A note on noncompact collectively coincidence point theory

Abstract

In this paper we present a variety of collectively fixed point and coincidence point results where the maps are not necessarily compact. Our arguments are based on a very general fixed point result in the literature.

1 Introduction

Some collectively coincidence type results are presented in this paper. In particular we consider maps with continuous selections, maps with upper semicontinuous selections and multivalued maps which are admissible with respect to Gorniewicz. We also note that we do not assume that the maps are compact so in some sense the noncompact case is considered in this paper. The arguments presented in this paper are based on a general result of the author [3, 11] in the literature.

Now we describe the maps considered in this paper. Let $H\,$ be the C̆ech 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 C̆ech homology group with compact carriers of $\,X$. For a continuous map $f:X \rightarrow X$, H(f) is the induced linear map $f_{\star }=\{ f_{\star \,q} \}$ where $\,f_{\star \,q}:H_q(X) \rightarrow H_q(X)$. 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 $\Gamma$ be Hausdorff topological spaces. A continuous single-valued map $p:\Gamma \rightarrow X$ is called a Vietoris map (written $p: \Gamma \Rightarrow X$) if the following two conditions are satisfied:

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

• (ii) 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 $\phi :X \rightarrow Y$ be a multivalued map (note for each $x\in X$ we assume $\phi (x)$ is a nonempty subset of $\,Y$). A pair (p, q) of single-valued continuous maps of the form $X {\mathop {\leftarrow }\limits ^{p}} \Gamma {\mathop {\rightarrow }\limits ^{q}} Y$ is called a selected pair of $\,\phi$ (written $(p,q) \subset \phi$) if the following two conditions hold:

• (i) p is a Vietoris map and

• (ii) $q\,(p^{-1}(x))\subset \phi (x)$ for any $x\in X$.

Now we define the admissible maps of Gorniewicz [6]. An upper semicontinuous map $\phi : X \rightarrow Y$ with compact values is said to be admissible (and we write $\phi \in \text {Ad}(X,Y)$) provided there exists a selected pair (p, q) of $\phi$. An example of an admissible map is a Kakutani map. An upper semicontinuous map $\phi : X \rightarrow K(Y)$ is said to be Kakutani (and we write $\phi \in \text {Kak}(X,Y)$); 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 F a multifunction. We say $F \in HLPY(Z,W)$ [9, 10] if W is convex and there exists a map $S:Z \rightarrow W$ with $\text {co}\,(S(x)) \subseteq F(x)$ for $x\in Z$, $S(x) \ne \emptyset$ for each $x\in Z$ and $Z=\bigcup \,\{ \,\text {int}\,S^{-1}(w):\,\,w\in W\}$; here $S^{-1}(w)=\{z\in Z:\,w \in S(z)\}$ and note $S(x) \ne \emptyset$ for each $x\in Z$ is redundant since if $z\in Z$ then there exists a $w\in W$ with $z \in \text {int}\,S^{-1}(w) \subseteq S^{-1}(w)$ so $w \in S(z)$, i.e., $S(z) \ne \emptyset$.

These maps are quite related to the DKT maps in the literature [5]. We say $F \in DKT(Z,W)$ [5] if W is convex and there exists a map $S:Z \rightarrow W$ with $\text {co}\,(S(x)) \subseteq F(x)$ for $x\in Z$, $S(x) \ne \emptyset$ 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 $F:X \rightarrow 2^Y$ (nonempty subsets of Y) belonging to $\,{{{\mathcal {X}}}}$, and ${{{\mathcal {X}}}}_c$ the set of finite compositions of maps in ${{{\mathcal {X}}}}$. We let

$$\begin{aligned} {{{\mathcal {F}}}}({{{\mathcal {X}}}})=\left\{ Z:\,\,\text {Fix}\,F \ne \emptyset \,\,\hbox { for all }\,\, F \in {{{\mathcal {X}}}}(Z,Z) \right\} , \end{aligned}$$

where $\text {Fix}\,F$ denotes the set of fixed points of F.

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

• (i) ${{{\mathcal {U}}}}$ contains the class $\,\textbf{C}$ of single-valued continuous functions;

• (ii) each $F \in {{{\mathcal {U}}}}_c$ is upper semicontinuous and compact valued; and

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

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.

The Kakutani maps and the admissible maps of Gorniewicz are in the above class.

For a subset K of a topological space X, we denote by $\text {Cov}_X\,(K)$ the directed set of all coverings of K by open sets of X (usually we write $\text {Cov}\,(K)= \text {Cov}_X\,(K)$). Given two maps $F,G:X \rightarrow 2^Y$ and $\alpha \in \text {Cov}\,(Y)$, F and G are said to be $\alpha$-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 $Y \in ES(Q)$) if for any pair (X, K) in Q with $K \subseteq X$ closed, any continuous function $f_0:K \rightarrow Y$ extends to a continuous function $f:X \rightarrow Y$. A space Y is an approximate extension space for Q (written $Y \in AES(Q)$) if for any $\alpha \in \text {Cov}\,(Y)$ and any pair (X, K) in Q with $K \subseteq X$ closed, and any continuous function $f_0:K \rightarrow Y$ there exists a continuous function $f:X \rightarrow Y$ such that $f|_K$ is $\alpha$-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 $\alpha \in \text {Cov}_V\,(K)$ there exists a continuous functions $\pi _{\alpha }:K \rightarrow V$ such that

• (i) $\pi _{\alpha }$ and $i:K \rightarrow V$ are $\alpha$-close;

• (ii) $\pi _{\alpha }(K)$ is contained in a subset $C \subseteq V$ with $C \in AES(\hbox {compact})$.

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

Theorem 1.1

[3, 11] Let X be a Schauder admissible subset of a Hausdorff topological vector space and $\Psi \in PK(X,X)$ a compact upper semicontinuous map with closed values. Then there exists $x\in X$ with $x\in \Psi (x)$.

Remark 1.2

Other variations of Theorem 1.1 can be found in [12].

