Fixed Point Theorem and Related Nonlinear Analysis by the Best Approximation Method in p-Vector Spaces

Abstract The goal of this paper is to develop some new and useful tools for nonlinear analysis by applying the best approximation approach for classes of semiclosed 1-set contractive set-valued mappings in locally p-convex or p-vector spaces for In particular, we first develop general fixed point theorems for both set-valued and single-valued condensing mappings which provide answers to the Schauder conjecture in the affirmative way under the setting of (locally p-convex) p-vector spaces, then the best approximation results for upper semi-continuous and 1-set contractive set-valued are established, which are used as tools to establish some new fixed points for non-self set-valued mappings with either inward or outward set conditions under various situations. These results unify or improve corresponding results in the existing literature for nonlinear analysis.