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The goal of this paper is to develop some fundamental and important nonlinearanalysis for single-valued mappings under the framework of p-vector spaces, inparticular, for locally p-convex spaces for p in (0, 1]. More precisely, basedon the fixed point theorem of single-valued continuous condensing mapping inlocally p-convex spaces as the starting point, we first establish bestapproximation results for (single-valued) continuous condensing mappings whichare then used to develop new results for three classes of nonlinear mappingsconsisting of 1) condensing; 2) 1-set contractive; and 3) semiclosed 1-setcontractive mappings in locally p-convex spaces. Next they are used toestablish general principle for nonlinear alternative, Leray - Schauderalternative, fixed points for non-self mappings with different boundaryconditions for nonlinear mappings from locally p-convex spaces, to nonexpansivemappings in uniformly convex Banach spaces, or locally convex spaces with Opialcondition. The results given by this paper not only include the correspondingones in the existing literature as special cases, but also expected to beuseful tools for the development of new theory in nonlinear functional analysisand applications to the study of related nonlinear problems arising frompractice under the general framework of p-vector spaces for p in (0, 1].  Finally, the work presented by this paper focuses on the development ofnonlinear analysis for single-valued (instead of set-valued) mappings forlocally p-convex spaces, essentially, is indeed the continuation of theassociated work given recently by Yuan [134] therein, the attention is given tothe study of nonlinear analysis for set-valued mappings in locally p-convexspaces for p in (0, 1].

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