In this paper we will consider more general maps than compact and condensing maps in the literature. These maps will be motivated by the following well-known result in the literature (see for example [2]): Let $\Omega$ be a closed convex subset of a Hausdorff topological vector space E with $x_0 \in \Omega$ and consider the multifunction $F: \Omega \rightarrow 2^{\Omega }$ satisfying the following property:

$$\begin{aligned} A \subseteq \Omega ,\,\,A=\overline{\text {co}}\,(\{x_0\} \cup F(A))\,\,\hbox { implies}\, A \,\,\hbox {is compact}. \end{aligned}$$

Then there exists a convex compact subset C of $\Omega$ with $F(C) \subseteq C$.

2 Fixed and coincidence type theory

We begin by establishing a general collective fixed point result.

Theorem 2.1

Let $\{X_i\}_{i=1}^{N}$ be a family of convex sets each in a Hausdorff topological vector space $E_i$. For each $i\in \{1,\ldots ,N\}$ suppose $F_i: X \equiv \prod _{i=1}^{N} X_i \rightarrow X_i$ and $F_i \in HLPY(X,X_i)$. In addition assume there is a compact convex subset K of X with $F(K) \subseteq K$ where $F(x)=\prod _{i=1}^{N}\,F_i(x)$ for $x\in X$. Then there exists $x\in X$ with $x_i \in F_i(x)$ for $i\in \{1,\ldots ,N\}$ (here $x_i$ is the projection of x on $X_i$).

Proof

For $i\in \{1,\ldots ,N\}$ let $S_i:X \rightarrow X_i$ with $S_i(x) \ne \emptyset$ for $x\in X$, $\text {co}\,(S_i(x)) \subseteq F_i(x)$ for $x\in X$ and $X=\bigcup \,\{ \,\text {int}\,S_i^{-1}(w):\,\,w\in X_i\}$. Let $K_i$ be the projection of K into $E_i$ and note $F_i(K) \subseteq K_i$ and $K_i$ is convex and compact. Let $F_i^{\star }$ (respectively, $S_i^{\star }$) denote the restriction of $F_i$ (respectively, $S_i$) to K and we claim $F_i^{\star } \in HLPY(K,K_i)$ for each $i\in \{1,\ldots ,N\}$. Note that since $K \subseteq X$, we have $S_i^{\star }(x) \ne \emptyset$ and $\text {co}\,(S_i^{\star }(x)) \subseteq F_i^{\star }(x)$ for $x\in K$. We will now show that $K=\bigcup \,\{ \,\text {int}_K\,S_i^{-1}(w):\,\,w\in K_i\}$. Note

$$\begin{aligned} K=K \cap X=K \cap \left( \bigcup \,\{ \,\text {int}\,S_i^{-1}(w):\,\,w\in X_i\} \right) = \bigcup \,\{ K \cap \,\text {int}\,S_i^{-1}(w):\,\,w\in X_i\}, \end{aligned}$$

so $K \subseteq \bigcup \,\{ \,\text {int}_K\,S_i^{-1}(w):\,\,w\in X_i\}$ since for each $w\in X_i$ we have that $K \cap \,\text {int}\,S_i^{-1}(w)$ is open in K. On the other hand clearly $\bigcup \,\{ \,\text {int}_K\,S_i^{-1}(w):\,\,w\in X_i\} \subseteq K$ so as a result

$$\begin{aligned} K = \bigcup \,\{ \,\text {int}_K\,S_i^{-1}(w):\,\,w\in X_i\}. \end{aligned}$$

Next note for any $y\in K$ from above there exists a $w\in X_i$ with $y\in \text {int}_K\,S_i^{-1}(w) \subseteq S_i^{-1}(w)$ so $w \in S_i(y) \subseteq K_i$ since $\text {co}\,(S_i^{\star }(y)) \subseteq F_i^{\star }(y)$ and $F_i(K) \subseteq K_i$, i.e., for any $y\in K$ there exists a $w\in K_i$ with $y\in \text {int}_K\,S_i^{-1}(w)$. Thus,

$$\begin{aligned} K = \bigcup \,\{ \,\text {int}_K\,S_i^{-1}(w):\,\,w\in K_i\}, \end{aligned}$$

so $F_i^{\star } \in HLPY(K,K_i)$ for $i\in \{1,\ldots ,N\}$. Now for each $i\in \{1,\ldots ,N\}$ from [9, 10] there exists a continuous (single-valued) selection $f_i: K \rightarrow K_i$ of $F_i^{\star }$ with $f_i(x) \in \text {co}\,(S_i^{\star }(x)) \subseteq F_i^{\star }(x)$ for $x\in K$ and also there exists a finite set $C_i$ of $K_i$ with $f_i(K) \subseteq \text {co}\,(C_i) \equiv D_i$. Note $D_i \subseteq \text {co}\,(K_i)=K_i$. Let

$$\begin{aligned} D=\prod _{i=1}^N\,D_i\,\,\hbox { and }\,\,f(x)=\prod _{i=1}^N \,f_i(x), \,x\in K. \end{aligned}$$

Note that $f:K \rightarrow K$ is continuous and also note $f(K)\subseteq D$ since $f_i(K) \subseteq D_i$ for each $i\in \{1,\ldots ,N\}$ and since $D=\prod _{i=1}^N\,D_i \subseteq \prod _{i=1}^N\,K_i=K$, we have $f:D \rightarrow D$ with f(D) lying in a finite-dimensional subspace of $E=\prod _{i=1}^N\,E_i$. Note that $D_i = \text {co}\,(C_i) \subseteq K_i$ is compact and D is compact and convex. Brouwer’s fixed point theorem (or alternatively Theorem 1.1) guarantees that there exists $x\in D \,(\subseteq K)$ with $x=f(x)$. Thus, $x_j=f_j(x) \in \text {co}\,(S_j^{\star }(x)) \subseteq F_j^{\star }(x)$ for each $j\in \{1,\ldots ,N\}$, i.e., $x_j \in F_j^{\star }(x)$ for each $j\in \{1,\ldots ,N\}$. $\square$

Remark 2.2

In Theorem 2.1, if $F_i \in HLPY(X,X_i)$ is replaced by $F_i \in DKT(X,X_i)$, then the proof in Theorem 2.1 is easier since to show $F_i^{\star } \in DKT(K,K_i)$ we just need to note that if $y\in X_i$, then

$$\begin{aligned} (S_i^{\star })^{-1}(y)= & {} \{ z\in K:\,y\in S_i^{\star }(z)\}= \{ z\in K:\,y\in S_i(z)\}\\= & {} K \cap \{ z\in X:\,y\in S_i(z)\}=K \cap S_i^{-1}(y) \end{aligned}$$

which is open in $K \cap X=K$.

