The Nonlinear Analysis in p-Vector Spaces by Applying Best Approximation Method

Abstract It is known that the class of p-vector spaces (0 < p ≤ 1) is an important generalization of usual norm spaces with rich topological and geometrical structure, but the 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 semiclosed 1-set contractive set-valued mappings in p-vector spaces. In particular, we first develop the general fixed point theorems of condensing mappings which provide answer to Schauder conjecture in 1930’s in the affirmative way under the setting of p-vector spaces by taking p = 1 for a p-vector space being a topological vector space. Then one best approximation result for upper semi-continuous and 1-set contractive set-valued is established, which is used as a useful tool to establish fixed points of non-self set-valued mappings with either inward or outward set conditions. Finally, we develop fixed points and related principle of nonlinear alterative for the classes of semiclosed set-valued mappings including nonexpansive set-valued mappings as special cases under uniformly convex Banach spaces, or locally convex topological spaces with Opial condition. The results in this paper under the category of nonlinear analysis not only include the corresponding results in the existing literature as special cases, but also expected to be useful tools for the study of nonlinear problems arising from theory to practice under the general framework of p-vector spaces for 0 < p ≤ 1.