Fixed point results in generalized metric spaces without Hausdorff property

Abstract

It is well known that generalized metric spaces in the sense of Branciari might not be Hausdorff and, hence, there may exist sequences in them having more than one limit. Thus, in most of the fixed point results obtained recently in such spaces, Hausdorffness was additionally assumed. We show in this article that, nevertheless, most of these results remain valid without this additional assumption.

Pata-type common fixed point results in b-metric and b-rectangular metric spaces

“…Moreover, similarly as in [21,23], y n = y m whenever n = m. Let us prove that the sequence c n = d(y 0 , y n ) is bounded.…”

“…were introduced by A. Branciari in 2000 [6]. Some of the papers where the structure of such spaces has been discussed and some fixed point results have been obtained are [11,14,21,22,28,30,31,38,39,41].…”

Fixed point theorems by combining Jleli and Samets, and Branciaris inequalities

“…However, if N ≥ 2, it was proved that B N -spaces can satisfy some properties that are not metrically desirable (see [14,29]). For instance, in a B N -space,…”

“…However, the presented proofs became incorrect because these spaces have metrically non-intuitive properties: for instance, there exist convergent sequences that are not Cauchy, or there exist convergent sequences with two different limits (see [29]). Nevertheless, these drawbacks have not been a limitation for developing fixed point theory in this environment (see [14,15,17,31,32]). On the other hand, Jleli and Samet [13] introduced a kind of generalized metric spaces which are not endowed with a proper triangle inequality: it was replaced by a weaker condition involving convergent sequences.…”

Fixed point theorems for generalized contraction mappings on b-rectangular metric spaces

“…Definition 1.1. Let X = ∅, d : X × X → [0, ∞), such that for each x, y ∈ X and u, v ∈ X (each distinct from x and y), we have that Furthermore, from [10] we mention that convergent sequences and Cauchy sequences can be introduced in a similar manner as in metric spaces. Also, from the same paper, we know that if (X, d) is a rectangular metric space and if (x n ) is a b-rectangular Cauchy sequence with the property that x n = x m , for each n = m, then (x n ) converge to at most one point, i.e.…”