Theorem 2.3

Let I be an index set and $\{X_i\}_{i\in I}$ be a family of convex sets each in a Hausdorff topological vector space $E_i$. For each $i\in I$ suppose $F_i: X \equiv \prod _{i\in I} X_i \rightarrow X_i$ and $F_i \in HLPY(X,X_i)$. In addition, assume there is a compact convex subset K of X with $F(K) \subseteq K$, where $F(x)=\prod _{i\in I}\,F_i(x)$ for $x\in X$. 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 $x\in X$ with $x_i \in F_i(x)$ for $i\in I$.

Proof

For $i\in I$ let $S_i$ be as in Theorem 2.1, $F_i^{\star }$ the restriction of $F_i$ to K and $K_i$ the projection of K into $E_i$ and as in Theorem 2.1 we have $F_i^{\star } \in HLPY(K,K_i)$ for $i\in I$, so for each $i\in I$ there exists a continuous (single-valued) selection $f_i: K \rightarrow K_i$ of $F_i^{\star }$ with $f_i(x) \in \text {co}\,(S_i^{\star }(x)) \subseteq F_i^{\star }(x)$ for $x\in K$ and also there exists a finite set $C_i$ of $K_i$ with $f_i(K) \subseteq \text {co}\,(C_i) \equiv D_i$; note $D_i \subseteq K_i$. Let

$$\begin{aligned} D=\prod _{i\in I}\,D_i\,\,\hbox { and }\,\,f(x)=\prod _{i\in I} \,f_i(x), \,x\in K, \end{aligned}$$

and as in Theorem 2.1 note $f:D \rightarrow D$ is continuous. Now D is Schauder admissible (since X is q-Schauder admissible) so Theorem 1.1 guarantees the existence of $x\in D \,(\subseteq K)$ with $x=f(x)$ and we conclude as in Theorem 2.1. $\square$

Now we will consider general collective coincidence type 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 $i\in \{1,\ldots ,N_0\}$ suppose $F_i: X \equiv \prod _{i=1}^{N} X_i \rightarrow Y_i$ and $F_i \in HLPY(X,Y_i)$ and for each $j\in \{1,\ldots ,N\}$ suppose $G_j: Y \equiv \prod _{i=1}^{N_0} Y_i \rightarrow X_j$ and $G_j \in HLPY(Y,X_j)$. In addition, assume there is a convex subset $\Omega$ of X and a compact subset D of Y with $F(\Omega ) \subseteq D$ and $G(D) \subseteq \Omega$, where $F(x)=\prod _{i=1}^{N_0}\,F_i(x)$, $x\in X$ and $G(y)=\prod _{i=1}^{N}\,G_i(x)$, $y\in Y$. Then there exist $x\in X$ and $y\in Y$ with $x_i \in G_i(y)$ for $i\in \{1,\ldots ,N\}$ and $y_j \in F_j(x)$ for $j\in \{1,\ldots ,N_0\}$.

Proof

For $i\in \{1,\ldots ,N_0\}$ let $T_i:X \rightarrow Y_i$ with $T_i(x) \ne \emptyset$ for $x\in X$, $\text {co}\,(T_i(x)) \subseteq F_i(x)$ for $x\in X$ and $Y=\bigcup \,\{ \,\text {int}\,T_i^{-1}(w):\,\,w\in Y_i\}$. For $i\in \{1,\ldots ,N\}$ let $S_i:Y \rightarrow X_i$ with $S_i(y) \ne \emptyset$ for $y\in Y$, $\text {co}\,(S_i(y)) \subseteq G_i(y)$ for $y\in Y$ and $X=\bigcup \,\{ \,\text {int}\,S_i^{-1}(w):\,\,w\in X_i\}$. Let $D_i$ (for $i\in \{1,\ldots ,N_0\}$) be the projection of D on $E_i$ and let $\Omega _i$ (for $i\in \{1,\ldots ,N\}$) be the projection of $\Omega$ on $E_i$; note $\Omega _i$ is a convex subset of $E_i$ and $D_i$ is a compact subset of $E_i$. Let $G_i^{\star }$ (respectively, $S_i^{\star }$) denote the restriction of $G_i$ (respectively, $S_i$) to D. We claim $G_i^{\star } \in HLPY(D,\Omega _i)$ for $i\in \{1,\ldots ,N\}$. This will be immediate once we show $D=\bigcup \,\{ \,\text {int}_D\,S_i^{-1}(w):\,\,w\in \Omega _i\}$. Note

$$\begin{aligned} D=D \cap Y&= D \cap \left( \bigcup \,\{ \,\text {int}\,S_i^{-1}(w):\,\,w\in X_i\} \right) = \bigcup \,\{ D \cap \,\text {int}\,S_i^{-1}(w):\,\,w\in X_i\} \\&\subseteq \bigcup \,\{ \,\text {int}_D\,S_i^{-1}(w):\,\,w\in X_i\} \end{aligned}$$

