Completeness of b∼Metric Spaces and the Fixed Points of Generalized Multivalued Quasicontractions

Abstract

In this article, we present a completeness characterization of b∼metric space via existence of fixed points of generalized multivalued quasicontractions. The purpose of this paper is twofold: (a) to establish the existence of fixed points of multivalued quasicontractions in the setup of b∼ metric spaces and (b) to establish completeness of a b∼ metric space which is a topological property in nature with existence of fixed points of generalized multivalued quasicontractions. Further, a comparison of our results with comparable results shows that the results obtained herein improve and unify the existing results in the literature applicable to the case where existing results fail.

Common Best Proximity Points and Completeness of ℱ−Metric Spaces

“…The study of the characterization of the completeness of a metric space can be traced to Subrahmanyam [29] in 1975, who proved that Kannan's contraction characterizes the metric completeness; that is, a metric space (X, d) is complete if and only if every Kannan's contraction on X has a fixed point. For more on the Completeness Problem in various contexts, we refer the readers to [30,31] and the references therein. The "Completeness Problem" is equivalent to another problem in Behavioral Sciences known as the "End Problem" (see [32]).…”

Completeness of b−Metric Spaces and Best Proximity Points of Nonself Quasi-Contractions

“…Ciric et al [18] obtained Suzuki type fixed point theorems for generalized multivalued mappings on a set endowed with two b−metrics. Alo et al [19] and Ali et al [20] obtained the existence of fixed points of multivalued quasi-contractions along with a completeness characterization of underlying b−metric spaces.…”

“…For details on the completeness problem and the end problem, we refer to [33,34] and references therein. In 1959, Connel presented an example ( [35], (Example 3)) (also compare [20]) which shows that BCP does not characterize metric (b−metric) completeness. That is, there exists an incomplete metric (b−metric) space W such that every Banach contraction on W has a fixed point.…”

“…Suzuki [36] presented a fixed point theorem that generalized BCP and characterized metric completeness as well. Recently, Ali et al [20] (compare with [19]) obtained completeness characterizations of b−metric spaces via the fixed point of Suzuki type contractions.…”

“…Let (W, p) be a b−metric space, ϑ : [0, 1) → (0, 1] and A, B ∈ C(W ). Let A r,ϑ be a class of mappings F : A → B that satisfies (a)-(b) (a) for x, y ∈ A, ϑ(r)p(x, F x) ≤ k(p(x, y) + p(A, B)) implies p(F x, F y) ≤ rp(x, y) − p(A, B),(20) …”