Fixed sets and fixed points in $\Lim$--spaces]{Fixed sets and fixed points for mappings in generalized $\Lim$--spaces of Fr\'echet

TL;DR

This paper explores fixed point and fixed set theorems in generalized $ ext{Lim}$-spaces, employing axiomatic approaches to convergence and contractiveness, extending classical contraction principles to broader contexts.

Contribution

It introduces a framework for fixed point theorems in generalized $ ext{Lim}$-spaces using axiomatic definitions of convergence and contractiveness, including new conditions with distance-like functions.

Findings

Established fixed set and point theorems for generalized contractions.

Extended classical contraction principles to $ ext{Lim}$-spaces.

Provided axiomatic approaches to convergence and contractiveness.

Abstract

In this article we discuss a possibility to implement a well-known scheme of proof for contraction mapping theorems in a situation, when convergence, families of Cauchy sequences, and contractiveness of mappings are defined axiomatically. We also consider ways to specify families of Cauchy sequences and contractiveness conditions using distance-like functions with values in some partially ordered set and establish fixed set and point theorems for generalized contractions of the \'Ciri\'c and Caristi types.