Common Best Proximity Points and Completeness of F-Metric Spaces

Citation: Zhou, M.; Saleem, N.; Ali, B.; Mohsin, M.; López de Hierro, A.F.R. Common Best Proximity Points and Completeness of F−Metric Spaces. Mathematics 2023, 11, 281. https://doi.org/10.3390/ math11020281 Academic Editors: Mircea Balaj and Vasile Berinde Received: 15 December 2022 Revised: 26 December 2022 Accepted: 27 December 2022 Published: 5 January 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). mathematics Article Common Best Proximity Points and Completeness of F −Metric Spaces Mi Zhou 1,2,3,4 , Naeem Saleem 5, Basit Ali 5,* , Misha Mohsin 5 and Antonio Francisco Roldán López de Hierro 6,* 1School of Science and Technology, Sanya University, Sanya 572000, China 2Center for Mathematical Research, University of Sanya, Sanya 572022, China 3Academician Guoliang Chen Team Innovation Center, University of Sanya, Sanya 572022, China 4Academician Chunming Rong Workstation, University of Sanya, Sanya 572022, China 5Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan 6Department of Statistics and Operations Research, University of Granada, 18071 Granada, Spain *Correspondence: [email protected] (B.A.); [email protected] (A.F.R.L.d.H.) Abstract: In this paper, we introduce three classes of proximal contractions that are called the proximally λ−ψ− dominated contractions, generalized ηγ β− proximal contractions and Berinde-type weak proximal contractions, and obtain common best proximity points for these proximal contractions in the setting of F− metric spaces. Further, we obtain the best proximity point result for generalized α−ϕ− proximal contractions in F− metric spaces. As an application, fixed point and coincidence point results for these contractions are obtained. Some examples are provided to support the validity of our main results. Moreover, we obtain a completeness characterization of the F− metric spaces via best proximity points. Keywords: common best proximity point; proximally λ−ψ− dominated contraction; generalized ηγ β− proximal contraction; Berinde-type weak proximal contraction; generalized α−ϕ− proximal contraction; F−metric spaces MSC: 47H10; 54H25 1. Introduction and Preliminaries In 1922, Banach [ 1 ] introduced his well-known contraction principle, which states that any single-valued self-mapping T defined on a complete metric space satisfying the contractivity condition admits a unique fixed point. Afterward, fixed point theory appeared as a fundamental and broad subject in nonlinear analysis, which is still developing at a rapid pace. It has useful applications in mathematics as well as various scientific disciplines, such as physics, chemistry, computer science, and others. In fact, many practical and research problems in science and engineering can be reduced to fixed point problems. As a result, this theory has offered a remarkable scope of research. Most of the research on this theory can be roughly divided into two directions: generalizing the contraction conditions, the underlying metric spaces and extending the applications. We refer readers to [ 2 – 12 ] and the references therein. Very recently, a new generalization of the notion of metric space, called an F− metric space, was given in [ 6 ], for which the authors used a certain class of auxiliary functions to establish the idea of such abstract spaces. It is obvious that any metric space is an F− metric space, but the converse is not valid in general. A comparison of the F− metric with the existing generalizations of metric in [ 6 ] illustrates that any s− relaxed p− metric is an F− metric, but the converse is not true, which confirms that the class of F− metric spaces is larger than the class of s− relaxed p− metric spaces. Moreover, some examples in [ 6 ] also figured out that F− metric and b− metric are two distinct notions. For Mathematics 2023,11, 281. https://doi.org/10.3390/math11020281 https://www.mdpi.com/journal/mathematics Mathematics 2023,11, 281 2 of 21 more details about other generalizations of metric, we refer readers to the book [ 13 ] written by Kirk. Metric fixed point theory gives sufficient conditions that ensure the existence of