Caristi-Kirk type and Boyd and Wong-Browder-Matkowski-Rus type fixed point results in b-metric spaces

Abstract

In this paper, based on a lemma giving a sufficient condition for a sequence with elements from a b-metric space to be Cauchy, we obtain Caristi-Kirk type and Boyd&Wong-Browder-Matkowski-Rus type fixed point results in the framework of b-metric spaces. In addition, we extend Theorems 1, 2 and 3 from [M. Bota,V. Ilea, E. Karapinar, O. Mle?ni?e, On ?*-?-contractive multi-valued operators in b-metric spaces and applications, Applied Mathematics & Information Sciences, 9 (2015), 2611-2620].

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).…”

A general common fixed point result for two pairs of maps in b-metric spaces

“…It is easy to see that every converging sequence is a Cauchy sequence. The following fundamental lemma was improved in 2017 by R. Miculescu and A. Mihail in their paper [16] and by T. Suzuki in his paper [20] by a different method. Lemma 1.1 ([16] and [20]).…”

Iterated Function Systems Consisting of Generalized Convex Contractions in the Framework of Complete Strong b-metric Spaces

“…A. Bakhtin (see [8]) and S. Czerwik (see [15] and [16]). In the last years a lot of fixed point results in the framework of b-metric spaces have been obtained (see, for example, [1], [7], [9], [11], [12], [17], [27], [28], [29], [31], [40], [41], [42], [44], [46], [47], [49], [50], [51], [55], [56] and [59]). …”