A note on the paper “Contraction mappings in b-metric spaces” by Czerwik

Abstract

In this paper we correct an inaccuracy that appears in the proof of Theorem 1. in Czerwik [1].

Fixed point and common fixed point theorems for (γ, s, q)-F-contraction mappings in b-metric like spaces with application

“…Take F 1 (z) = ln z, q = 2, s = 2 and Γ = 1 s , γ(t) = t 1+40t with η = 1 4 , ζ = 1 6 , ξ = 1 3 . We shall prove that µ and ν satisfy the condition (25).…”

“…Next, we shall show that z is a unique common fixed point of µ and ν. We claim that σ b (z, νz) > 0, then σ b (z, νz) > 0 for all n ∈ N. Applying contraction condition (25) with ι = ι n and κ = z, we get…”

“…Assume that µ(Υ) ⊆ ν(Υ). We construct a sequence {ι n } by ι n = νι n = µι n−1 for all n ∈ N and ι n+1 = νι n+1 = µι n for all n ∈ N. Now, denote (25) with ι = ι n−1 and κ = ι n , we have…”

On the Correlation between Banach Contraction Principle and Caristi’s Fixed Point Theorem in b-Metric Spaces

“…Let us recall that many authors have contributed to the development of a consistent theory of fixed point for b-metric spaces (the bibliographies of [1], and [3][4][5] contain a high account of references to this respect). In particular, the Banach contraction principle [6] admits, mutatis mutandis, a full extension to b-metric spaces [7] (Theorem 2.1) (see also [3,8,9]), and regarding the extension of Caristi's fixed point theorem [10] to b-metric spaces, significant contributions are given, among others, in [11] (Theorem 2.4), as well as in [3] (Corollary 12.1), [7] (Example 2.8) and [12] (Theorem 3.1).…”

Graphical contractions and common fixed points in b-metric spaces

“…For both notions and results see [1,[4][5][6]8]. Definition 1.7 Let (X, d) be a b-metric space with constant s ≥ 1.…”