ with $S_j(y)\subseteq G_j(y)$ for $y\in Y,$ $S_j(y)$ has convex values for each $y\in Y$ and $S_j^{-1}(w)$ is open (in Y) for each $w\in X_j$. Finally suppose for each $y\in Y$ there exists a $j\in \{1,\dots ,N\}$ with $S_j(y)\ne \mathrm{\varnothing }$. Then there exists a $x\in X,$ a $y\in Y,$ a $i_0\in \{1,\dots ,N\}$ with $y_j\in F_j(x)$ for all $j\in \{1,\dots ,N_0\}$ and $x_{i_0}\in G_{i_0}(y)$.

Proof.

Let $C_i$, $D_i$, $M_i$ and $L_i$ be as in Theorem 2.9. The reasoning in Theorem 2.9 guarantees for each $i\in \{1,\dots ,N\}$ that $M_i\in {\mathrm{\Phi }}^{\star }(Y,X_i)$. Let $K=\prod _{i=1}^N{K}_i$ (note K is compact) and let $M_i^{\star }$ (respectively, $L_i^{\star }$) denote the restriction of $M_i$ (respectively, $L_i$) to K and as in Theorem 2.9 we have for $i\in \{1,\dots ,N\}$ that $M_i^{\star }\in {\mathrm{\Phi }}^{\star }(K,X_i)$ so there exists a continuous (single-valued) selection $q_i:K\to X_i$ of $M_i^{\star }$ with $q_i(y)\in L_i^{\star }(y)\subseteq M_i^{\star }(y)$ for $y\in K$ and there exists a finite subset $R_i$ of $X_i$ with $q_i(K)\subseteq \mathrm{co}(R_i)\equiv Q_i$. Let $Q=\prod _{i=1}^N{Q}_i(\subseteq X)$ and let $F_i^{\star }$ denote the restriction of $F_i$ to Q and let $F^{\star }(x)=\prod _{i=1}^{N_0}{F}_i^{\star }(x)$ for $x\in Q$. Note $F^{\star }\in Ad(Q,Y)$ and note since $F_i^{\star }(Q)\subseteq F_i(X)\subseteq K_i$ for $i\in \{1,\dots ,N_0\}$ then $F^{\star }(Q)\subseteq K$. Let $q(y)=\prod _{i=1}^N{q}_i(y)\text{for}y\in K$and note $q:K\to Q$ since $q_i:K\to Q_i$ for $i\in \{1,\dots ,N\}$. Now $q{F}^{\star }\in Ad(Q,Q)$ and note Q is a compact convex subset in a finite-dimensional subspace of $E=\prod _{i=1}^N{E}_i$ so Theorem 1.1 guarantees a $x\in Q$ with $x\in q(F^{\star }(x))$. Now let $y\in F^{\star }(x)$ with $x=q(y)$. Note $y\in F^{\star }(x)\subseteq F^{\star }(Q)\subseteq K$ and $y_j\in F_j(x)$ for all $j\in \{1,\dots ,N_0\}$. Also since $x\in Q$ we have $x_i=q_i(y)\in L_i^{\star }(y)\subseteq M_i^{\star }(y)=M_i(y)$ for $i\in \{1,\dots ,N\}$. Now since $\{D_i{\}}_{i=1}^N$ is a covering of Y then there exists a $i_0\in \{1,\dots ,N_0\}$ with $y\in D_{i_0}$ so $x_{i_0}\in M_{i_0}(y)=G_{i_0}(y)$.

Remark 2.14

It is easy to adjust the above argument so Ad maps can be replaced by PK maps in Theorem 2.13 once we assume $F\in PK(X,Y)$ where $F(x)=\prod _{i=1}^{N_0}{F}_i(x)$ for $x\in X$. Also there is an analogue result when $\{X_i{\}}_{i=1}^N$, $\{Y_i{\}}_{i=1}^{N_0}$ are replaced by $\{X_i{\}}_{i\in I}$, $\{Y_i{\}}_{i\in J}$ where I and J are index sets.

Our next result considers the coercive case.