solutions to the equation T(x) = x , where T is a self mapping defined on a metric space (X , d) . On the other hand, for a non-self mapping T:A→B , if T(A)∩A=∅ , the mapping T has no fixed point. In this case, it is vital to find an element u0 from the domain spaces whose distance from its image is a minimum. This type of problem is often stated as follows: “Does there exist a point u0 in the metric space (X , d) such that d(u0 , Tu0) = d(A , B) , where A , B are two nonempty subsets of X , T:A→B is a non-self mapping and d(A , B) = inf{d(a , b) , (a , b)∈A×B} .” Recall that the point u0 is called the best proximity point, which was introduced by Basha and Veeramani [ 14 ]. By the definition of the best proximity point, it is clear that every best proximity point is a fixed point of the mapping T when A∩B6=∅ . This new setting is richer and more general than the metric fixed point theory. A noteworthy best approximation theorem (see [ 15 ]) states that if A is a nonempty compact convex subset of a Hausdorff locally convex topological vector space X and f:A→X is a continuous single-valued function, then either f has a fixed point in A , or there exists an element x0∈A and a continuous semi-norm p on X such that 0 <p(x0−f x0) = minx∈Ap(x−f(x0)) . In best proximity point theory, many authors attempted to find minimum conditions on the non-self mapping T to ensure the existence and uniqueness of the best proximity point. For more details, we refer to [16–20] and the references therein. The aim of this paper is to present some common best proximity point results in the setting of F− metric spaces. Based on the notion of commuting mappings introduced by Jungck [ 5 ], a common fixed point theorem, due to Das and Naik [ 4 ], is a special case of our common best proximity point theorem for commuting self-mappings. In the following discussion, the concepts of proximally λ−ψ− dominated contraction, ηγ β− proximal contraction and Berinde-type weak proximal contraction are introduced. Further, some common best proximity point results are proven, which generalize the main results of [ 21 , 22 ] in the setting of F− metric spaces. Moreover, we will introduce the notion of generalized α−ϕ− proximal contractions and prove the best proximity point result for such contractions. As an application, coincidence point and fixed point theorems corresponding to the above proximal contractions are also presented. Some examples are also presented to support our main results. Moreover, a completeness characterization of an F− metric space will be studied in connection with the best proximity points. Given two nonempty subsets, A and B , of a metric space (X , D) , the following notations and notions will be used. D(A,B) = inf{D(x,y):x∈A,y∈B}. A0={x∈A:D(x,y) = D(A,B), for some y∈B}. B0={y∈B:D(x,y) = D(A,B), for some x∈A}. If A∩B6=∅ , then A0 and B0 are nonempty. Further, it is interesting to notice that A0 and B0 are included in the boundaries of A and B , respectively, provided that A and B are closed subsets of a normed linear space such that d(A,B)>0 (see [14]). In 2018, the concept of F− metric space was presented by Jleli and Samet in [ 6 ] as a generalization of the notion of metric space. More precisely, let F be the set of functions f:(0, +∞)→Rsatisfying the following conditions: (F1)fis non-decreasing, i.e., 0 <s<t⇒f(s)≤f(t). (F2) For every sequence {tn} ⊂ ( 0, +∞) , we have limn→+∞tn= 0 ⇔limn→+∞f(tn) = −∞. Here are some examples of fbelonging to F. (i)f1(t) = −1 t,t∈(0, +∞); (ii)f2(t) = ln t,t∈(0, +∞); (iii)f3(t) = −e1 t,t∈(0, +∞). Mathematics 2023,11, 281 3 of 21 Using such functions, Jleli and Samet [ 6 ] generalized the concept of ordinary metric space and introduced the notion of F−metric space as follows: Definition 1 ([ 6 ]) . Let X be a nonempty set, and D:X×X→[0, +∞) be a given mapping. Assume that there exists (f,α)∈ F × [0, +∞)such that (D1) (x,y)∈X×X,D(x,y) = 0⇔x=y; (D2)D(x,y) = D(y,x),∀(x,y)∈X×X; (D3) for every (x , y)∈X×X , N∈N with N≥ 2and (ui)N i=1⊂X with (u1 , uN) = (x , y) , we have D(x,y)>0⇒f(D(x,y))≤f N−1 ∑ i=1 D(ui,ui+1)!