## Applications of two matching theorems in generalized convex spaces.(English)Zbl 1034.54016

The KKM theory is the study of applications of various equivalent formulations of the classical Knaster-Kuratowski-Mazurkiewicz theorem. Recently, S. Park and H. Kim [J. Math. Anal. Appl. 197, 173–187 (1996; Zbl 0851.54039)] have introduced the concept of generalized convex space which is a common generalization of the usual convexity in a topological vector space and many abstract convexities which have been developed in connection mainly with the fixed point theory and KKM theory.
Given a set $$A$$, let $$\langle A\rangle$$ denote the set of all nonempty finite subsets of $$A$$, and $$| A|$$ the cardinality of $$A$$. Let $$\Delta_n$$ be the standard simplex of dimension $$n$$. A generalized convex space or a $$G$$-convex space $$(X,D;\Gamma)$$ consists of a topological space $$X$$, a nonempty subset $$D$$ of $$X$$, and a map $$\Gamma: \langle D\rangle\multimap X$$ with nonempty values such that for each $$A\in\langle D\rangle$$ with $$| A|= n+1$$ there exists a continuous function $$\varphi_A:\Delta_n\to \Gamma(A)$$ satisfying $$\varphi_A(\Delta_I)\subset \Gamma(J)$$ for every $$I\in\langle A\rangle$$, where $$\Delta_I$$ denotes the face of $$\Delta_n$$ corresponding to $$I\in\langle A\rangle$$. Given a $$G$$-convex space $$(X,D;\Gamma)$$, a subset $$K$$ of $$X$$ is said to be $$G$$-convex if for each $$A\in \langle D\rangle$$, $$A\subset K$$ implies $$\Gamma(A)\subset K$$.
Let $$(X,D;\Gamma)$$ be a $$G$$-convex space, $$Y$$ a topological space, $$F: X\multimap Y$$, $$T: D\multimap Y$$ maps. We say that $$T$$ is a $$G$$-KKM map with respect to $$F$$ if for any $$A\in\langle D\rangle$$ we have $$F(\Gamma(A))\subset T(A)$$.
In this paper two matching theorems in generalized convex spaces are derived from some KKM type theorem presented in [S. Park, Korean J. Comput. Appl. Math. 7, 1–28 (2000; Zbl 0959.47035)]. From these results the author proves some fixed point theorems, geometric properties for sets with $$G$$-convex sections and a coincidence theorem. The main results are the following theorems:
Theorem 2. Let $$G: D\multimap Y$$, $$T: Y\multimap Y$$ be two maps where $$(X,D;\Gamma)$$ is a $$G$$-convex space, with $$D\in\langle X\rangle$$, $$Y$$ a $$T_1$$-regular space and $$F$$ is an admissible map of $$X$$ into $$Y$$. Suppose that: (i) $$G$$ is $$G$$-KKM map with respect to $$F$$, (ii) for each $$y\in Y$$ there exists $$x\in d$$ such that $$G(x)\subset T(y)$$, and (iii) for each $$z\in G(D)$$, $$T^-(z)$$ is closed, where $$T^-(z)= \{x\in Y: z\in Tx\}$$.
Theorem 8. Let $$(X; \Gamma)$$ be a $$G$$-convex space, $$Z$$ a nonempty set and $$F,T: X\multimap Z$$ two maps such that: (i) for each $$y\in X$$, the set $$\{x\in X: F(x)\cap F(y)\neq\emptyset\}$$ is a nonempty and $$G$$ convex, and (ii) for each $$z\in F(X)$$, $$T^-(z)$$ is open. Then there exists $$x_0\in X$$ such that $$F(x_0)\cap T(x_0)\neq\emptyset$$.
Reviewer: V. Popa (Bacau)

### MSC:

 54H25 Fixed-point and coincidence theorems (topological aspects) 54C60 Set-valued maps in general topology 47H10 Fixed-point theorems

### Citations:

Zbl 0851.54039; Zbl 0959.47035