Some Equivalences between Coneb-Metric Spaces andb-Metric Spaces

Abstract

We introduce ab-metric on the coneb-metric space and then prove some equivalences between them. As applications, we show that fixed point theorems on coneb-metric spaces can be obtained from fixed point theorems onb-metric spaces.

Common fixed point theorems of generalized Lipschitz mappings in cone b-metric spaces over Banach algebras and applications

“…A large number of works are noted in [1,2,3,8,11,12,15,17,18,20] and the relevant literature therein. Unfortunately, recently these problems became not attractive since some scholars found the equivalence of fixed point results between cone metric spaces and metric spaces, also between cone b-metric spaces and b-metric spaces (see [4,5,6,7,13,14]). However, quite fortunately, very recently, Liu and Xu [16] introduced the notion of cone metric space over Banach algebra and considered fixed point theorems in such spaces in a different way by restricting the contractive constants to be vectors and the relevant multiplications to be vector ones instead of usual real constants and scalar multiplications.…”

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

“…It is easy to see that b-metrics in the previous two examples are (sequentially) continuous functions (in both variables). However, examples were provided [16,17,29] showing that, in general, this might not be the case. We present here the following Example 2.6.…”

“…A. Bakhtin in 1989 [4] and S. Czerwik in 1993 [10]. There is a vast literature concerning this type of spaces, we mention just some of them [1,2,3,16,17,18,25,26,27,29,33,37,42].…”

SOME FIXED POINT THEOREMS FOR GENERALIZED $\alpha$-GERAGHTY CONTRACTION TYPE MAPPINGS IN $B$-METRIC~SPACES AND SOME APPLICATIONS TO THE NONLINEAR INTEGRAL EQUATION

“…Note that b-metric is a generalization of a metric that was introduced by Czerwik in [7] and then extensively used by Czerwik in [12,13]. The first important difference between a metric and a b-metric is that the b-metric need not be a continuous function in its two variables, see [22,Example 13]. This led to many fixed point theorems on b-metric spaces being stated, so the readers may refer to [1,3,6,8,17,18,19,20,24,25,28,29] and references therein.…”