+α. Then D is said to be an F− metric on X , and the pair (X , D) is said to be an F− metric space. Example 1 ([ 6 ]) . Let X=N and let D:X×X→[ 0, +∞) be a mapping defined for all (x,y)∈X×X, D(x,y) = (exp(|x−y|),if x 6=y 0, if x =y. For f (t) = −1 t,t>0and α=1,Dis an F− metric. Definition 2 ([ 6 ]) . Let (X , D) be an F− metric space. A subset O of X is said to be F− open if for ∀x∈ O,∃r>0such that B(x,r)⊂ O, where B(x,r) = {y∈X:D(x,y)<r}. We say that a subset C of X is F− closed if X\C is F− open. We denote by τF the family of all F−open subsets of X. Proposition 1 ([6]).Let (X,D)be an F−metric space. Then τFis a topology on X. Proposition 2 ([ 6 ]) . Let (X , D) be an F− metric space. Then for any nonempty subset A of X , the following statements are equivalent: (i) A is F−closed. (ii) For any sequence {xn} ⊂ A, we have lim n→+∞ D(xn,x) = 0, x∈X⇒x∈A. Definition 3 ([6]).Let (X,D)be an F−metric space and {xn}be a sequence in X. Then (i) {xn} is F− convergent to x∈X ,if {xn} is convergent to x with respect to the topology τF , i.e., for every F− open subset Ox of X containing x , there exists some N∈N such that xn∈ Oxfor ∀n≥N. In this case, we say that x is the limit of {xn}. (ii) {xn}is F−Cauchy, if lim n,m→+∞ D(xn,xm) = 0. (iii) (X , D) is F− complete, if every F− Cauchy sequence in X is F− convergent to a certain element in X. Proposition 3 ([ 6 ]) . Let (X , D) be an F− metric space and {xn} be a sequence in X . Then we have (1) a sequence {xn}that is F−convergent to x ⇒lim n→+∞ D(xn,x) = 0. (2) the limit of an F− convergent sequence is unique, i.e., (x , y)∈X×X , lim n→+∞ D(xn , x) = lim n→+∞ D(xn,y) = 0⇒x=y. (3) if a sequence is {xn}is F−convergent, then it is F−Cauchy. Mathematics 2023,11, 281 4 of 21 Definition 4 ([ 6 ]) . Let X be a nonempty set and D:X×X→[ 0, ∞) be a given mapping satisfying (D1) and (D2) . Then the pair (X , D) is F− metric bounded with respect to (f , α)∈ F × [0, +∞)if there exists a metric d on X such that (x,y)∈X×X,D(x,y)>0⇒f(d(x,y)) ≤f(D(x,y)) ≤f(d(x,y)) + α. Definition 5 ([ 23 ]) . Let α:X×X→R be a given function. We say that T:X→X is α−admissible if (x,y)∈X×X,α(x,y)≥1⇒α(Tx,Ty)≥1. Based on the existing definitions introduced in normal metric spaces, we will introduce the analogous definitions in the setting of an F−metric space as follows. Definition 6 ([ 21 ]) . Let A and B be two nonempty subsets of an F− metric space (X , D) . A mapping T:A→B is said to be a proximal contraction if there exists a non-negative real number λ<1such that D(u1,Tx1) = D(A,B) = D(u2,Tx2)⇒D(u1,u2)≤λD(x1,x2),∀u1,u2,x1,x2∈A. Definition 7 ([ 21 ]) . Let A and B be two nonempty subsets of an F− metric space (X , D) . Two mappings T,S:A→B are said to proximally commute if the following holds true D(u,Sx) = D(v,Tx) = D(A,B)⇒Sv =Tu,∀u,v,x∈A. Definition 8 ([ 21 ]) . Let A and B be two nonempty subsets of an F− metric space (X , D) . A mapping T:A→B is said to proximally dominate a mapping S:A→B if there exists a non-negative real number ξ<1such that D(u1,Sx1) = D(v1,Tx1) = D(A,B) = D(u2,Sx2) = D(v2,Tx2) ⇒D(u1,u2)≤ξD(v1,v2),∀u1,u2,x1,x2,v1,v2∈A. Definition 9 ([ 16 ]) . Let (X , D) be an F− metric space and T:A→B and α:A×A→[ 0, +∞) be two mappings. We say that T is α−proximal admissible if α(x1,x2)≥1, D(u1,Tx1) = D(A,B) = D(u2,Tx2)⇒α(u1,u2)≥1, ∀x1,x2,u1,u2∈A. Definition 10 ([ 14 ]) . Let A and B be nonempty subsets of a metric space F− metric space (X , D) and T:A→B be a given mapping. A point x0∈A is said to be the best proximity point of T if d(x0,Tx0) = D(A,B). Definition 11 ([ 24 ]) . Let A and B be two nonempty subsets of an F− metric space (X , D) . An element x ∈A is said to be a common best proximity point of the pair (T,S)if x satisfies D(x,Sx) = D(x,Tx) = D(A,B), where T,S:A→B are two non-self mappings. Definition 12 ([ 17 ]) . Let A and B be two nonempty subsets of an F− metric space (X , D) with A06=∅ . Then the pair (A , B) is said to satisfy the P− property ⇔ for all x1 , x2∈A0 , y1 , y2∈B0 , D(x1,y1) = D(A,B) = D(x2,y2)⇒D(x1,x2) = D(y1,y2). Mathematics 2023,11, 281 5 of 21 Definition 13 ([ 18 ]) . Let A and B be two nonempty subsets of an F− metric space (X , D) with A06=∅ . Then the pair (A , B) is said to satisfy the weak P− property ⇔ for all x1,x2∈A0,y1,y2∈B0 , D(x1,y1) = D(A,B) = D(x2,y2)⇒D(x1,x2)≤D(y1,y2). Here