Applications of a fixed point theorem in \(G\)-convex space. (English) Zbl 1001.47041

For a nonempty subset \(D\) of a set \(X\), let \(\langle D\rangle\) denote the set of all nonempty finite subset of \(D\). Let \(\Delta_n\) denote the standard \(n\)-simplex with vertices \(e_1,e_2, \dots, e_{n+1}\), where \(e_i\) is the \(i\)th unit vector in \(R^{n+1}\). A generalized convex space [S. Park, H. Kim, J. Math. Anal. Appl. 197, No. 1, 173-187 (1996; Zbl 0851.54039)] or a \(G\)-convex space \((X,D;\Gamma)\) consists of a topological space \(X\), a nomepty set \(D\) of \(X\) and a function \(\Gamma:\langle D\rangle\to X\) with nonempty values such that:
1. for each \(A,B\in\langle D\rangle\), \(A \subset B\) implies \(T(A)\subset \Gamma(B)\);
2. for each \(A\in\langle D \rangle\) with \(|A|=n+1\), there exists a continuous function \(\varphi: \Delta_n\to \Gamma(A)\), such that \(\varphi_A(\Delta_j) \subset\Gamma(J)\), where \(\Delta_J\) denotes the faces of \(\Delta_n\) corresponding to \(J\in\langle A \rangle\).
For a \(G\)-convex space \((X,D;\Gamma)\) a subset \(C\) of \(X\) is said to be \(G\)-convex if for each \(A\in\langle D\rangle\), \(A\subset C\) implies \(\Gamma (A)\subset C\). The following theoren which improves a result from [L. J. Lin, Z. T. Yu, Nonlinear Anal. Theory, Methods Appl. 43A, No. 8, 987-999 (2001; Zbl 0989.47051)] and is a generalization of Fan-Browder theorem is the main result of this paper.
Theorem Let \((X,\Gamma)\) be a \(G\)-convex space and \(S,T:X\to X\) be two maps satisfying the conditions:
(i) for each \(x\in X\), \(M\in\langle S(x)\rangle\), implies \(\Gamma(M)\subset T(x)\);
(ii) \(X=\cup \{\text{Int} (S^{-1}(y): y\in X\}\) and
(iii) there exists a compact subset \(K\) of \(X\) such that \(M\in\langle X\rangle\), there exists a \(G\)-convex subset \(L_M\subset X\) containing \(M\) such that \(L_M\cap \{X\setminus \text{Int} (S^{-1}(y)\); \(y\in L_M\}\subset K\). Then there exists an \(\overline x\in X\) such that \(\overline x\in T(\overline x)\).
Reviewer: V.Popa (Bacau)


47H10 Fixed-point theorems
52A01 Axiomatic and generalized convexity
Full Text: DOI


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