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**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 \(F(x_0)\cap T(x_0)\neq\emptyset\).

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 |