is an example to show that (A , B) satisfies the weak P− property but not the P−property. Example 2. Consider an F−metric space illustrated in Example 4 in [25] as follows. Define g :R2→[0, +∞)by g(x,y) = (2|x|y=0, |x|+|y|y6=0. Let D((x1 , y1) , (x2 , y2)) = g(x1−x2 , y1−y2) , where (x1 , y1) , (x2 , y2)∈R2 . Then, it can be easily checked that (R2,D)is an F−metric space with f (t) = ln t∈ F and α=ln 2. Set A={( 0, 0 )} and B={(x , y):y= 1 − |x|} . Obviously, A0={( 0, 0 )} , B0={(x , y): y=1− |x|,|x| ≤ 1}and D(A,B) = 1. Furthermore, D(( 0, 0 ) , (1 2 , 1 2)) = D(( 0, 0 ) , (−1 2 , 1 2)) = 1,0 =D(( 0, 0 ) , ( 0, 0 )) <D((1 2 , 1 2) , (−1 2 , 1 2)) = 2. We can conclude that (A,B)satisfies the weak P−property but not the P−property. Definition 14 ([ 26 ]) . Let (X , D) be an F− metric space and (A , B) be a pair of nonempty closed subsets of X . An F− metric space (X , D) has the R− property with respect to the pair (A , B) , that is, if xn is a sequence in A such that α(xn , xn+1)≥ 1and limn→+∞xn=x∗∈A , then there exists a subsequence {xnk} of {xn} such that α(xnk , x∗)≥ 1for all k∈N . If A=B=X ,then we say that (X,D)satisfies property (A). 2. Common Best Proximity Points for New Proximal Contractions First, we introduce the following auxiliary functions [ 27 ] that will be used in the main discussions. Let Ψdenote the family of all functions ψ:[0, +∞)→[0, +∞), where (1) ψis increasing and continuous; (2) t≤ψ(t)and ψ(0) = 0; (3) ψ(x+y)≤ψ(x) + ψ(y),∀x,y∈[0, +∞). Let Υdenote the family of all functions ϕ:[0, +∞)→[0, +∞), where (1) ϕis non-decreasing and continuous; (2) ϕ(t)<t,∀t>0; (3) ∑+∞ n=1ϕn(t)<+∞,∀t>0. Let Λdenote the family of all functions λ:[0, +∞)→[0, 1), where (1) λis non-decreasing; (2) λ(tn)→1⇒tn→0. Next, we will introduce the notions of proximally λ−ψ− dominated contraction, generalized ηγ β− proximal contraction and Berinde-type weak proximal contraction in the setting of an F−metric space as follows. Mathematics 2023,11, 281 6 of 21 Definition 15. Let A and B be two nonempty subsets of an F− metric space (X , D) . The pair (T , S) of non-self mappings T , S:A→B is said to be a proximally λ−ψ− dominated contraction if there exist λ∈Λand ψ∈Ψsuch that for all u1,u2,v1,v2,x1,x2∈A D(u1,Sx1) = D(v1,Tx1) = D(A,B) = D(u2,Sx2) = D(v2,Tx2) ⇒ψ(D(u1,u2)) ≤λ(D(v1,v2))ψ(D(v1,v2)). Definition 16. Let A and B be two nonempty subsets of an F− metric space (X , D) . The pair (T , S) of non-self mappings T , S:A→B is said to be a generalized ηγ β− proximal contraction if there exist η∈(0, 1)and β,γ∈[0, 1)with η+β+γ<1such that for all x,y∈A D(Tx,Ty)≤ηD(Sx,Sy) + βD(Sx,Tx) + γD(Sy,Ty). Definition 17. Let A and B be two nonempty subsets of an F− metric space (X , D) . The pair (T , S) of non-self mappings T , S:A→B is said to be Berinde-type weak proximal contraction, if there exist ξ∈(0, 1)and L ≥0such that for all u1,u2,v1,v2,x1,x2∈A D(u1,Sx1) = D(v1,Tx1) = D(A,B) = D(u2,Sx2) = D(v2,Tx2)⇒ D(u1,u2)≤ξmax{D(u1,v1),D(u1,u2)}+Lmin{D(u2,Sx2)−D(A,B)D(u1,v1)}. Definition 18. Let (X , D) be an F− metric space and (A , B) be a pair of nonempty subsets of X . A mapping T:A→B is said to be a generalized α−φ− proximal contraction if there exists φ∈Υ such that for all x,y∈A α(x,y)D(Tx,Ty)≤φ(D(x,y)), where α:A×A→[0, ∞)is a mapping. Theorem 1. Let (A , B) be a pair of nonempty subsets of an F− complete metric space (X , D) . Assume that A0 is a nonempty and closed subset of X . Suppose that the pair (T , S) of non-self mappings T,S:A→B satisfies the following conditions: (1) the pair (T,S)of non-self mappings is a proximally λ−ψ−dominated contraction; (2) T and S proximally commute; (3) T and S are continuous; (4) S(A0)⊆B0and S(A0)⊆T(A0). Then the pair (T,S)admits a unique common best proximity point. Proof. Let x0 be a fixed element in A0 . Since S(A0)⊆T(A0) , there exists an element x1∈A0 such that Sx0=Tx1 . Repeating this process, having chosen xn∈A0 , we can find an element xn+1∈A0satisfying Sxn=Txn+1,∀n∈N∪ {0}. (1) Further, since S(A0)⊆B0 , correspondingly, there exists an element un∈A0 such that D(un,Sxn) = D(A,B),∀n∈N∪ {0}. (2) Further, it follows from the choice of xn and un and from (1) and (2) that for