Fixed point theory in modular function spaces

Abstract

The author would like to thank Colegio El Pinar for their hospitality during the preparation of this survey.

Remarks on Caristi’s fixed point theorem

“…This is not the first time that the author has to deal with this kind of limitations. Indeed in [12] the authors dealt with a metric-like structure that fails the triangle inequality.…”

On Reich fixed point theorem of G -contraction mappings on modular function spaces

“…Before giving the results, we need to lie out some definitions, notations and facts about the background spaces. More information can be found in [9].…”

Uniformly Lipschitzian mappings in modular function spaces

“…We can assume that k > 1; otherwise T will be nonexpansive and the existence of a fixed point is a consequence of [8,Theorem 3.5…”

“…We recall that the function Φ is said to be uniformly convex [8] if for all ε > 0 there exists δ(ε) ∈ (0, 1) such that:…”

“…Another direction is based on considering an abstractly given functional which controls the growth of the functions. Even though a metric is not defined, many problems in fixed point theory for nonexpansive mappings can be reformulated in modular spaces (see, for instance, [8] and references therein). In this paper, we study the existence of fixed points for a more general class of mappings: uniformly Lipschitzian mappings.…”