Research Article Coincidence Best Proximity Point Results in Branciari Metric Spaces with Applications Naeem Saleem , 1 Manuel De la Sen , 2 and Sadia Farooq 1 1 Department of Mathematics, University of Management and Technology, Lahore, Pakistan 2 Institute of Research and Development of Processes, University of the Basque Country, Campus of Leioa (Bizkaia), P.O. Box 644, Bilbao, Barrio Sarriena, 48940 Leioa, Spain Correspondence should be addressed to Naeem Saleem; naeem.saleem2@gmail.com Received 16 July 2020; Revised 30 August 2020; Accepted 27 November 2020; Published 14 December 2020 Academic Editor: Nawab Hussain Copyright © 2020 Naeem Saleem et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper is aimed at studying the uniqueness of coincidence best proximity point for ðϑ, α + , gÞ-proximal contractions in complete Branciari metric space. Throughout this article, discontinuity of the Branciari metric space is used and we obtained the desired results without assuming it as a continuous. Some examples are provided to validate the results proved herein. As an application, we derive the best proximity point results in the setup of complete Branciari metric space endowed with graph. Further, our results extend and generalize the existing ones in literature. 1. Introduction and Preliminaries Let F : X ⟶ X be a mapping, where X be any nonempty set. “An element q ∗ ∈ X is a fixed point of F if q ∗ satisfies the equation Fq ∗ = q ∗ (known as a fixed point equation) or dðq ∗ , Fq ∗ Þ =0: } A collection of all “fixed points” of F will be represented as FðXÞ, that is, F X ð Þ = q ∗ ∈ X : d q ∗ , Fq ∗ ð Þ =0 f g: ð1Þ In this direction, Banach [1] gives the existence and uniqueness of the “fixed point” of the self mapping F, if map- ping F is a contraction and ðX, dÞ is a complete, but it becomes more interesting, if F is a nonself mapping then it is not necessary that the operator equation Fq ∗ = q ∗ has a solution. In this situation, we can find a point q ∗ ∈ X which is closest to Fq ∗ and we have the following minimization/op- timization problem min q∈X d q, Fq ð Þ: ð2Þ Now, consider X = ðX, dÞ be a metric space, Q and P are nonempty subsets of X, and consider a mapping F : Q ⟶ P , we can find a point q ∗ in Q such that dðq ∗ , Fq ∗ Þ is mini- mum. In other words, we have to minimize dðq ∗ , Fq ∗ Þ for all q ∗ in Q and Fq ∗ in P : It is important to see that the min q∈Q dðq, FqÞ = dðQ, P Þ, where dðQ, P Þ = inf fdðq, pÞ: q ∈ Q, p ∈ P g which cannot be further reduced. If such point q ∗ in Q exists then q ∗ is called an “approximate fixed point” of F [2]. Later, several authors studied the results dealing with “approximate fixed points” in different spaces (for detail, see [3–14]). The best proximity point of the mapping F : Q ⟶ P is actually “a point q ∗ ∈ X such that dðq ∗ , Fq ∗ Þ = dðQ, P Þ.” Note that if Q ∩ P ≠ ϕ then dðQ, P Þ =0; in this case, every “approx- imate fixed point” becomes “fixed point” of the mapping F . From this perspective, we can say that “the best proximity point results” are natural generalization of “fixed point results.” The concept of “coincidence best proximity point” was introduced in [5] for a pair of mappings in metric space. “A point q ∗ ∈ Q is called the coincidence best proximity point of a pair of mappings F : Q ⟶ P and g : Q ⟶ Q if dðgq ∗ , F q ∗ Þ = dðQ, P Þ.” We denote the set of all “coincidence best proximity points” of a pair of mappings F and g by FgðQÞ, that is, Fg Q ð Þ = q ∗ ∈ Q : dgq ∗ , Fq ∗ ð Þ = d Q, P ð Þ f g: ð3Þ Hindawi Journal of Function Spaces Volume 2020, Article ID 4126025, 17 pages https://doi.org/10.1155/2020/4126025