Nonlinear analysis by applying best approximation method in p-vector spaces

Abstract

It is known that the class of p-vector spaces (0 < p leq 1) is an important generalization of the usual norm spaces with rich topological and geometrical structure, but most tools and general principles with nature in nonlinearity have not been developed yet. The goal of this paper is to develop some useful tools in nonlinear analysis by applying the best approximation approach for the classes of 1-set contractive set-valued mappings in p-vector spaces. In particular, we first develop general fixed point theorems of compact (single-valued) continuous mappings for closed p-convex subsets, which also provide an answer to Schauder’s conjecture of 1930s in the affirmative way under the setting of topological vector spaces for 0 < p leq 1. Then one best approximation result for upper semicontinuous and 1-set contractive set-valued mappings is established, which is used as a useful tool to establish fixed points of nonself set-valued mappings with either inward or outward set conditions and related various boundary conditions under the framework of locally p-convex spaces for 0 < p leq 1. In addition, based on the framework for the study of nonlinear analysis obtained for set-valued mappings with closed p-convex values in this paper, we conclude that development of nonlinear analysis and related tools for singe-valued mappings in locally p-convex spaces for 0 < p leq 1 seems even more important, and can be done by the approach established in this paper.