all n∈N , D(un+1,Sxn+1) = D(un,Txn+1) = D(un−1,Txn) = D(A,B). (3) Since the pair (T , S) is a proximally λ−ψ− dominated contraction, from (1) to (3) , we have ψ(D(un,un+1)) ≤λ(D(un−1,un))ψ(D(un−1,un)) <ψ(D(un−1,un)). (4) Mathematics 2023,11, 281 7 of 21 Since ψ is increasing, {D(un−1 , un)} is non-increasing and bounded. Therefore, lim n→+∞ D(un−1,un)exists. Let lim n→+∞ D(un−1,un) = r≥0. Assume that r>0. Then from (4) we have ψ(D(un,un+1)) ψ(D(un−1,un)) ≤λ(D(un−1,un)). Since ψis continuous, the above inequality yields lim n→+∞λ(D(un−1,un)) = 1, which implies that r=0, that is, lim n→+∞ D(un−1,un) = 0. Keeping in mind that {D(un−1 , un)} is non-increasing and λ is a non-decreasing function, we have λ(D(u0,u1)) ≥λ(D(u1,u2)) ≥... ≥λ(D(un−1,un)). Further, from inequality (4), we have ψ(D(un+1,un)) <ψ(D(un−1,un)) ≤λ(D(un−2,un−1))ψ(D(un−2,un−1)) ≤λ(D(un−2,un−1))λ(D(un−3,un−2))ψ(D(un−3,un−2)) ≤λ(D(un−2,un−1))λ(D(un−3,un−2)) · · · λ(D(u1,u2))λ(D(u0,u1))ψ(D(u0,u1)) ≤λ(D(u0,u1))λ(D(u0,u1)) · · · λ(D(u0,u1))λ(D(u0,u1))ψ(D(u0,u1)) ≤λn(D(u0,u1))ψ(D(u0,u1)) =µnψ(D(u0,u1)), where µ=λ(D(u0,u1)) ∈[0, 1). Therefore, ψ(D(un,un+1)) <µnψ(D(u0,u1)), which yields that m−1 ∑ i=n ψ(D(ui,ui+1)) <µn 1−µψ(D(u0,u1)). (5) Next, we will prove that {un}is an F−Cauchy sequence. Without a loss in the generality, we may suppose that D(u0 , u1)> 0. Otherwise, from the construction of {un} , we can conclude that un=u0 for all n∈N and so {un} is an F−Cauchy. First, let (f , α)∈ F × [ 0, ∞) be such that (D3) is satisfied. For any given e> 0, by (F2), there exists δ>0 such that 0<t<δ⇒f(t)<f(e)−α. (6) Since lim n→+∞ µn 1−µψ(D(u0,u1)) = 0, then for δ>0, there exists some N∈Nsuch that 0<µn 1−µψ(D(u0,u1)) <δ,∀n≥N. (7) Mathematics 2023,11, 281 8 of 21 In addition, since ψsatisfies that t≤ψ(t), we have m−1 ∑ i=n D(ui,ui+1)≤ m−1 ∑ i=n ψ(D(ui,ui+1)) <µn 1−µψ(D(u0,u1)). (8) Hence, by (7)–(8) and (F1), we have f m−1 ∑ i=n D(ui,ui+1)!≤fµn 1−µψ(D(u0,u1))<f(e)−α,∀m>n≥N. (9) By (D3)and inequality (9), we have D(un,um)>0, m>n>N⇒f(D(un,um)) ≤f m−1 ∑ i=n D(ui,ui+1)!+α<f(e), which implies by (F1)that D(un,um)<e,∀m>n≥N. This proves that {un} is an F− Cauchy sequence. Since (X , D) is a complete F− metric space and A0 is closed, there exists some u∈A0 such that limn→+∞un=u . Because of the fact that the mappings S and T proximally commute and from inequalities (2) and (3) , we have Tun=Sun−1,∀n∈N. Therefore, the continuity of the mappings Sand Tensures that Tu =lim n→+∞Tun=lim n→+∞Sun−1=Su. Since S(A0)⊆B0, there exists an element x∈A0⊆Asuch that D(x,Su) = D(x,Tu) = D(A,B), It follows from Definition 7that Sx =Tx . Again, since S(A0)⊆B0 , there exists z∈A such that D(z,Sx) = D(z,Tx) = D(A,B). Since the pair of mappings (T , S) is λ−ψ− dominate proximally contractive, we have ψ(D(x,z)) ≤λ(D(x,z))ψ(D(x,z)), which implies x=z. Thus, it follows that D(x,Sx) = D(z,Sx) = D(A,B) = D(x,Tx) = D(z,Tx). Therefore, xis a common best proximity point of the mappings Sand T. Suppose that y is another common best proximity point of the mappings S and T such that x6=y. We have D(x,Sx) = D(A,B) = D(x,Tx). (10) D(y,Sy) = D(A,B) = D(y,Ty). By the definition of λ−ψ−dominate proximal contractivity and (10), we have ψ(D(x,y)) ≤λ(D(x,y))ψ(D(x,y)) <ψ(D(x,y), Mathematics 2023,11, 281 9 of 21 which implies that x=y . Hence, x is a unique common best proximity point that satisfies D(x,Sx) = D(A,B) = D(x,Tx). Example 3. Let X∗= [ 0, 1 ]×[ 0, 1 ] be endowed with a metric D:[ 0, 1 ]2×[ 0, 1 ]2→[ 0, ∞) defined by for all x = (x1,x2),y= (y1,y2)∈X∗ D(x,y) = (exp∑2 i=1|xi−yi|,if xi6=yi, 0, if xi=yi. It can be easily checked that Dis an F−metric with f (t) = −1 t∈ F and α=1. Let A={( 0, x∗): 0 ≤x∗≤ 1 } and B={( 1, y∗): 0 ≤y∗≤ 1 } .Notice that we can have that A0=A , B0=B and D(A , B) = e . Consider the mappings T , S:A→B defined by S(0, x∗) = (1, ln(1+x∗)), and T(0, x∗) = (1, x∗). Now, we claim that the pair (T , S) is a λ−ψ− dominate proximal contraction. Consider λ∈Λ and ψ∈Ψ defined by λ(t) = 1 −ln2t 2t , ∀t∈( 0, ∞) and ψ(t) = t , ∀t∈[ 0, ∞) . Choosing u1= (0, u),v1= (0, v),w1= (0, w),s1= (0, s), x1= (0, x), y1= (0, y)in A satisfying D(u1,Sx1) = D(v1,Sy1) = D(A,B) = D(w1,Tx1) = D(s1,Ty1) = e. We have w =x,s=y and u =ln(1+x),v=ln(1+y). Thus, we can write ψ(D(u1,v1)) = e|ln(1+x)−ln(1+y)| =e|ln(1+w)−ln(1+s)| ≤eln(1+|w−s|) <e|w−s|−1 2|w−s|2 =e|w−s| 1−ln2e|w−s| 2e|w−s|! =λ(D(w1,s1))ψ(D(w1,s1)), which shows that the pair (T , S) commutes proximally and is a λ−ψ− dominate proximal contraction. Therefore, all conditions of Theorem 1are satisfied. From the conclusion of Theorem 1, (T , S) has a unique common best proximity point, which is (0, 0)∈A. Theorem 2. Suppose that (A , B) is a pair of nonempty subsets of a complete F− metric space (X , D) that satisfies the P− property. Assume that A0 is a nonempty and closed subset of A . Further, suppose that S , T:A→B are continuous mappings, where T and S proximally commute. Further, assume that the pair (T , S) is a generalized ηγ β− proximal contraction satisfying S(A0)⊆T(A0) and S(A0)⊆B0. Then the pair (T,S)admits a unique common best proximity point. Proof. Let x0 be a fixed element in A0 . Since S(A0)⊆T(A0) , there exists an element x1∈A0 such that Sx0=Tx1 . Repeating this process, having chosen xn∈A0 , we can find an element xn+1∈A0satisfying Txn=Sxn+1,∀n∈N∪ {0}. (11) Further, since S(A0)⊆B0 , correspondingly, there exists an element un∈A0 such that D(un,Txn) = D(A,B),∀n∈N∪ {0}. (12) Further, it follows from the choice of xnand unand from (11) and (12) that D(un+1,Txn+1) = D(un,Sxn+1) = D(un−1,Sxn) = D(A,B),∀n∈N. (13) Mathematics 2023,11, 281 16 of 21 Inductively, we can construct a sequence {xn} ⊂ A0such that D(xn+1,Txn) = D(A,B),α(xn,xn+1)≥1, ∀n∈N∪ {0}. (37) We claim that {xn} is an F− Cauchy sequence. Using the P− property, we deduce from (37) that D(xn,Txn−1) = D(A,B) D(xn+1,Txn) = D(A,B)⇒D(xn,xn+1) = D(Txn−1,Txn),∀n∈N. (38) Since T is generalized α−ϕ− proximally contractive, there exists a function ϕ∈Λ such that D(Txn−1,Txn)≤α(xn−1,xn)D(Txn−1,Txn)≤ϕ(D(xn−1,xn)), which implies D(xn,xn+1) = D(Txn−1,Txn)≤ϕ(D(xn−1,xn)),∀n∈N. (39) Since the mapping ϕis increasing, inequality (39) becomes D(xn,xn+1)) ≤ϕ2(D(xn−2,xn−1)) ≤ϕ3(D(xn−3,xn−2)) . . . ≤ϕn(D(x0,x1)),∀n∈N∪ {0}. The second property of ϕyields that m−1 ∑ i=n D(ui,ui+1) =D(xn,xn+1) + D(xn+1,xn+2) + D(xn+2,xn+3) + · · · +D(xm−1,xm) ≤ϕn(D(x0,x1)) + ϕn+1(D(x0,x1)) + ϕn+2(D(x0,x1)) + · · · +ϕm−1(D(x0,x1)) =ϕn(D(x0,x1))(1+ϕ(D(x0,x1)) + ϕ2(D(x0,x1)) + · · · +ϕm−n−1(D(x0,x1)) <ϕn(D(x0,x1))1+1 1−ϕ(D(x0,x1)). We conclude from taking the limit as n→+∞in the above inequality that lim n→+∞ϕn(D(x0,x1))(1+1 1−ϕ(D(x0,x1))) = 0. (40) Let (f , α)∈ F × [ 0, +∞) satisfy (D3) and ε> 0 be fixed. By (F2) , there exists δ> 0 such that 0<t<δ⇒f(t)<f(e)−α. (41) For the δ>0 in (41), by (40), there exists N∈Nsuch that 0<ϕn(D(x0,x1))(1+1 1−ϕ(D(x0,x1)))<δ,∀m>n≥N. Mathematics 2023,11, 281 17 of 21 Hence, by (41) and (F2), we have f m−1 ∑ i=n D(xi,xi+1)!≤fϕn(D(x0,x1))(1+1 1−ϕ(D(x0,x1))(42) <f(e)−α,∀m>n≥N. Using (D3)and (42), we have D(xn,xm)>0⇒f(D(xn,xm))≤f m−1 ∑ i=n D(ui,ui+1)!+α<f(e),m>n≥N. This implies by (F1)that D(xn,xm)<e,m>n≥N. Hence, the sequence {xn} is an F− Cauchy sequence. Since (X , D) is a complete F− metric space and A0 is a closed subset of (X , D) , there exists x∗∈A0 such that xn converges to x∗. By R− Property, there exists a subsequence {xnk} of {xn} such that α(xnk , x∗)≥ 1 for all k∈N. Since Tis a generalized α−ϕ−proximal contraction, we have D(Txnk,Tx∗)≤α(xnk,x∗)D(Txnk,Tx∗)(43) ≤ϕ(D(xnk,x∗)),∀k∈N. We will prove that x∗ is the best proximity point of T . Suppose that D(x∗,Tx∗)>D(A,B). By (D3), we have f(D(x∗,Tx∗)−D(A,B)) ≤f(D(x∗,Txnk) + D(Txnk,Tx∗)−D(A,B)) + α,∀k∈N. Together with (43) and (F1), we have f(D(x∗,Tx∗)) ≤f(D(x∗,Txnk) + ϕ(D(xnk,x∗))) + α. Taking the limit as k→+∞in the above inequality, it follows from (F2)that f(D(x∗,Tx∗)−D(A,B)) ≤lim k→+∞f(D(x∗,Txnk) + ϕ(D(xnk,x∗))) + α =−∞, which implies that D(x∗ , Tx∗)−D(A , B) = 0. Hence x∗ is the best proximity point of T . Remark 1. The conclusions of Theorems 2,3, and 4and Corollary 1still hold if we replace the P−property assumption imposed on (A,B)by the weak P−property. 