Theorem 2.15

Let $\{X_i{\}}_{i=1}^N,$ $\{Y_i{\}}_{i=1}^{N_0}$ be families of convex sets each in a Hausdorff topological vector space $E_i$ with $\prod _{i=1}^N{X}_i$ and $\prod _{i=1}^{N_0}{Y}_i$ paracompact. For each $i\in \{1,\dots ,N_0\}$ suppose $F_i:X\equiv \prod _{i=1}^N{X}_i\to Y_i$ and there exists a map $T_i:X\to Y_i$ with $T_i(x)\subseteq F_i(x)$ for $x\in X,$ $T_i(x)$ has convex values for each $x\in X$ and $T_i^{-1}(w)$ is open $($in $X)$ for each $w\in Y_i$. For each $j\in \{1,\dots .N\}$ suppose $G_j:Y\equiv \prod _{i=1}^{N_0}{Y}_i\to X_j$ and there exists a map $S_j:Y\to X_j$ with $S_j(y)\subseteq G_j(y)$ for $y\in Y$, $S_j(y)$ has convex values for each $y\in Y$ and $S_j^{-1}(w)$ is open $($in $Y)$ for each $w\in X_j$. In addition assume there is a compact subset K of Y and for each $i\in \{1,\dots ,N\}$ a convex compact subset $Z_i$ of $X_i$ such that for each $y\in Y\mathrm{\setminus }K$ there exists a $i\in \{1,\dots ,N\}$ with $S_i(y)\cap Z_i\ne \mathrm{\varnothing }$. Finally suppose for each $x\in X$ there exists a $i\in \{1,\dots ,N_0\}$ with $T_i(x)\ne \mathrm{\varnothing }$ and suppose for each $y\in Y$ there exists a $j\in \{1,\dots ,N\}$ with $S_j(y)\ne \mathrm{\varnothing }$. Then there exists a $x\in X,$ a $y\in Y,$ a $j_0\in \{1,\dots ,N_0\}$ and a $i_0\in \{1,\dots ,N\}$ with $y_{j_0}\in F_{j_0}(x)$ and $x_{i_0}\in G_{i_0}(y)$.

Proof.

Let $A_i$, $B_i$, $C_i$, $D_i$, $H_i$, $J_i$, $M_i$ and $L_i$ be as in Theorem 2.9. The same reasoning as in Theorem 2.9 guarantees that $H_i\in {\mathrm{\Phi }}^{\star }(X,Y_i)$ for $i\in \{1,\dots ,N_0\}$ and $M_i\in {\mathrm{\Phi }}^{\star }(Y,X_i)$ for $i\in \{1,\dots ,N\}$. Let K be as in the statement of Theorem 2.15 and let $M_i^{\star }$ (respectively, $L_i^{\star }$) denote the restriction of $M_i$ (respectively, $L_i$) to K. The same reasoning as in Theorem 2.9 guarantees that for $i\in \{1,\dots ,N\}$ we have $M_i^{\star }\in {\mathrm{\Phi }}^{\star }(K,X_i)$ so from [2, 4] there exists a continuous (single-valued) selection $q_i:K\to X_i$ of $M_i^{\star }$ with $q_i(y)\in L_i^{\star }(y)\subseteq M_i^{\star }(y)$ for $y\in K$ and there exists a finite subset $R_i$ of $X_i$ with $q_i(K)\subseteq \mathrm{co}(R_i)$. Let ${\mathrm{\Omega }}_i=\mathrm{co}(\mathrm{co}(R_i)\cup Z_i)\text{for}i\in \{1,\dots ,N\}$which is a convex compact subset of $X_i$. Let $G_i^{\star \star }(y)=G_i(y)\cap {\mathrm{\Omega }}_i\text{and}{S}_i^{\star \star }(y)=S_i(y)\cap {\mathrm{\Omega }}_i\text{for}y\in Y\text{and}i\in \{1,\dots ,N\}.$Note for $i\in \{1,\dots ,N\}$ and $y\in Y$ that $S_i^{\star \star }(y)=S_i(y)\cap {\mathrm{\Omega }}_i\subseteq G_i(y)\cap {\mathrm{\Omega }}_i=G_i^{\star \star }(y)$ and if $x\in {\mathrm{\Omega }}_i$ then $\begin{array}{rl}(S_i^{\star \star }{)}^{-1}(x)& =\{z\in Y:x\in S_i^{\star \star }(z)\}=\{z\in Y:x\in S_i(z)\cap {\mathrm{\Omega }}_i\}\\ & =\{z\in Y:x\in S_i(z)\}=S_i^{-1}(x)\end{array}$which is open in Y. Next fix $y\in Y$. We now claim there exists a $j\in \{1,\dots ,N\}$ with $S_j^{\star \star }(y)\ne \mathrm{\varnothing }$. This is immediate if $y\in Y\mathrm{\setminus }K$ since from one of our assumptions in the statement of Theorem 2.15 there exists a $j\in \{1,\dots ,N\}$ with $S_j(y)\cap Z_j\ne \mathrm{\varnothing }$ so $S_j^{\star \star }(y)=S_j(y)\cap {\mathrm{\Omega }}_j\ne \mathrm{\varnothing }$ since $Z_j\subseteq {\mathrm{\Omega }}_j$. It remains to consider the case when $y\in K$. Since $\{D_i{\}}_{i=1}^N$ is a covering of Y there exists a $j_0\in \{1,\dots ,N\}$ with $y\in D_{j_0}$, and note $q_{j_0}(y)\in L_{j_0}^{\star }(y)=S_{j_0}(y)$ since $y\in D_{j_0}$ and $q_{j_0}(y)\in \mathrm{co}(R_{j_0})\subseteq {\mathrm{\Omega }}_{j_0}$, so $S_{j_0}^{\star \star }(y)=S_{j_0}(y)\cap {\mathrm{\Omega }}_{j_0}\ne \mathrm{\varnothing }$. Combining all the above we see there exists a $j\in \{1,\dots ,N\}$ with $S_j^{\star \star }(y)\ne \mathrm{\varnothing }$.

