@ Appl. Gen. Topol. 23, no. 2 (2022), 405-424 doi:10.4995/agt.2022.14000 © AGT, UPV, 2022 Best proximity point (pair) results via MNC in Busemann convex metric spaces This paper is dedicated in the memory of Prof. Hans-Peter Künzi Moosa Gabeleh a and Pradip Ramesh Patle b a Department of Mathematics, Ayatollah Boroujerdi University, Boroujerd, Iran (Gabeleh@abru.ac.ir, gab.moo@gmail.com) b Department of Mathematics, Amity School of Engineering and Technology, Amity University Madhya Pradesh, Gwalior, India. (pradip.patle@gmail.com) Communicated by M. Abbas Abstract In this paper, we present a new class of cyclic (noncyclic) α-ψ and β-ψ condensing operators and survey the existence of best proximity points (pairs) as well as coupled best proximity points (pairs) in the setting of reflexive Busemann convex spaces. Then an application of the main existence result to study the existence of an optimal solution for a system of differential equations is demonstrated. 2020 MSC: 53C22; 47H09; 34A12. Keywords: coupled best proximity point (pair); cyclic (noncyclic) condens- ing operator; optimum solution; Busemann convex space. 1. Introduction In the present paper, we mainly focus on the study of best proximity points for certain classes of mappings T : A ∪ B → A ∪ B for which T(A) ⊆ B and T(B) ⊆ A. Such mappings are called cyclic mappings. Likewise, if T(A) ⊆ A and T(B) ⊆ B, then T is said to be a noncyclic mapping. For the noncyclic case, the point (p,q) ∈ A×B is said to be a best proximity pair for the mapping T provided that p and q are two fixed points of T which estimates the distance between two sets A and B, that is, p = Tp, q = Tq and d(p,q) = dist(A,B). Received 10 July 2020 – Accepted 01 May 2022 http://dx.doi.org/10.4995/agt.2022.14000 https://orcid.org/0000-0001-5439-1631 https://orcid.org/0000-0002-9650-6107 M. Gabeleh and P. R. Patle The first existence theorem of best proximity points (pairs) was established in [7] for cyclic (noncyclic) relatively nonexpansive mappings. We recall that the mapping T : A∪B → A∪B is called relatively nonexpansive, if d(Tx,Ty) ≤ d(x,y) for all (x,y) ∈ A×B. In [11] the main result of [7] was extended from Banach spaces to reflexive Busemann convex spaces. Recently Gabeleh and Markin introduced a class of cyclic (noncyclic) con- densing operators by using a notion of measure of noncompactness and then studied the existence of best proximity points (pairs) for such mappings in (strictly convex) Banach spaces (see Theorem 3.4 and Theorem 4.3 of [12]). Subsequently Gabeleh and Künzi generalized the main conclusions of [12] from two points of view. One by considering a class of cyclic (noncyclic) con- densing operators of integral type and another one by generalizing (strictly con- vex) Banach spaces to reflexive Busemann convex spaces (see Theorem 3.9 and Theorem 3.11 of [13]). We refer to articles [14, 18] in this direction. For more information about the existence of best proximity points for various classes of non-self mappings, one can see [17, 20, 21, 22]. In the current article, we introduce two new classes of cyclic (noncyclic) condensing operators, called α−ψ and β −ψ condensing operators which are properly contain the class of cyclic (noncyclic) condensing operators introduced by Gabeleh and Markin in [12]. We establish the existence of best proximity points (pairs) for such mappings in the framework of reflexive Busemann convex spaces and then apply our conclusions to present a coupled best proximity point theorem by the notion of measure of noncompactness. Finally, the existence of an optimal solution for a system of second order differential equations by using the existence result of best proximity points for the considered extension of cyclic condensing operators is studied. 2. Preliminaries In this section, we compile the main notions and notations we will work with along this paper. 