since for each $w\in X_i$ we have that $D \cap \,\text {int}\,S_i^{-1}(w)$ is open in D. On the other hand, clearly $\bigcup \,\{ \,\text {int}_D\,S_i^{-1}(w):\,\,w\in X_i\} \subseteq D$ so as a result $D = \bigcup \,\{ \,\text {int}_D\,S_i^{-1}(w):\,\,w\in X_i\}$. For any $y\in D$ from the above there exists a $w\in X_i$ with $y\in \text {int}_D\,S_i^{-1}(w) \subseteq S_i^{-1}(w)$ so $w \in S_i(y) \subseteq \Omega _i$ since $\text {co}\,(S_i^{\star }(y)) \subseteq G_i^{\star }(y)$ and $G_i(D) \subseteq \Omega _i$, i.e., for any $y\in D$ there exists a $w\in \Omega _i$ with $y\in \text {int}_D\,S_i^{-1}(w)$. Thus, $D = \bigcup \,\{ \,\text {int}_D\,S_i^{-1}(w):\,\,w\in \Omega _i\}$, so $G_i^{\star } \in HLPY(D,\Omega _i)$ for $i\in \{1,\ldots ,N\}$. Now for each $i\in \{1,\ldots ,N\}$ from [9, 10] there exists a continuous (single-valued) selection $g_i: D \rightarrow \Omega _i$ of $G_i^{\star }$ with $g_i(y) \in \text {co}\,(S_i^{\star }(y)) \subseteq G_i^{\star }(y)$ for $y\in D$ and also there exists a finite set $R_i$ of $\Omega _i$ with $g_i(D) \subseteq \text {co}\,(R_i) \equiv Q_i$; note $Q_i \subseteq \Omega _i$. Let $Q=\prod _{i=1}^N\,Q_i\,\,(\subseteq \Omega \subseteq X)$ and note Q is compact. Let $F_i^{\star }$ (respectively, $T_i^{\star }$) denote the restriction of $F_i$ (respectively, $T_i$) to Q. We next note that $F_i^{\star } \in HLPY(Q,Y_i)$ for $i\in \{1,\ldots ,N_0\}$ since

$$\begin{aligned} Q=Q \cap X&= Q \cap \left( \bigcup \,\{ \,\text {int}\,T_i^{-1}(w):\,\,w\in Y_i\} \right) = \bigcup \,\{ Q \cap \,\text {int}\,T_i^{-1}(w):\,\,w\in Y_i\} \\&\subseteq \bigcup \,\{ \,\text {int}_Q\,T_i^{-1}(w):\,\,w\in Y_i\} \end{aligned}$$

and clearly $\{ \,\text {int}_Q\,T_i^{-1}(w):\,\,w\in Y_i\} \subseteq Q$ so as a result $Q=\{ \,\text {int}_Q\,T_i^{-1}(w):\,\,w\in Y_i\}$. Now for each $i\in \{1,\ldots ,N_0\}$ from [9, 10] there exists a continuous (single-valued) selection $f_i: Q \rightarrow Y_i$ of $F_i^{\star }$ and note since $Q \subseteq \Omega$ that $f_i(Q) \subseteq F_i^{\star }(Q) \subseteq F_i^{\star }(\Omega ) \subseteq D_i$ since $F(\Omega ) \subseteq D$. Let

$$\begin{aligned} f(x)=\prod _{i=1}^{N_0} \,f_i(x) \,\,\hbox { for }\,\,x\in Q\,\,\hbox { and }\,\,g(y)=\prod _{i=1}^N \,g_i(y)\,\,\hbox { for }\,\,y\in D \end{aligned}$$

and note since $g_i:D \rightarrow Q_i$, $f_i:Q \rightarrow D_i$ (see above) that $g:D \rightarrow Q$ and $f:Q \rightarrow D$. Consider the continuous map $h:Q \rightarrow Q$ given by $h(x)=g(f(x))$ for $x\in Q$ and since Q is a compact convex subset in a finite-dimensional subspace of $E=\prod _{i=1}^N\,E_i$ the Brouwer fixed point theorem (or alternatively Theorem 1.1) guarantees that there exists $x\in Q$ with $x=h(x)=g(f(x))$. Let $y=f(x)$ so $x=g(y)$. Then since $x\in Q$ and $y=f(x) \in f(Q) \subseteq D$, we have $y_j=f_j(x) \in F_j^{\star }(x)=F_j(x)$ for $j\in \{1,\ldots ,N_0\}$ and $x_i=g_i(y) \in G_i^{\star }(y)=G_i(y)$ for $i\in \{1,\ldots ,N\}$. $\square$

Remark 2.5

In Theorem 2.4 we could replace $\{X_i\}_{i=1}^{N}$, $\{Y_i\}_{i=1}^{N_0}$ with $\{X_i\}_{i\in I}$, $\{Y_i\}_{i\in J}$, where I and J are index sets and in this case we will use Theorem 1.1.

Other classes of maps could also be considered (see our next result).

Theorem 2.6

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 $i\in \{1,\ldots ,N_0\}$ suppose $F_i: X \equiv \prod _{i=1}^{N} X_i \rightarrow Y_i$ and $F_i \in \text {Ad}(X,Y_i)$ and for each $j\in \{1,\ldots ,N\}$ suppose $G_j: Y \equiv \prod _{i=1}^{N_0} Y_i \rightarrow X_j$ and $G_j \in HLPY(Y,X_j)$. In addition, assume there is a convex subset $\Omega$ of X and a compact subset D of Y with $F(\Omega ) \subseteq D$ and $G(D) \subseteq \Omega$, where $F(x)=\prod _{i=1}^{N_0}\,F_i(x)$, $x\in X$ and $G(y)=\prod _{i=1}^{N}\,G_i(x)$, $y\in Y$. Then there exist $x\in X$ and $y\in Y$ with $x_i \in G_i(y)$ for $i\in \{1,\ldots ,N\}$ and $y_j \in F_j(x)$ for $j\in \{1,\ldots ,N_0\}$.

