ON FIXED POINT THEORY AND CONTROL FUNCTIONS. STATE OF THE ART.

Authors

Edwin Alexander Loor Andrade Universidad Técnica de Manabí https://orcid.org/0000-0002-1035-6778
Wilmer Eduardo Barrera Yayes, Wilmer Universidad Técnica de Manabí https://orcid.org/0000-0002-5736-1865

DOI:

https://doi.org/10.33936/revbasdelaciencia.v7iESPECIAL.4062

Keywords:

Altering distance function, compatible function, continuous function, decreasing function, fixed point.

Abstract

The fixed-point theory studies the conditions either one or two mappings and the space of definition them, such that the existence and uniqueness of fixed point can be guaranteed. During the first half of the 20th century, Stefan Banach proved a fixed-point theorem for a contractive function defined on a complete metric space. This theorem was developed in several

ways by many authors obtaining results that involve two commutative applications, as is the case of theorem introduced by Gerald Jungck. From 1984 appear extensions of Banach’s contraction principle where the contractive inequality depends on a control function called: altering distance function between points. The goal of this paper is to make an updated review of conditions that guarantee the existence and uniqueness for one and two compatible functions, defined on a complete metric space and considering the altering distance functions and replacing the contraction constant by a function. Some examples that show the importance of generalizations of fixed-point theorems both Banach and Jungck, are given. We provide conditions on a pair of mappings that guarantee the existence and uniqueness of common fixed point by using altering distance functions.

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