Generalized 𝜓-Geraghty-Zamfirescu Contraction Pairs in b-metric Spaces

Abstract

The purpose of this paper is to introduce a class of contractive pairs of mappings satisfying a Zamfirescu-type inequality, but controlled with altering distance functions and with parameters satisfying the so-called Geraghty condition in the framework of b-metric spaces. For this class of mappings we prove the existence of points of coincidence, the convergence and stability of the Jungck, Jungck-Mann and Jungck-Ishikawa iterative processes and the existence and uniqueness of its common fixed points. 1. Motivation In 1922, S. Banach [4] established his famous and fundamental result in the metric fixed point theory as follows: Theorem 1.1.(Banach Contraction Principle) Let (M, d) be a complete metric space and let S : M −→ M be a Banach contraction, that is, S satisfies that there exists α ∈ (0, 1) such that d(Sx, Sy) ≤ αd(x, y) (z1) for all x, y ∈M. Then, S has a unique fixed point in M. Notice that Banach’s contractions are continuous mappings, so, in the spirit to extend the BCP, in 1968, R. Kannan [11] introduced a new class of contractive mappings admitting discontinuous functions, as follows. * Corresponding Author. Received September 13, 2020; revised January 15, 2021; accepted January 19, 2021. 2020 Mathematics Subject Classification: 47H09, 47H10, 47J25.