Proof

Let $S_i,\,D_i,\,\Omega _i,\,G_i^{\star }$ and $S_i^{\star }$ be as in Theorem 2.4. The same reasoning as in Theorem 2.4 guarantees that $G_i^{\star } \in HLPY(D,\Omega _i)$ for $i\in \{1,\ldots ,N\}$ and from [9, 10] there exists a continuous (single-valued) selection $g_i: D \rightarrow \Omega _i$ of $G_i^{\star }$ and also a finite set $R_i$ of $\Omega _i$ with $g_i(D) \subseteq \text {co}\,(R_i) \equiv Q_i$; note $Q_i \subseteq \Omega _i$. Let $Q=\prod _{i=1}^{N}\,Q_i\,\,(\subseteq \Omega \subseteq X)$ and let $F_i^{\star }$ denote the restriction of $F_i$ to Q; here $i\in \{1,\ldots ,N_0\}$. Since the composition of admissible maps of Gorniewicz is an admissible map of Gorniewicz, $F_i^{\star } \in \text {Ad}(Q,Y_i)$. Let $F^{\star }(x)=\prod _{i=1}^{N_0} \,F_i^{\star }(x)$ for $x\in Q$. Since a finite product of admissible maps of Gorniewicz is an admissible map of Gorniewicz [6], $F^{\star } \in \text {Ad}(Q,Y)$. Note $F_i^{\star }(Q) \subseteq F_i(\Omega ) \subseteq D_i$ for each $i\in \{1,\ldots ,N_0\}$ so $F^{\star }(Q) \subseteq D$. Let $g(y)=\prod _{i=1}^N \,g_i(y)$ for $y\in D$. Note $g:D \rightarrow Q$ is continuous since $g_i:D \rightarrow Q_i$ is continuous and also note $F^{\star }(Q) \subseteq D$, so it is easy to check that $F^{\star } \in \text {Ad}(Q,D)$ so $g\,F^{\star } \in \text {Ad}(Q,Q)$. Now since Q is a compact convex subset in a finite-dimensional subspace of $E=\prod _{i=1}^N\,E_i$, Theorem 1.1 guarantees that there exists $x\in Q$ with $x\in g\,(F^{\star }(x))$. Now let $y\in F^{\star }(x)$ with $x=g(y)$. Note $y \in F^{\star }(Q) \subseteq D$ and $y_j \in F_j^{\star }(x)= F_j(x)$ for $j\in \{1,\ldots ,N_0\}$ and $x_i=g_i(y) \in G_i^{\star }(y)=G_i(y)$ for $i\in \{1,\ldots ,N\}$. $\square$

Remark 2.7

In the statement of Theorem 2.6 we could replace the statement “In addition assume there is a convex subset $\Omega$ of X and a compact subset D of Y with $F(\Omega ) \subseteq D$ and $G(D) \subseteq \Omega$” with the statement “In addition assume there is a convex compact subset $\Omega$ of X and a paracompact subset D of Y with $F(\Omega ) \subseteq D$ and $G(D) \subseteq \Omega$ and D (or $\Omega$) is a Schauder admissible subset of $\prod _{i=1}^{N_0}\,E_i$ (or $\prod _{i=1}^N\,E_i$)” and the result in Theorem 2.6 holds. To see this let $S_i,\,D_i,\,\Omega _i,\,G_i^{\star }$ and $S_i^{\star }$ be as in Theorem 2.4. The same reasoning as in Theorem 2.4 guarantees that $G_i^{\star } \in HLPY(D,\Omega _i)$ for $i\in \{1,\ldots ,N\}$. Now from [9, 10] (note D is paracompact) there exists a continuous (single-valued) selection $g_i: D \rightarrow \Omega _i$ of $G_i^{\star }$. Let $g(y)=\prod _{i=1}^N \,g_i(y)$ for $y\in D$ and note $g:D \rightarrow \Omega$ is continuous. Let $F_i^{\star \star }$ denote the restriction of $F_i$ to $\Omega _i$ for $i\in \{1,\ldots ,N_0\}$ (note $F_i^{\star \star } \in \text {Ad}(\Omega , Y_i)$) and let $F^{\star \star }(x)=\prod _{i=1}^{N_0} \,F_i^{\star \star }(x)$ for $x\in \Omega$. Note $F^{\star \star } \in \text {Ad}(\Omega ,Y)$ with $F^{\star \star }(\Omega ) \subseteq D$ since $F(\Omega ) \subseteq D$. Note $g\,F^{\star \star } \in \text {Ad}(\Omega ,\Omega )$ and $F^{\star \star }\,g \in \text {Ad}(D,D)$ are compact maps. Now apply Theorem 1.1.

Next we will consider types of maps other than $\text {Ad}$ and HLPY maps. We refer the reader to [1, 14] for the following class of maps. Let X and Y be subsets of Hausdorff topological vector spaces $E_1$ and $E_2$ and let F be a multifunction. We say $F \in W(X,Y)$ if $F:X \rightarrow 2^Y$ and there exists $\theta :X \rightarrow 2^Y$ which is lower semicontinuous with ${\overline{co}}\,(\theta (x)) \subseteq F(x)$ for each $x\in X$. The following result was established in [1, 14].

Theorem 2.8

Let X be a paracompact subset of a Hausdorff topological vector space $E_1$ and Y a metrizable, complete subset of a Hausdorff locally convex linear topological space $E_2$. If $F \in W(X,Y)$, then there exists an upper semicontinuous map $G:X \rightarrow K(Y)$ with $G(x) \subseteq F(x)$ for $x\in X$.

To establish our results recall the following fixed point result of Himmelberg [8].

Theorem 2.9

Let Y be a convex subset of a Hausdorff locally convex linear topological space, X a nonempty subset of Y, and let $G:Y \rightarrow K(X)$ be an upper semicontinuous compact map. Then there exists an $x\in X$ with $x\in G(x)$.

Theorem 2.10

Let $\{X_i\}_{i=1}^{N}$ be a family of convex metrizable subsets each in a Hausdorff locally convex topological vector space $E_i$. For each $i\in \{1,\ldots ,N\}$ suppose $F_i: X \equiv \prod _{i=1}^{N} X_i \rightarrow X_i$ and $F_i \in W(X,X_i)$. In addition, assume there is a compact convex subset D of X with $F(D) \subseteq D$, where $F(x)=\prod _{i=1}^{N}\,F_i(x)$ for $x\in X$. Then there exists $x\in X$ with $x_i \in F_i(x)$ for $i\in \{1,\ldots ,N\}$.