3. Coincidence Point Results in F−Metric Spaces In this section, we will discuss some coincidence point results for proximal contractions endowed with an F−metric. From Theorems 1–3, we can obtain the following coincidence/fixed point results by using our previous proximal contraction results. Theorem 5. Let (X , D) be a complete F− metric space. Suppose that T , S:X→X are continuous and commuting; also the pair (T , S) is a λ−ψ− dominated contraction. Then (T , S) has a unique coincidence point. Mathematics 2023,11, 281 18 of 21 Proof. If we take A=B=X in Theorem 1, then D(A , B) = 0. In addition to this, every proximally λ−ψ− dominated contraction becomes a λ−ψ− dominated contraction. Analysis similar to that in the proof of Theorem 1shows that there exists x∈Xsuch that D(x,Sx) = D(x,Tx) = D(A,B) = 0, which implies that Sx =Tx . Hence, x is a coincidence point of the pair (T , S) . Moreover, the uniqueness of the coincidence point can be deduced from the same arguments presented in the proof of Theorem 1. Theorem 6. Let (X , D) be a complete F− metric space. Suppose that T , S:X→X are continuous and commuting; also the pair (T , S) is a generalized ηγ β− contraction. Then (T , S) has a unique coincidence point. Proof. The conclusion can be drawn by applying the same argument in the proof of Theorem 5by replacing the λ−ψ− dominate proximal contraction with the generalized ηγ β−contraction. Theorem 7. Let (X , D) be a complete F− metric space. Suppose that T , S:X→X are continuous and commuting; also the pair (T , S) is a Berinde-type weak contraction. Then (T , S) has a unique coincidence point. Proof. The conclusion can be drawn by applying the same argument in the proof of Theorem 5by replacing the λ−ψ− dominate proximal contraction with the Berinde-type weak contraction. Definition 19. A mapping T :X→X is said to be a generalized α−ϕ−contraction, if α(x,y)D(Tx,Ty)≤ϕ(D(x,y)),∀x,y∈A, where α:A×A→[0, +∞),ϕ∈Υ. Taking A=B=Xin Theorem 4, we get the following fixed point result. Corollary 2. Let A be a nonempty F− closed subset of a complete F -metric space (X , D) . Suppose that an α− admissible mapping T:X→X is a generalized α−ϕ− contraction and there exist elements x0 , x1∈X such that α(x0 , x1)≥ 1. Further, (X , D) satisfies property (A) . Then T has a unique fixed point x∗∈A. 4. Completeness of F−Metric Spaces via the Best Proximity Points In Mathematics, the “Completeness Problem ”is an essential issue that concerns when a space is complete. In such a case, a Cauchy sequence converges. The famous Banach Contraction Principle holds in complete metric spaces, but completeness is not a necessary condition; that is, there are incomplete metric spaces on which every contraction has a fixed point (see [ 28 ]). The Banach contraction principle does not characterize metric completeness. As every metric is F− metric, the Banach contraction principle does not characterize the 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 characterizations have further been studied in connection with best proximity points [33,34]. Mathematics 2023,11, 281 19 of 21 In this section, we obtain a completeness characterization of an F− metric space via the best proximity points. Definition 20. Let (X , D) be an F− metric space and (A , B) be a pair of nonempty closed subsets of X . A generalized α−ϕ− proximal contraction T:A→B is an α−ϕ−SVV proximal contraction if (1) A0is nonempty; (2) A and B satisfy the P−property; (3) T(A0)⊆B0; (4) there exist elements x0,x1∈A0such that D(x1,Tx0) = D(A,B)and α(x0,x1)≥1. (5) T is α−proximal admissible; (6) (X,D)satisfies the R−property with respect to the pair (A,B). If, in the above definition, we set A=B=X, then we obtain the following. Definition 21 ([ 26 ]) . Let (X , D) be an F− metric space. A generalized α−ϕ− contraction T:X→X is an α−ϕ−SVV contraction if (1) there exist elements x0∈X such that α(x0,Tx0)≥1; (2) T is α−admissible; (3) (X,D)satisfies property (A). Romaguera and Tirado [ 26 ] obtained the following characterization of completeness of metric spaces via α−ϕ−contractions T:X→Xas follows. Theorem 8 ([ 26 ]) . A metric space is complete if and only if every α−ϕ−SVV contraction has a fixed point. Now we get the following characterization of F− completeness of an F− metric space (X,D). Theorem 9. Let (X , D) be an F− metric space and (A , B) a pair of nonempty closed subsets of X . Then