Nonlinear Analysis in p-Vector Spaces for Singe-Valued 1-Set Contractive Mappings

TL;DR

This paper develops nonlinear analysis tools for single-valued mappings in p-vector spaces, establishing fixed point theorems and approximation results that extend existing theories and facilitate applications to nonlinear problems.

Contribution

It introduces new fixed point and approximation results for single-valued mappings in locally p-convex spaces, extending nonlinear analysis in p-vector space frameworks.

Findings

Established fixed point theorems for condensing and 1-set contractive mappings.

Developed general principles for nonlinear alternatives and fixed points.

Extended existing results to broader classes of nonlinear mappings in p-vector spaces.

Abstract

The goal of this paper is to develop some fundamental and important nonlinear analysis for single-valued mappings under the framework of p-vector spaces, in particular, for locally p-convex spaces for p in (0, 1]. More precisely, based on the fixed point theorem of single-valued continuous condensing mapping in locally p-convex spaces as the starting point, we first establish best approximation results for (single-valued) continuous condensing mappings which are then used to develop new results for three classes of nonlinear mappings consisting of 1) condensing; 2) 1-set contractive; and 3) semiclosed 1-set contractive mappings in locally p-convex spaces. Next they are used to establish general principle for nonlinear alternative, Leray - Schauder alternative, fixed points for non-self mappings with different boundary conditions for nonlinear mappings from locally p-convex spaces, to nonexpansive mappings in uniformly convex Banach spaces, or locally convex spaces with Opial condition. The results given by this paper not only include the corresponding ones in the existing literature as special cases, but also expected to be useful tools for the development of new theory in nonlinear functional analysis and applications to the study of related nonlinear problems arising from practice under the general framework of p-vector spaces for p in (0, 1]. Finally, the work presented by this paper focuses on the development of nonlinear analysis for single-valued (instead of set-valued) mappings for locally p-convex spaces, essentially, is indeed the continuation of the associated work given recently by Yuan [134] therein, the attention is given to the study of nonlinear analysis for set-valued mappings in locally p-convex spaces for p in (0, 1].