## 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)

### MSC:

 47H10 Fixed-point theorems 52A01 Axiomatic and generalized convexity

### Citations:

Zbl 0851.54039; Zbl 0989.47051
Full Text:

### References:

 [1] Browder, F. E., The fixed point theory of multivalued mappings in topological vector spaces, Math. Ann, 177, 283-301 (1968) · Zbl 0176.45204 [2] Chang, S. S.; Lee, B. S.; Wu, X.; Cho, Y. J.; Lee, M., The generalized quase-variational inequalitied problems, J. Math. Ann. Appl., 203, 686-711 (1996) · Zbl 0867.49008 [3] Lin, L. J.; Park, S., On some generalized quasi-equilibrium problems, J. Math. Anal. Appl., 224, 167-181 (1998) · Zbl 0924.49008 [4] L.J. Lin, Z.T. Yu, Fixed point theorems and equilibrium problems, Nonlinear Anal. TMA (1999).; L.J. Lin, Z.T. Yu, Fixed point theorems and equilibrium problems, Nonlinear Anal. TMA (1999). [5] Park, S.; Kim, H., Coincidence theorems for admissible multifunctions on generalized convex spaces, J. Math. Anal. Appl., 197, 173-187 (1996) · Zbl 0851.54039 [6] Tan, K. K.; Zhang, X. L., Fixed point theorems on G-convex spaces and applications, Proc. of Nonlinear Funct. Anal. Appl., 1, 1-19 (1996) [7] Tarafdar, E., A fixed point theorem equivalent to Fan-Knaster-Kuratowski-Mazurkiewica’s theorem, J. Math. Anal. Appl., 128, 475-479 (1987) · Zbl 0644.47050 [8] Tian, G. Q., Generalizations of KKM theorem and the Ky Fan minimax inequality with applibrium and complementarity, J. Math. Anal. Appl., 170, 457-471 (1992) · Zbl 0767.49007 [9] Wu, X.; Shen, S., A further generalization of Yannelis-Prabhaker’s continuous selection and its applications, J. Math. Anal. Appl., 197, 61-74 (1996) · Zbl 0852.54019 [10] Yannelis, N. C.; Prabhabakar, N. D., Existence of maximal elements and equilibria in linear topological spaces, J. Math. Econom., 12, 233-245 (1983) · Zbl 0536.90019 [11] Zhang, H.; Wu, X., A new existence theorem of maximal elements in non-compact H-space with applications to minimax inequalities and variational inequalities, Acta Math. Hungary, 80, 115-127 (1998) · Zbl 0926.49005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.