the following statements are equivalent: (i) (X,D)is F−complete. (ii) Every α−ϕ−SVV proximal contraction T:A→B has the best proximity point in (X , D) . (iii) Every α−ϕ−SVV contraction T :X→X has a fixed point in (X,D). Proof. (i) =⇒(ii): It follows directly from Theorem 4. (ii) =⇒(iii): This follows by setting A=B=Xin (ii). (iii) =⇒(i) : Suppose, on the contrary, that (X , D) is not F -complete, that is, there is an F−Cauchy sequence {wn}(of distinct points) in (X,D)that does not converge. Set B={wn:n∈N} . As D(w1,B\ {w1})> 0, there exists h1∈N with h1> 1 such that Dwj,wk<1 2D(w1,B\ {w1}) for all k≥j≥h1. Similarly, there exists h2∈Nwith h2>max{2, h1}such that Dwj,wk<1 2D(w2,B\ {w2}) for all k≥j≥h2 . Repeating this argument, we get a subsequence {hn} of N such that hn>max{n,hn−1}and Dwj,wk<1 2D(wn,B\ {wn}) Mathematics 2023,11, 281 20 of 21 for all k≥j≥hn. Define the mappings T:X→Xand α:X×X→[0, ∞)as Tw =whn, if w=wnfor n∈N w1, if w∈X\B and α(w,z) = 1, if w=wnand z=wmfor m,n∈Nwith n<m, 0, otherwise. Note that α(w1 , Tw1) = 1 as h1> 1. If α(w , z)≥ 1 for w=wn , and z=wm , then α(Tw , Tz) = α(whn , whm) = 1, as hm>hn . Hence T is α− admissible. Further, (X , D) satisfies property ( A ) because any F− convergent sequence {zn} satisfying α(zn , zn+1)≥ 1 is a constant sequence. Now for ϕ(t) = t 2 and from the construction of α , it is sufficient to check the α−ϕ−contraction condition for w=wnand z=wmwith n<m. Thus α(w,z)D(Tw,Tz) = α(wn,wm)D(whn,whm) <1 2D(wn,B\ {wn}) <1 2D(wn,wm)=1 2D(w,z). Hence T is an α−ϕ−SVV contraction, which does not have a fixed point. This is a contradiction. Hence (X,D)is F−complete. This completes the proof. 5. Conclusions and Future Work This article proves the existence of common best proximity points for proximally λ−ψ− dominated contractions, generalized ηγ β− contractions and Berinde-type weak contractions in the setting of F− complete metric spaces. As an application, fixed point and coincidence point results for such generalized proximal contraction are obtained. Moreover, a completeness characterization of F− metric spaces is obtained via the existence of the best proximity points of a certain proximal contraction. One can consider the results in this paper for further study in the setup of more general spaces such as metric-like spaces, quasi-metric spaces, fuzzy metric spaces and so on. Besides this, one can strive to obtain weaker conditions to ensure the existence of the best proximity points for some generalized proximal contractions in several classical metric spaces; for instance, the existence of a proximity point without P− property in a fuzzy metric space would be worth investigating. Moreover, Ghasab et al. [35] introduced F− quasi-metric spaces, which is viewed as a generalization of F− metric spaces. One could extend our main results to the F− quasi-metric spaces for furnishing the best proximity theory. Author Contributions: Conceptualization, M.Z., N.S., B.A. and M.M.; formal analysis, M.Z., B.A. and M.M.; investigation, M.Z., N.S. and A.F.R.L.d.H.; writing original draft preparation, M.Z., B.A. and M.M.; writing review and editing, M.Z., N.S. and A.F.R.L.d.H. All authors have read and agreed to the published version of the manuscript. Funding: This work is partially supported by the High Level Project of Hainan Provincial Natural Science Foundation (Grant No. 621RC602) and Key Special Project of University of Sanya (Grant No. USY22XK-04). Data Availability Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest. References 1. Banach, S. Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fund. Math. 1922 ,3, 133–181. [CrossRef] 2. Boyd, D.W.; Wong, J.S.W. On Nonlinear Contractions. Proc. Am. Math. Soc. 1969,20, 458–464. [CrossRef] 3. Czerwik, S. Contraction mappings in b−metric spaces. Acta Math. Univ. Ostrav. 1993,1, 5–11. Mathematics 2023,11, 281 21 of 21 4. Das, K.M.; Viswanatha Naik, K. Common fixed point theorems for commuting maps on a metric space. Proc. Am. Math. Soc. 1979 , 77, 369–373. 5. Jungck, G. Commuting mappings and fixed points. Amer. Math. Monthly 1976,83, 261–263. [CrossRef] 6. Jleli, M.; Samet, B. 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