Contractions, inwardness, tool theorems

TL;DR

This paper advances fixed point theory by generalizing contraction conditions, applying broader spaces for nonexpansive mappings, and replacing Caristi's theorem with specialized tools to derive stronger results for inward mappings.

Contribution

It introduces a more general contraction condition, broadens the class of spaces for nonexpansive mappings, and replaces Caristi's theorem with new tools for inward mappings.

Findings

New generalized contraction condition for multivalued mappings

Broader class of spaces for nonexpansive mappings beyond uniformly convex spaces

Stronger fixed point theorems for inward mappings using specialized tools

Abstract

The paper is devoted to the fixed point theory in four aspects: of contractions, nonexpansive mappings, generalized inward mappings, and of the tool theorems. The manuscript was written about ten years ago. At first Nadler's concept of contraction for multivalued mappings is replaced here by a more general, and yet elegant condition: for some $\alpha + \epsilon <1$, and each $x \in X$ there exists a $y \in F(x)$ such that $d(F(y),y) \leq \alpha d(y,x) \leq (\alpha + \epsilon) d(F(x),x)$}. For ``nonexpansive'' mappings we apply bead spaces that are more general than uniformly convex spaces, and our requirements on mappings are weaker than nonexpansivity in the sense of the Hausdorff distance. In the last, third section the Caristi theorem is replaced by more specialized ``tools'', and we apply them to obtain stronger fixed point theorems on generalized inward mappings. In particular, if for each $x \in X$ a nearest point of $F(x)$ belongs to the generalized inward set, then the values of $F$ need not to be closed.