Contractive conditions in b-metric spaces

Abstract

Abstract:The purpose of this paper is to consider various contractive conditions in b-metric spaces which have been recently published. Our results improve and complement many recent results from this field. Using the recently obtained result by R. Miculescu and A. Mihail (Miculescu & Mihail, 2017, pp.1-11) the authors of this article show that the proofs of the majority of known results in the context of b-metric spaces can be shortened.

Some Remarks and New Results in Ordered Partial b-Metric Spaces

“…where k = 1 s ∈ (0, 1) because s > 1. Now the result follows by known recent results (for example, see [11,12,21,35]).…”

“…either for each ∈ X, or for each ∈ X with = T . The result then follows by [Proposition 2] [11]. If = T , then (16) clearly holds.…”

“…for all n ∈ N. Fix n ∈ N. If n = n−1 , then k = n−1 for all k ≥ n and (11) holds. If n = n−1 , then we get Now, in the sequel we show that it is possible to simplify the proof of Theorem 3 if s > 1.…”

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

“…In [10], T. M. Došenović, M. V. Pavlović and S. N. Radenović made several important observations concerning Lemma 1.1 above and showed how various known fixed point results in b-metric spaces can be shortened by using this lemma.…”