Abstract
We solve a question posed by E. Karapinar, F. Khojasteh and Z.D. Mitrović in their paper “A Proposal for Revisiting Banach and Caristi Type Theorems in b-Metric Spaces”. We also characterize the completeness of b-metric spaces with the help of a variant of the contractivity condition introduced by the authors in the aforementioned article.
Highlights
In order to investigate correlations between the Banach contraction principle and results of Caristi type in the realm of b-metric spaces, Karapinar, Khojasteh and Mitrovicproved in [1] (Theorem 1) the following interesting result by using a new type of contractions.Citation: Romaguera, S
A b-metric space is a triple (X, b, s), where X is a set, s is a real number with s ≥ 1, and b : X × X → R+ is a function satisfying, for every u, v, w ∈ X, the following conditions: (b1) b(u, v) = 0 if and only if u = v; (b2) b(u, v) = b(v, u); (b3) b(u, v) ≤ s(b(u, w) + b(w, v))
Motivated by the preceding example, in Definition 1 below we present a modification of the notion of a correlation contraction from which a characterization of b-metric completeness will be obtained via a fixed point result
Summary
In order to investigate correlations between the Banach contraction principle and results of Caristi type in the realm of b-metric spaces, Karapinar, Khojasteh and Mitrovicproved in [1] (Theorem 1) the following interesting result by using a new type of contractions. Let T be a self mapping of a complete b-metric space (X , b, s) such that there is a function F : X → R (the set of real numbers) satisfying the following two conditions: (c1) F is bounded from below, i.e., there is an a ∈ R such that inf F(X ) > a; (c2) for every u, v ∈ X : b(u, T u) > 0 ⇒ b(T u, T v) ≤ (F(u) − F(T u))b(u, v).
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