Proof

Let $i\in \{1,\ldots ,N\}$ and let $D_i$ be the projection of D on $E_i$ and note $F_i(D) \subseteq D_i$ with $D_i$ convex and compact. Note D is paracompact and $F_i \in W(D,X_i)$ for $i\in \{1,\ldots ,N\}$ (we can write $F_i$ as $F_i\,j$, where $j:D \rightarrow X$ is given by $j(x)=x$ for $x\in D$). Also since $F_i(D) \subseteq D_i$, it is easy to see that $F_i \in W(D,D_i)$. Now we apply Theorem 2.8 (recall a compact subset of a Hausdorff topological vector space is complete [13, p. 53] so $D_i$ is complete) for each $i\in \{1,\ldots ,N\}$. Thus, for each $i\in \{1,\ldots ,N\}$ there exists an upper semicontinuous map $G_i:D \rightarrow K(D_i)$ with $G_i(x) \subseteq F_i(x)$ for $x\in D$. Define an upper semicontinuous map [4] $G:D \rightarrow K(D)$ by $G(x)= \prod _{i=1}^N \,G_i(x)$ for $x\in D$ (note $G(D) \subseteq D$ with D compact). Now Theorem 2.9 guarantees the existence of $x\in D \subseteq X$ with $x\in G(x)=\prod _{i=1}^N \,G_i(x) \subseteq \prod _{i=1}^N \,F_i(x)$. $\square$

Remark 2.11

In Theorem 2.10 we could replace $\{X_i\}_{i=1}^{N}$ with $\{X_i\}_{i\in I}$, where I is an index set.

Now we consider a variety of collective coincidence results.

Theorem 2.12

Let $\{X_i\}_{i=1}^{N}$, $\{Y_i\}_{i=1}^{N_0}$ be families of convex sets each in a Hausdorff locally convex topological vector space $E_i$ and let $Y_i$ be a metrizable subset of $E_i$ for each $i\in \{1,\ldots ,N_0\}$. For each $i\in \{1,\ldots ,N_0\}$ suppose $F_i: X \equiv \prod _{i=1}^{N} X_i \rightarrow Y_i$ and $F_i \in W(X,Y_i)$ and for each $j\in \{1,\ldots ,N\}$ suppose $G_j: Y \equiv \prod _{i=1}^{N_0} Y_i \rightarrow X_j$ and $G_j \in HLPY(Y,X_j)$. In addition, assume there is a convex subset $\Omega$ of X and a compact subset D of Y with $F(\Omega ) \subseteq D$ and $G(D) \subseteq \Omega$, where $F(x)=\prod _{i=1}^{N_0}\,F_i(x)$, $x\in X$ and $G(y)=\prod _{i=1}^{N}\,G_i(x)$, $y\in Y$. Then there exist $x\in X$ and $y\in Y$ with $x_i \in G_i(y)$ for $i\in \{1,\ldots ,N\}$ and $y_j \in F_j(x)$ for $j\in \{1,\ldots ,N_0\}$.

Proof

Let $S_i,\,D_i,\,\Omega _i,\,G_i^{\star }$ and $S_i^{\star }$ be as in Theorem 2.4. The same reasoning as in Theorem 2.4 guarantees that $G_i^{\star } \in HLPY(D,\Omega _i)$ for $i\in \{1,\ldots ,N\}$ and from [9, 10] there exists a continuous (single-valued) selection $g_i: D \rightarrow \Omega _i$ of $G_i^{\star }$ with $g_i(y) \in \text {co}\,(S_i^{\star }(y)) \subseteq G_i^{\star }(y)$ for $y\in D$ and also there exists a finite set $R_i$ of $\Omega _i$ with $g_i(D) \subseteq \text {co}\,(R_i) \equiv Q_i$; note $Q_i \subseteq \Omega _i$. Let $Q=\prod _{i=1}^{N}\,Q_i\,\,(\subseteq \Omega \subseteq X)$ and let $F_i^{\star }$ denote the restriction of $F_i$ to Q; here $i\in \{1,\ldots ,N_0\}$. Note $F_i^{\star } \in W(Q,Y_i)$ for $i\in \{1,\ldots ,N_0\}$ (we can write $F_i^{\star }$ as $F_i\,j$, where $j:Q \rightarrow X$ is given by $j(x)=x$ for $x\in Q$) and also since $Q \subseteq \Omega$ and $F_i(\Omega ) \subseteq D_i$, it is easy to see that $F_i^{\star } \in W(Q,D_i)$. Now Theorem 2.8 (note Q is paracompact, $D_i$ is complete and $Y_i$ is metrizable) guarantees that for each $i\in \{1,\ldots ,N_0\}$ there exists an upper semicontinuous map $\theta _i:Q \rightarrow K(D_i)$ with $\theta _i(x) \subseteq F_i^{\star }(x)$ for $x\in Q$. Define an upper semicontinuous map $\theta :Q \rightarrow K(D)$ by $\theta (x)= \prod _{i=1}^{N_0} \,\theta _i(x)$ for $x\in Q$, so in particular $\theta \in \text {Ad}(Q,D)$. Let $g(y)=\prod _{i=1}^N \,g_i(y)$ for $y\in D$ and note since $g_i:D \rightarrow Q_i$ that $g:D \rightarrow Q$ is continuous and so $g\,\theta \in \text {Ad}(Q,Q)$. Now since Q is a compact convex subset in a finite-dimensional subspace of $E=\prod _{i=1}^N\,E_i$, Theorem 1.1 guarantees that there exists $x\in Q$ with $x\in g(\theta (x))$. Let $y\in \theta (x)$ with $x=g(y)$. Note $y\in \theta (x)= \prod _{i=1}^{N_0} \,\theta _i(x) \subseteq \prod _{i=1}^{N_0} \,F_i^{\star }(x)$ so $y_j \in F_j(x)$ for $j\in \{1,\ldots ,N_0\}$ and $x_i=g_i(y) \in G_i^{\star }(y)=G_i(y)$ for $i\in \{1,\ldots ,N\}$. $\square$

Remark 2.13

(i) In Theorem 2.12 we could replace $\{X_i\}_{i=1}^{N}$, $\{Y_i\}_{i=1}^{N_0}$ with $\{X_i\}_{i\in I}$, $\{Y_i\}_{i\in J}$, where I and J are index sets.

