Sehie Park

Continuous Selection Theorems in Generalized Convex Spaces

Any multimap from a Hausdorff compact space to a generalized convex space has a continuous selection whenever it has nonempty generalized convex values and open fibers. Some related new results are obtained as well as generalizations of many known results. Furthermore, applications to fixed point theorems and existence of equilibria are given.

Generalizations of Ky Fan's matching theorems and their applications

Abstract We obtain generalizations of Ky Fans matching theorems for open [or closed] coverings and their applications. Generalized forms of the KKM theorem, the Fan-Browder fixed point theorem, and the Schauder fixed point theorem and other new results are obtained.

Fixed point theorems in hyperconvex metric spaces

The notion of hyperconvex spaces was introduced by Aronszajn and Panitchpakdi [1] in 1956. In 1979, independently Sine [13] and Soardi [16] proved the xed point property for nonexpansive maps on bounded hyperconvex spaces. Since then many interesting works have appeared for hyperconvex spaces. For example, see [2–8,14,15]. It is known that the space C(E) of all continuous real functions on a Stonian space E (extremally disconnected compact Hausdor space) with the usual norm is hyperconvex, and that every hyperconvex real Banach space is a space C(E) for some Stonian space E. Then (Rn; ‖ · ‖∞); l∞ and L∞ are concrete examples of hyperconvex spaces. Until recently, the study of hyperconvex spaces was concentrated to the relationship with nonexpansive maps. However, recently, Khamsi [5] established the Knaster– Kuratowski–Mazurkiewicz theorem (in short, KKM theorem) for hyperconvex spaces and applied it to prove an analogue of Ky Fan’s best approximation theorem extending the Brouwer and the Schauder xed point theorems. This seems to be the rst attempt to prove such results in a hyperconvex space setting. In this paper, we obtain a Ky Fan type matching theorem for open covers, a coincidence theorem, a Fan–Browder type xed point theorem, a Brouwer–Schauder–Rothe type xed point theorem, and other results for hyperconvex spaces. Those are usually

Fixed point theorems in locally G -convex spaces

It is well-known that the Brouwer /xed point theorem, the Sperner lemma, the Knaster–Kuratowski–Mazurkiewicz theorem (simply, the KKM principle), and many results in nonlinear analysis are equivalent. In particular, it was shown in [13] that the KKM principle implies the Brouwer theorem. In this paper, we show that the KKM theorem implies far-reaching generalizations of the Brouwer theorem including well-known /xed point theorems due to Schauder, Tychono9, Kakutani, Himmelberg, and many others. For the literature, see [17,26]. In a recent work, Tarafdar [34] obtained a /xed point theorem for a continuous compact multimap T :X (X with closed H -convex values, where X is a locally H -convex uniform space. In the present paper, we show that his theorem holds for u.s.c. maps instead of continuous maps and for the class of generalized convex spaces (or G-convex spaces) containing properly that of H -spaces and many other types of spaces. Our main result (Theorem 2) is applied to various /xed point theorems for LG-spaces, LC-spaces, hyperconvex spaces, and normed vector spaces. Section 2 deals with a new KKM theorem for generalized convex spaces. In Section 3, we obtain our main /xed point theorem for LG-spaces and some of its simple consequences, especially, a generalization of the Himmelberg theorem. Section 4 deals with lower semicontinuous multimaps and >-maps de/ned on paracompact LC-spaces. In fact, the selection theorems due to Ben-El-Mechaiekh and Oudadess [3] and Park [22] are used to deduce new /xed point theorems for such multimaps. In Section 5, our

ELEMENTS OF THE KKM THEORY FOR GENERALIZED CONVEX SPACES

In the present paper, we introduce fundamental results in the KKM theory for G-convex spaces which are equivalent to the Brouwer theorem, the Sperner lemma, and the KKM theorem. Those results are all abstract versions of known corresponding ones for convex subsets of topological vector spaces. Some earlier applications of those results are indicated. Finally, we give a new proof of the Himmelberg fixed point theorem andG-convex space versions of the von Neumann type minimax theorem and the Nash equilibrium theorem as typical examples of applications of our theory.

On generalizations of the Meir-Keeler type contraction maps

In 1969, Meir and Keeler [29] obtained a remarkable generalization of the Banach contraction principle. Since then, there have appeared a number of generalizations of their result. In 1981, the second author and Bae [33] extended the Meir-Keeler theorem to two commuting maps by adopting Jungck’s method. This influenced many authors, and, consequently, a number of new results in this line followed. Recent works of Sessa and others [46,47] contain common fixed point theorems of four maps satisfying certain contractive type conditions. In the present paper, we give a new result which encompasses most of such generalizations of the Meir-Keeler theorem. Further our result also includes many other generalizations of the Banach contraction principle. Some authors have obtained their results on 2-metric spaces. However, 2-metric versions are easily obtained from metric ones by an obvious

Fixed points and quasi-equilibrium problems

A fixed-point theorem for compact acyclic maps defined on convex subsets of not-necessarily locally convex topological vector spaces is applied to the existence of solutions of quasi-equilibrium problems. Such existence theorems extend known ones which were used for unified approaches to quasi-variational inequalities in [1-5] and others.

ON MINIMAX INEQUALITIES ON SPACES HAVING CERTAIN CONTRACTIBLE SUBSETS

The concept of a convex space is extended to an if-space; that is, a space havingcertain family of contractible subsets. For such spaces the KKM type theorems,the Fan-Browder fixed point theorem, the Ky Fan type matching theorem, andminimax inequalities are given. Moreover, applications to a von Neumann-Siontype minimax theorem, a saddle point theorem, a quasi-variational inequality, anda Kakutani typ fixede point theorem are obtained.

Fixed points, intersection theorems, variational inequalities, and equilibrium theorems.

From a fixed point theorem for compact acyclic maps defined on admissible convex sets in the sense of Klee, we first deduce collectively fixed point theorems, inter- section theorems for sets with convex sections, and quasi-equilibrium theorems. These quasi-equilibrium theorems are applied to give simple and unified proofs of the known variational inequalities of the Hartman-Stampacchia-Browder type. Moreover, from our new fixed point theorem, we deduce new variational inequalities which can be used to obtain fixed point results for convex-valued maps. Finally, various general economic equi- librium theorems are deduced in the forms of the Nash type, the Tarafdar type, and the Yannelis-Prabhakar type. Our results are stated for not-necessarily locally convex topolog- ical vector spaces and for abstract economies with arbitrary number of commodities and agents. Our new results extend a lot of known works with much simpler proofs. Keywords and phrases. Multimap (map), closed map, compact map, upper semicontin- uous (u.s.c.), lower semicontinuous (l.s.c.), acyclic map, quasiconcave, quasiconvex, ad- missible subset of a topological vector space (t.v.s.), fixed point, convex space, polytope, quasi-equilibrium problem, variational inequality, economic equilibrium theorem, abstract economy, equilibrium point, maximal point.

Acyclic versions of the von Neumann and Nash equilibrium theorems

Applying a fixed point theorem for compact compositions of acyclic maps, we obtain acyclic versions of the von Neumann intersection theorem, the minimax theorem, the Nash equilibrium theorem, and others.