Let $\mathrm{\Omega }=\prod _{i=1}^N{\mathrm{\Omega }}_i(\subseteq X)$ and let $H_i^{\star }$ (respectively, $J_i^{\star }$) denote the restriction of $H_i$ (respectively, $J_i$) to Ω. Now for $i\in \{1,\dots ,N_0\}$ since $H_i\in {\mathrm{\Phi }}^{\star }(X,Y_i)$ then it is immediate that $H_i^{\star }\in {\mathrm{\Phi }}^{\star }(\mathrm{\Omega },Y_i)$ (note for $y\in Y_i$ we have $(J_i^{\star }{)}^{-1}(y)=\{z\in \mathrm{\Omega }:y\in J_i^{\star }(z)\}=\{z\in \mathrm{\Omega }:y\in J_i(z)\}=\mathrm{\Omega }\cap J_i^{-1}(y)$) so from [2, 4] there exists a continuous (single-valued) selection $h_i:\mathrm{\Omega }\to Y_i$ of $H_i^{\star }$ with $h_i(x)\in J_i^{\star }(x)\subseteq H_i^{\star }(x)$ for $x\in \mathrm{\Omega }$ and also there exists a finite set $V_i$ of $Y_i$ with $h_i(\mathrm{\Omega })\subseteq \mathrm{co}(V_i)\equiv W_i$ (note $W_i$ is a convex compact subset of $Y_i$). For $i\in \{1,\dots .,N_0\}$ let $F_i^{\star \star }(x)=F_i(x)\cap W_i\text{and}{T}_i^{\star \star }(x)=T_i(x)\cap W_i\text{for}x\in \mathrm{\Omega }.$Note for $i\in \{1,\dots ,N_0\}$ and $x\in \mathrm{\Omega }$ that $T_i^{\star \star }(x)=T_i(x)\cap W_i\subseteq F_i(x)\cap W_i=F_i^{\star \star }(x)$ and if $y\in W_i$ then $\begin{array}{rl}(T_i^{\star \star }{)}^{-1}(y)& =\{z\in \mathrm{\Omega }:y\in T_i^{\star \star }(z)\}=\{z\in \mathrm{\Omega }:y\in T_i(z)\cap W_i\}\\ & =\{z\in \mathrm{\Omega }:y\in T_i(z)\}=\mathrm{\Omega }\cap T_i^{-1}(y)\end{array}$which is open in Ω. Next fix $x\in X$. We now claim there exists a $i\in \{1,\dots ,N_0\}$ with $T_i^{\star \star }(x)\ne \mathrm{\varnothing }$. Note since $\{B_i{\}}_{i=1}^{N_0}$ is a covering of X and $\mathrm{\Omega }\subseteq X$ there exists a $i_0\in \{1,\dots ,N_0\}$ with $x\in B_{i_0}$, and note $h_{i_0}(\mathrm{\Omega })\subseteq \mathrm{co}(V_{i_0})=W_{i_0}$ and $h_{i_0}(x)\in J_{i_0}^{\star }(x)=T_{i_0}(x)$ since $x\in B_{i_0}$ and so $T_{i_0}^{\star \star }(x)=T_{i_0}(x)\cap W_{i_0}\ne \mathrm{\varnothing }$.

