Abstract
In this paper, we will consider the coincidence point problem for a pair of single-valued operators satisfying to some contraction and expansion type conditions. Existence, uniqueness and qualitative properties of the solution will be presented. The results are based on some fixed point theorems for nonlinear contractions in complete b-metric spaces. An application illustrates the theoretical results.
Highlights
An extension of the Banach’s contraction principle was given, in the framework of b-metric spaces, by S
It is worth to mention that the b-metric structure produces some differences to the classical case of metric spaces: the b-metric on a nonempty set X need not be continuous, open balls in such spaces need not be open sets and so on
2) If (X, d) is a complete metric space and the contraction condition holds for all x, y ∈ X, (without the assumptions (ii) and (iii) in the above theorem) we obtain Czerwik’s fixed point theorem in [4]
Summary
An extension of the Banach’s contraction principle was given, in the framework of b-metric spaces ( called, in some papers, quasi-metric spaces or metric type spaces), by S. For several fixed point results in this framework see [1], [2], [7]. Let (X, d) and (Y, ρ) be two metric spaces and g, t : X → Y be two operators. The coincidence point problem for t and g means to find x∗ ∈ X such that t(x∗) = g(x∗). We will denote by CP (g, t) the coincidence point set for g and t. The aim of this paper is to present, in the context of b-metric spaces, two types of coincidence point theorems under some contraction and expansion type conditions. The method is based on the application of some fixed point point theorems of Ran-Reurings type in ordered b-metric spaces. Our coincidence results are in connection with some nice previous theorems given in A.
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