(ii) In Theorem 2.12 we need only the $E_i$’s to be locally convex for $i\in \{1,\ldots ,N_0\}$.

Theorem 2.14

Let $\{X_i\}_{i=1}^{N}$, $\{Y_i\}_{i=1}^{N_0}$ be families of convex sets each in a Hausdorff locally convex topological vector space $E_i$ and let $Y_i$ be a metrizable subset of $E_i$ for each $i\in \{1,\ldots ,N_0\}$. For each $i\in \{1,\ldots ,N_0\}$ suppose $F_i: X \equiv \prod _{i=1}^{N} X_i \rightarrow Y_i$ and $F_i \in W(X,Y_i)$ and for each $j\in \{1,\ldots ,N\}$ suppose $G_j: Y \equiv \prod _{i=1}^{N_0} Y_i \rightarrow X_j$ and $G_j \in \text {Ad}(Y,X_j)$. In addition, assume there is a convex paracompact subset $\Omega$ of X and a compact subset D of Y with $F(\Omega ) \subseteq D$ and $G(D) \subseteq \Omega$ and also assume $\Omega$ is a Schauder admissible subset of $\prod _{i=1}^{N} E_i$; here $F(x)=\prod _{i=1}^{N_0}\,F_i(x)$, $x\in X$ and $G(y)=\prod _{i=1}^{N}\,G_i(x)$, $y\in Y$. Then there exist $x\in X$ and $y\in Y$ with $x_i \in G_i(y)$ for $i\in \{1,\ldots ,N\}$ and $y_j \in F_j(x)$ for $j\in \{1,\ldots ,N_0\}$.

Proof

Let $D_i$ ($i\in \{1,\ldots ,N_0\}$) be the projection of D on $E_i$, let $\Omega _i$ ($i\in \{1,\ldots ,N\}$) be the projection of $\Omega$ on $E_i$ and let $F_i^{\star }$ ($i\in \{1,\ldots ,N_0\}$) be the restriction of $F_i$ to $\Omega$ and note $F_i^{\star } \in W(\Omega , Y_i)$. Now since $F_i(\Omega ) \subseteq D_i$, we have $F_i^{\star } \in W(\Omega ,D_i)$ for $i\in \{1,\ldots ,N_0\}$. Now Theorem 2.8 (note $\Omega$ is paracompact, $D_i$ is complete and $Y_i$ is metrizable) guarantees that for each $i\in \{1,\ldots ,N_0\}$ there exists an upper semicontinuous map $\theta _i:\Omega \rightarrow K(D_i)$ with $\theta _i(x) \subseteq F_i^{\star }(x)$ for $x\in \Omega$. Define an upper semicontinuous map $\theta :\Omega \rightarrow K(D)$ by $\theta (x)= \prod _{i=1}^{N_0} \,\theta _i(x)$ for $x\in \Omega$, so in particular $\theta \in \text {Ad}(\Omega ,D)$. For $j\in \{1,\ldots ,N\}$ let $G_j^{\star }$ denote the restriction of $G_j$ to D and note $G_j^{\star } \in \text {Ad}(D,X_j)$ and since $G_j(D) \subseteq \Omega _j$ we have $G_j^{\star } \in \text {Ad}(D,\Omega _j)$. Let $G^{\star }(x)=\prod _{i=1}^N \,G_j^{\star }(x)$, $x\in D$ and note $G^{\star } \in \text {Ad}(D, \Omega )$. Now $G^{\star }\,\theta \in \text {Ad}(\Omega , \Omega )$ is a compact map and Theorem 1.1 guarantees a $x\in \Omega$ with $x\in G^{\star }(\theta (x))$ Now let $y\in \theta (x)$ with $x\in G^{\star }(y)$. Note $y\in \theta (x)= \prod _{i=1}^{N_0} \,\theta _i(x) \subseteq \prod _{i=1}^{N_0} \,F_i^{\star }(x)$ so $y_j \in F_j(x)$ for $j\in \{1,\ldots ,N_0\}$. Also $x\in G^{\star }(y)=G(y)$ (note $y\in \theta (x) \subseteq D$) so $x_i \in G_i(y)$ for $i\in \{1,\ldots ,N\}$. $\square$

Remark 2.15

There is a slight inaccuracy in the statement and proof of Theorem 3.3 in [1]. It is easy to correct those by noticing that $F\,\theta \in \text {Ad}(Q,Q)$ in [1].

Remark 2.16

If $\Omega$ or alternatively X is metrizable, then $\Omega$ is a Schauder admissible subset of $\prod _{i=1}^{N} E_i$ since from Dugundji’s extension theorem [7, p. 164] we know that $\Omega$ is an AR (absolute retract).

Theorem 2.17

Let $\{X_i\}_{i=1}^{N}$, $\{Y_i\}_{i=1}^{N_0}$ be families of convex metrizable sets each in a Hausdorff locally convex topological vector space $E_i$. For each $i\in \{1,\ldots ,N_0\}$ suppose $F_i: X \equiv \prod _{i=1}^{N} X_i \rightarrow Y_i$ and $F_i \in W(X,Y_i)$ and for each $j\in \{1,\ldots ,N\}$ suppose $G_j: Y \equiv \prod _{i=1}^{N_0} Y_i \rightarrow X_j$ and $G_j \in W(Y,X_j)$. In addition, assume there is a convex paracompact subset $\Omega$ of X and a compact subset D of Y with $F(\Omega ) \subseteq D$ and $G(D) \subseteq \Omega$ and assume $\Omega _j$ is a complete subset of $E_j$ for $j\in \{1,\ldots ,N\}$; here $\Omega _j$ is the projection of $\Omega$ on $E_j$, $F(x)=\prod _{i=1}^{N_0}\,F_i(x)$, $x\in X$ and $G(y)=\prod _{i=1}^{N}\,G_i(x)$, $y\in Y$. Then there exist $x\in X$ and $y\in Y$ with $x_i \in G_i(y)$ for $i\in \{1,\ldots ,N\}$ and $y_j \in F_j(x)$ for $j\in \{1,\ldots ,N_0\}$.