Now Theorem 2.9 (with $X_i$ replaced by ${\mathrm{\Omega }}_i$, X replaced by Ω, $F_i$ replaced by $F_i^{\star \star }$, $T_i$ replaced by $T_i^{\star \star }$, $K_i$ replaced by $W_i$, $G_i$ replaced by $G_i^{\star \star }$ and $S_i$ replaced by $S_i^{\star \star }$) and Remark 2.10 (note $F_i^{\star \star }:\mathrm{\Omega }\to W_i$ and $G_i^{\star \star }:Y\to {\mathrm{\Omega }}_i$) guarantees that there exists a $x\in \mathrm{\Omega }$, a $y\in Y$, a $j_0\in \{1,\dots ,N_0\}$ and a $i_0\in \{1,\dots ,N\}$ with $y_{j_0}\in F_{j_0}^{\star \star }(x)=F_{j_0}(x)\cap W_{j_0}$ and $x_{i_0}\in G_{i_0}^{\star \star }(y)=G_{i_0}(y)\cap {\mathrm{\Omega }}_{i_0}$.

Remark 2.16

There is an analogue Theorem 2.15 type result when $\{X_i{\}}_{i=1}^N$, $\{Y_i{\}}_{i=1}^{N_0}$ are replaced by $\{X_i{\}}_{i\in I}$, $\{Y_i{\}}_{i\in J}$ where I and J are index sets.

Our coincidence results will generate analytic alternatives. To illustrate this, we will obtain an analytic alternative off Theorems 2.9 and 2.15 which will then generate a minimax inequality.

Theorem 2.17

Let $\{X_i{\}}_{i=1}^N,$ $\{Y_i{\}}_{i=1}^{N_0}$ be families of convex sets each in a Hausdorff topological vector space $E_i$ with $X=\prod _{i=1}^N{X}_i$ and $Y=\prod _{i=1}^{N_0}{Y}_i$ paracompact. For each $i\in \{1,\dots ,N_0\}$ let $f_i,t_i:X\times Y_i\to \mathbf{R}$ with $t_i(x,y_i)\le f_i(x,y_i)$ for $(x,y_i)\in X\times Y_i$. For each $i\in \{1,\dots ,N\}$ let $g_i,s_i:X_i\times Y\to \mathbf{R}$ with $g_i(x_i,y)\le s_i(x_i,y)$ for $(x_i,y)\in X_i\times Y$. Let $\lambda \in \mathbf{R}$ and let $\begin{array}{rl}& F_i(x)=\{y_i\in Y_i:t_i(x,y_i)<\lambda \},T_i(x)=\{y_i\in Y_i:f_i(x,y_i)<\lambda \}\text{for}x\in X,i\in \{1,\dots ,N_0\}\\ & G_i(y)=\{x_i\in X_i:s_i(x_i,y)>\lambda \},S_i(y)=\{x_i\in X_i:g_i(x_i,y)>\lambda \}\text{for}y\in Y,i\in \{1,\dots ,N\}\end{array}$and assume $\underset{j\in \{1,\dots .,N\}}{sup}{s}_j(x_j,y)\le \underset{i\in \{1,\dots ,N_0\}}{inf}{t}_i(x,y_i)$ for $(x,y)\in X\times Y$ where $x_j$ $($respectively, $y_i)$ denote the projection of x $($respectively, $y)$ on $X_j$$($respectively, $Y_i)$. Suppose for each $i\in \{1,\dots .,N_0\}$ that $T_i(x)$ has convex values for each $x\in X$ and $T_i^{-1}(w)$ is open $($in $X)$ for each $w\in Y_i$ and also for each $i\in \{1,\dots .N\}$ that $S_i(y)$ has convex values for each $y\in Y$ and $S_i^{-1}(w)$ is open $($in $Y)$ for each $w\in X_i$. In addition suppose either

