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 $p\in (0,1].$ 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.

Keywords:

• Best approximation
• condensing mapping
• p-vector space
• locally p-convex space
• fixed point theorem
• graph-approximation
• p-inward and p-outward set
• measure of noncompactness
• nonexpansive mapping
• nonlinear alternative. Leray-Schauder alternative
• nonlinear analysis
• Schauder conjecture
• 1-set contractive mapping
• uniform convex space

AMS CLASSIFICATION:

• 46A16
• 46A55
• 52A07
• 54C60
• 47H04
• 47H10
• 54H25
• 49J27
• 49J35
• 55M20

Acknowledgement

The author thanks Professor S.S.Chang (Shi-Sheng Zhang), Professor K.K. Tan, Professor Bruce Smith, Professor Jian Yu, Professor Hong Ma, Professor Y.J. Cho, Professor S. Park, Professor M. Nashed for their always encouragements in the past more than two decades. My thanks go to Professor Hong-Kun Xu, Professor Tiexin Guo, Professor Xiao-Long Qin, Professor Ganshan Yang, Professor Xian Wu, Professor Nanjing Huang, Professor Mohamed Ennassik; and my colleagues and friends across China, Australia, Canada, UK, USA and else where. In particular, author thanks anonymous referees’ comment and suggest which led to the present version of the paper.

Compliance with ethical standards

The author declares that there is no conflict of interest.

Additional information

Funding

This research is partially supported by the National Natural Science Foundation of China [grant numbers 71971031 and U1811462].