Proof

Let $D_i,\,F_i^{\star },\,\theta _i$ and $\theta$ be as in Theorem 2.14. From the reasoning in Theorem 2.14 we obtain $\theta \in \text {Ad}(\Omega ,D)$. For $j\in \{1,\ldots ,N\}$ let $G_j^{\star }$ denote the restriction of $G_j$ to D and note $G_j^{\star } \in W(D,X_j)$ and since $G_j(D) \subseteq \Omega _j$, we have $G_j^{\star } \in W(D,\Omega _j)$. Now Theorem 2.8 (note D is paracompact, $\Omega _j$ is complete and $X_j$ is metrizable) guarantees that for each $j\in \{1,\ldots ,N\}$ there exists an upper semicontinuous map $\psi _j:D \rightarrow K(\Omega _j)$ with $\psi _j(x) \subseteq G_j^{\star }(x)$ for $x\in D$. Define an upper semicontinuous map $\psi :D \rightarrow K(\Omega )$ by $\psi (x)=\prod _{i=1}^N\,\psi _i(x)$ for $x\in D$, so in particular $\Psi \in \text {Ad}(D,\Omega )$. Note $\psi \,\theta \in \text {Ad}(\Omega , \Omega )$ is a compact map. Theorem 1.1 (note $\Omega$ is an AR) guarantees an $x\in \Omega$ with $x\in \psi (\theta (x))$ and now use the usual argument to finish the proof. $\square$

It is of interest to note that the maps considered in this paper (e.g., HLPY, $\text {Ad}$ and W maps) (a) include and generalize all maps in the literature and in addition (b) the “compactness condition” on the maps is extremely general and new as we noted at the end of Sect. 1. Indeed our results are new even if only one of (a) or (b) occur in their full generality.

In applications many problems for systems of differential an integral inclusions can be rewritten as operator inclusions so fixed point and coincidence point theorems here can be used to guarantee the existence of a solution to the inclusion considered. However, here we will illustrate our theory with new minimax type inequalities of Ky–Fan, von Neumann–Sion type. Each theorem above will generate an inequality and we will illustrate the strategy by just considering Theorem 2.1.

Theorem 2.18

Let $\{X_i\}_{i=1}^{N}$ be a family of convex sets each in a Hausdorff topological vector space. For each $i\in \{1,\ldots ,N\}$ let $f_i: X \times X_i \rightarrow \textbf{R}$, where $X= \prod _{i=1}^{N} X_i$, let $\lambda \in \textbf{R}$ and let $F_i(x)=\{ z_i \in X_i:\,\,f_i(x,z_i)<\lambda \}$. Also for $i\in \{1,\ldots ,N\}$ suppose $F_i \in HLPY(X,X_i)$ and in addition assume there is a compact convex subset K of X with $F(K) \subseteq K$, where $F(x)=\prod _{i=1}^{N}\,F_i(x)$ for $x\in X$. Then

$$\begin{aligned} \inf _{x\in X}\,\,\sup _{i\in \{1,\ldots ,N\}}\,f_i(x,x_i)< \lambda . \end{aligned}$$

Proof

Theorem 2.1 guarantees that there exists $x\in X$ with $x_i \in F_i(x)$ for $i\in \{1,\ldots ,N\}$ so $f_i(x,x_i) <\lambda$ for $i\in \{1,\ldots ,N\}$, i.e., $\inf _{y\in X}\,\,\sup _{i\in \{1,\ldots ,N\}}\,f_i(y,y_i)< \lambda$. $\square$

To obtain our minimax inequality (assuming $\sup _{x\in X}\,\sup _{j\in \{1,\ldots ,N\}}\,\inf _{y_j \in X_j}\,f_j(x,y_j)<\infty$ since otherwise the inequality is trivial) let

$$\begin{aligned} \lambda =\sup _{x\in X}\,\sup _{j\in \{1,\ldots ,N\}}\,\inf _{y_j \in X_j}\,f_j(x,y_j)+\epsilon , \end{aligned}$$

where $\epsilon >0$ is sufficiently small.

Remark 2.19

Note $\lambda >\sup _{x\in X}\,\sup _{j\in \{1,\ldots ,N\}}\,\inf _{y_j \in X_j}\,f_j(x,y_j)$. Let $x\in X$. Then for all $j\in \{1,\ldots ,N\}$ there exists a $y_j \in X_j$ with $\lambda >f_j(x,y_j)$.

Let $F_i$ be as in Theorem 2.18 and suppose the conditions in Theorem 2.18 hold (for arbitrary $\epsilon >0$). Then

$$\begin{aligned} \inf _{x\in X}\,\,\sup _{i\in \{1,\ldots ,N\}}\,f_i(x,x_i)<\sup _{x\in X}\,\sup _{j\in \{1,\ldots ,N\}}\,\inf _{y_j \in X_j}\,f_j(x,y_j)+\epsilon \end{aligned}$$

so

$$\begin{aligned} \inf _{x\in X}\,\,\sup _{i\in \{1,\ldots ,N\}}\,f_i(x,x_i)\le \sup _{x\in X}\,\sup _{j\in \{1,\ldots ,N\}}\,\inf _{y_j \in X_j}\,f_j(x,y_j). \end{aligned}$$

Remark 2.20

If $N=1$, we have $\inf _{x\in X}\,f(x,x) \le \sup _{x\in X}\,\inf _{y\in X}\,f(x,y)$.