1. for each $i\in \{1,\dots ,N_0\}$ there exists a convex compact set $K_i$ with $F_i(X)\subseteq K_i\subseteq Y_i,$

   or

2. there is a compact subset K of Y and for each $i\in \{1,\dots ,N\}$ a convex compact subset $Z_i$ of $X_i$ such that for each $y\in Y\mathrm{\setminus }K$ there exists a $i\in \{1,\dots ,N\}$ with $S_i(y)\cap Z_i\ne \mathrm{\varnothing },$

   hold. Then either

1. there exists a $y\in Y$ with $\underset{i\in \{1,\dots ,N\}}{sup}\underset{x_i\in X_i}{sup}{g}_i(x_i,y)\le \lambda ,$

   or

2. there exists a $x\in X$ with $\underset{i\in \{1,\dots ,N_0\}}{inf}\underset{y_i\in Y_i}{inf}{f}_i(x,y_i)\ge \lambda ,$

   occurs.

Proof.

Now either there exists a $x\in X$ with $T_i(x)=\mathrm{\varnothing }$ for $i\in \{1,\dots ,N_0\}$ or it does not hold. Similarly either there exists a $y\in Y$ with $S_i(y)=\mathrm{\varnothing }$ for $i\in \{1,\dots ,N\}$ or it does not hold. So basically we have there cases to consider.

Suppose first that there exists a $y\in Y$ with $S_i(y)=\mathrm{\varnothing }$ for $i\in \{1,\dots ,N\}$. Then $g_i(x_i,y)\le \lambda$ for all $x_i\in X_i$ and $i\in \{1,\dots ,N\}$ so $\underset{i\in \{1,\dots ,N\}}{sup}\underset{x_i\in X_i}{sup}{g}_i(x_i,y)\le \lambda$ (i.e. (A1) occurs).

Suppose next that there exists a $x\in X$ with $T_i(x)=\mathrm{\varnothing }$ for $i\in \{1,\dots ,N_0\}$. Then $f_i(x,y_i)\ge \lambda$ for all $y_i\in Y_i$ and $i\in \{1,\dots ,N_0\}$ so $\underset{i\in \{1,\dots ,N_0\}}{inf}\underset{y_i\in Y_i}{inf}{f}_i(x,y_i)\ge \lambda$ (i.e. (A2) occurs).

Finally consider the case that for each $x\in X$ there exists a $i\in \{1,\dots ,N_0\}$ with $T_i(x)\ne \mathrm{\varnothing }$ and for each $y\in Y$ there exists a $j\in \{1,\dots ,N\}$ with $S_j(y)\ne \mathrm{\varnothing }$. Note $S_i$ is a selection of $G_i$ and $T_i$ is a selection of $F_i$ so Theorem 2.9 (if (1) occurs)) or Theorem 2.15 (if (2) occurs) guarantees a $x\in X$, a $y\in Y$, a $j_0\in \{1,\dots ,N_0\}$ and a $i_0\in \{1,\dots ,N\}$ with $y_{j_0}\in F_{j_0}(x)$ and $x_{i_0}\in G_{i_0}(y)$ i.e. $t_{j_0}(x,y_{j_0})<\lambda <s_{i_0}(x_{i_0},y)$. This contradicts $\underset{j\in \{1,\dots .,N\}}{sup}{s}_j(x_j,y)\le \underset{i\in \{1,\dots ,N_0\}}{inf}{t}_i(x,y_i)$ for $(x,y)\in X\times Y$.

Now we present a minimax inequality of Neumann–Sion-type from Theorem 2.17.

Theorem 2.18

Let $\{X_i{\}}_{i=1}^N,$ $\{Y_i{\}}_{i=1}^{N_0}$ be families of convex sets each in a Hausdorff topological vector space $E_i$ with $X=\prod _{i=1}^N{X}_i$ and $Y=\prod _{i=1}^{N_0}{Y}_i$ paracompact. For $i\in \{1,\dots ,N_0\}$ let $f_i,t_i:X\times Y_i\to \mathbf{R}$ with $t_i(x,y_i)\le f_i(x,y_i)$ for $(x,y_i)\in X\times Y_i$. For $i\in \{1,\dots ,N\}$ let $g_i,s_i:X_i\times Y\to \mathbf{R}$ with $g_i(x_i,y)\le s_i(x_i,y)$ for $(x_i,y)\in X_i\times Y$. For each $\lambda \in \mathbf{R}$ let $\begin{array}{rl}& F_{i,\lambda }(x)=\{y_i\in Y_i:t_i(x,y_i)<\lambda \},T_{i,\lambda }(x)=\{y_i\in Y_i:f_i(x,y_i)<\lambda \}\text{for}x\in X,i\in \{1,\dots ,N_0\}\\ & G_{i,\lambda }(y)=\{x_i\in X_i:s_i(x_i,y)>\lambda \},S_{i,\lambda }(y)=\{x_i\in X_i:g_i(x_i,y)>\lambda \}\text{for}y\in Y,i\in \{1,\dots ,N\}\end{array}$and assume $\underset{j\in \{1,\dots .,N\}}{sup}{s}_j(x_j,y)\le \underset{i\in \{1,\dots ,N_0\}}{inf}{t}_i(x,y_i)$ for $(x,y)\in X\times Y$ where $x_j$ (respectively, $y_i)$ denote the projection of x (respectively, $y)$ on $X_j$ $($respectively, $Y_i)$. For each $\lambda \in \mathbf{R}$ suppose for each $i\in \{1,\dots .,N_0\}$ that $T_{i,\lambda }(x)$ has convex values for each $x\in X$ and $T_{i,\lambda }^{-1}(w)$ is open $($in $X)$ for each $w\in Y_i$ and also for each $i\in \{1,\dots .N\}$ that $S_{i,\lambda }(y)$ has convex values for each $y\in Y$ and $S_{i,\lambda }^{-1}(w)$ is open $($in $Y)$ for each $w\in X_i,$ and in addition suppose either

1. for each $i\in \{1,\dots ,N_0\}$ there exists a convex compact set $K_i$ with $F_i(X)\subseteq K_i\subseteq Y_i,$

   or

2. there is a compact subset K of Y and for each $i\in \{1,\dots ,N\}$ a convex compact subset $Z_i$ of $X_i$ such that for each $y\in Y\mathrm{\setminus }K$ there exists a $i\in \{1,\dots ,N\}$ with $S_{i,\lambda }(y)\cap Z_i\ne \mathrm{\varnothing },$

   hold. Then $\alpha \equiv \underset{y\in Y}{inf}\underset{i\in \{1,\dots ,N\}}{sup}\underset{x_i\in X_i}{sup}{g}_i(x_i,y)\le \underset{x\in X}{sup}\underset{i\in \{1,\dots ,N_0\}}{inf}\underset{y_i\in Y_i}{inf}{f}_i(x,y_i)\equiv \beta .$

Proof.

Let $\beta <\mathrm{\infty }$ and $\alpha >-\mathrm{\infty }$ (otherwise we are finished). Suppose $\beta <\alpha$. Then there exists a λ with $\beta <\lambda <\alpha$. We will now apply Theorem 2.17. If there exists a $y\in Y$ with $\underset{i\in \{1,\dots ,N\}}{sup}\underset{x_i\in X_i}{sup}{g}_i(x_i,y)\le \lambda$ then $\alpha =\underset{y\in Y}{inf}\underset{i\in \{1,\dots ,N\}}{sup}\underset{x_i\in X_i}{sup}{g}_i(x_i,y)\le \lambda$ and this contradicts $\lambda <\alpha$. On the other hand if there exists a $x\in X$ with $\underset{i\in \{1,\dots ,N_0\}}{inf}\underset{y_i\in Y_i}{inf}{f}_i(x,y_i)\ge \lambda$ then $\beta =\underset{x\in X}{sup}\underset{i\in \{1,\dots ,N_0\}}{inf}\underset{y_i\in Y_i}{inf}{f}_i(x,y_i)\ge \lambda$ and this contradicts $\beta <\lambda$. In both cases (see Theorem 2.17) we get a contradiction. Thus $\beta \ge \alpha$.

Disclosure statement

No potential conflict of interest was reported by the author(s).