2.1. Geodesic spaces. Let (X,d) be a metric space. By B(x0; r) we denote the closed ball in the space X centered at x0 ∈ X with radius r > 0. Consider A and B two nonempty subsets of X. Define δx(A) = sup{d(x,y) : y ∈ A} for all x ∈ X, δ(A,B) = sup{δx(B) : x ∈ A}, diam(A) = δ(A,A). Throughout this article, we say that the pair (A,B) satisfies a property if both A and B satisfy that property. For example, (A,B) is closed if and only if both A and B are closed. Likewise, (A,B) ⊆ (C,D) if and only if A ⊆ C and B ⊆ D. The proximal pair of (A,B) is defined as A0 = {x ∈ A: d(x,y′) = dist(A,B) for some y′ ∈ B}, © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 406 Best proximity point (pair) results via MNC in Busemann convex metric spaces B0 = {y ∈ B : d(x′,y) = dist(A,B) for some x′ ∈ A}. Proximal pairs may be empty but, in particular, if A and B are nonempty, bonded, closed and convex in a reflexive Banach space X, then (A0,B0) is a nonempty, bounded, closed and convex pair in X. We say that (A,B) is proximinal provided that A = A0 and B = B0. In this paper we will mainly work with geodesic spaces. Let x,y ∈ X. A geodesic path from x to y is a mapping c : [0, l] ⊆ R → X such that c(0) = x, c(l) = y, and d(c(t),c(t′)) = |t− t′| for all t,t′ ∈ [0, l]. The image of the mapping c forms a geodesic segment which joins x and y and will be denoted by [x,y] whenever it is unique. (X,d) is said to be a (uniquely) geodesic space if every two points in X can be joined by a (unique) geodesic path. A point p ∈ X belongs to a geodesic segment [x,y] if and only if there exists t ∈ [0, 1] such that d(x,p) = td(x,y) and d(y,p) = (1 − t)d(x,y) and for convenience we write p = (1 − t)x⊕ ty. In this situation p = c(tl), where c : [0, l] → X is a geodesic path from x to y. A subset E of a geodesic space X is convex if it contains any geodesic segment joining each two points in E, that is, for any x,y ∈ E we have [x,y] ⊆ E. The (closed) convex hull of a subset E of a geodesic space X is the smallest (closed) convex set containing the set E which is denoted by con(E) and con(E), respectively. Lemma 2.1 ([5]). Let E be a nonempty subset of a geodesic space X. Let G1(E) denote the union of all geodesic segments with endpoints in E. Recursively, for n ≥ 2 put Gn(E) = G1(Gn−1(E)). Then con(E) = ∞⋃ n=1 Gn(E). It is remarkable to note that in a Busemann convex space X the closure of con(E) is convex and so, coincides with con(A) (see [10]). A metric d: X×X → R of a uniquely geodesic space (X,d) is called convex if for any x ∈ X and every geodesic path c : [0, l] → X we have d(x,c(tl)) ≤ (1 − t)d(x,c(0)) + td(x,c(l)), ∀t ∈ [0, 1]. Furthermore, (X,d) is said to be Busemann convex ([6]) if for any two geodesics c1 : [0, l1] → X and c2 : [0, l2] → X one has d ( c1(tl1),c2(tl2) ) ≤ (1 − t)d(c1(0),c2(0)) + td(c1(l1),c2(l2)) ∀t ∈ [0, 1]. Equivalently, a geodesic metric space (X,d) is convex in the sense of Busemann if d ( (1 − t)x⊕ ty, (1 − t)z ⊕ tw ) ≤ (1 − t)d(x,z) + td(y,w), for all x,y,z,w ∈ X and t ∈ [0, 1]. It is well-known that Busemann convex spaces are uniquely geodesic and with convex metric. A very important class of Busemann convex spaces are CAT(0) spaces, that is, metric spaces of non- positive curvature in the sense of Gromov (see [5, 6] for a detailed discussion on CAT(0) spaces). © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 407 M. Gabeleh and P. R. Patle In the sequel we say that a geodesic metric space (X,d) is strictly convex ([3]) if for every r > 0, a,x and y ∈ X with d(x,a) ≤ r, d(y,a) ≤ r and x 6= y, it is the case that d(a,p) < r, where p ∈ [x,y] − {x,y}. We mention that Busemann convex spaces are strictly convex with convex metric ([9]). In the next section we will also work with reflexive geodesic spaces which is a generalization of the notion of reflexivity from Banach to geodesic spaces. A geodesic space X is said to be reflexive if for every decreasing chain {Cα}⊆ X with α ∈ I such that Cα is nonempty, bounded, closed and convex for all α ∈ I we have that ⋂ α∈I Cα 6= ∅. It was announced in [8] that a reflexive and Busemann convex space is complete. The following property of reflexive and Busemann convex spaces plays an important role in our coming discussions. Proposition 2.2 ([11, Proposition 3.1]). If (A,B) is a nonempty, closed and convex pair in a reflexive and Busemann convex space X such that B is bounded, then (A0,B0) is nonempty, bounded, closed and convex. We are now ready to state a main existence result of [11]. Theorem 2.3 ([11, Theorem 3.3, Theorem 3.4]). Let (A,B) be a nonempty, compact and convex pair in a Busemann convex space X. Then every cyclic (noncyclic) relatively nonexpansive mapping defined on A∪B has a best prox- imity point (pair). We mention that the proof of Theorem 2.3 is based on the fact that any compact and convex pair in a geodesic space with convex metric has a geometric notion, called proximal normal structure (see Proposition 3.10 of [11]). To state an extended version of Theorem 2.3 we recall the following concept. Definition 2.4 ([12]). Let (A,B) be a nonempty and bounded pair in a metric space (X,d) and T : A∪B → A∪B be a cyclic (noncyclic) mapping. We say that T is compact whenever the pair (T(A),T(B)) is compact. Theorem 2.5 ([13, Theorem 3.2 and Theorem 3.4]). Let (A,B) be a nonempty, closed and convex pair in a reflexive and Busemann convex space (X,d) such that B is bounded and let closed convex hulls of finite sets be compact. Assume that T : A∪B → A∪B is a cyclic (noncyclic) relatively nonexpansive mapping. If T is compact, then T has a best proximity point (pair). We finish this section by stating the following auxiliary lemmas. Lemma 2.6 ([13, Lemma 3.7]). Let (A,B) be a nonempty, closed and convex pair in a reflexive and Busemann convex space (X,d). Assume that (E,F) ⊆ (A,B) is a nonempty and proximinal pair with dist(E,F) = dist(A,B). Then the pair (con(E), con(F)) is proximinal with dist(con(E), con(F)) = dist(A,B). Lemma 2.7 ([13, Lemma 3.8]). Let (A,B) be a nonempty, closed and convex pair in a reflexive and Busemann convex space (X,d) such that B is bounded. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 408 Best proximity point (pair) results via MNC in Busemann convex metric spaces Let T : A∪B → A∪B be a cyclic relatively nonexpansive mapping. Suppose U0 := A0 and V0 := B0 and for all n ∈ N define Un = con(T(Un−1)), Vn = con(T(Vn−1)). Then {(U2n,V2n)} is a descending sequence of nonempty, bounded, closed, con- vex and proximinal pairs which are T -invariant, that is, T is cyclic on U2n∪V2n for all n ∈ N. Moreover, Un+2 ⊆Vn+1 ⊆Un ⊆Vn−1, dist(U2n,V2n) = dist(A,B), ∀n ∈ N. 2.2. Measure of noncompactness on metric spaces. In this section we recall some basic notions of measure of noncompactness which will be used in introducing new classes of condensing operators. Let (X,d) be a metric space. By Σ we denote the collection of all bounded subsets of X. Definition 2.8. A function µ : Σ → [0,∞) is called a measure of noncompact- ness (MNC for brief) if it satisfies the following conditions: (i) µ(A) = 0 iff A is relatively compact, (ii) µ(A) = µ(A) for all A ∈ Σ, (iii) µ(A∪B) = max{µ(A),µ(B)} for all A,B ∈ Σ. We can survey the following useful properties of an MNC, easily; (1) If A ⊆ B, then µ(A) ≤ µ(B), (2) µ(A∩B) ≤ min{µ(A),µ(B)} for all A,B ∈ Σ, (3) If A is a finite set, then µ(A) = 0, (4) If {An} is a decreasing sequence of nonempty, bounded and closed subsets of X such that limn→∞µ(An) = 0, then A∞ := ∩n≥1An is nonempty and compact. In the case that (X,d) is a geodesic metric space, we say µ is invariant w.r.t. the convex hull, whenever µ(con(A)) = µ(A), ∀A ∈ Σ. From now on, we assume that the considered MNC is invariant w.r.t. convex hulls. Two well-known MNSs are due to Kuratowski and Hausdorff which are denoted by α and χ, respectively and define as below: α(A) = inf{ε > 0 | A ⊆∪nj=1Ej : Ej ∈ Σ, diam(Ej) ≤ ε, ∀ 1 ≤ j ≤ n < ∞}, χ(A) = inf{ε > 0 | A ⊆∪nj=1B(xj; rj) : xj ∈ X, rj ≤ ε, ∀ 1 ≤ j ≤ n < ∞}, for all A ∈ Σ (see [2] for more details). 3. Best proximity points (pairs) We begin our investigations by recalling the following class of functions which will be considered as control functions to introduce a new class of cyclic (non- cyclic) condensing operators. Let Ψ denote the class of functions ψ : [0,∞) → [0,∞) such that limn→∞ψn(t) = 0, for every t > 0, where ψn denotes nth iteration of ψ. It is evident that for every nondecreasing ψ ∈ Ψ, for each t > 0, ψ(t) < t. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 409 M. Gabeleh and P. R. Patle Let α : X ×X → [0, +∞) be a mapping. T : X → X is called α-admissible ([23]) if for every x,y ∈ X, α(x,y) ≥ 1 =⇒ α(Tx,Ty) ≥ 1. 3.1. Results for α-ψ-condensing operator. Let us introduce the following new class of cyclic (noncyclic) mappings. Definition 3.1. Let (A,B) be a nonempty and convex pair in a Banach space X and µ an MNC on X. A mapping T : A∪B → A∪B is said to be a cyclic (noncyclic) α − ψ-condensing operator if for any nonempty, bounded, closed, convex, proximinal and T-invariant pair (K1,K2) ⊆ (A,B) with dist(K1,K2) = dist(A,B) and x ∈ X, we have α(x,Tx)µ(T(K1) ∪T(K2)) ≤ ψ(µ(K1 ∪K2)) where ψ ∈ Ψ is a nondecreasing function and α : X ×X → [0, +∞). Example 3.2. Let X = BC(R+) be a Banach space and conisder A = {x ∈ BC(R+) : ||x|| ≤ 1} and B = {x ∈ BC(R+) : ||x|| > 1}. Let us define T : A∪B → A∪B as Tx =   x 2 , if x ∈ A; 2x− 2, if x ∈ B. Clearly T is noncyclic mapping. Let K1 = A and K2 = B. Then for µ(E) = diam(E), we have µ(T(K1) ∪T(K2)) = max{µ(T(K1)),µ(T(K2))} = max { sup { ‖Tx(t) −Ty(t)‖ : x(t),y(t) ∈ K1 } , sup { ‖Tx(t) −Ty(t)‖ : x(t),y(t) ∈ K2 }} = max { sup { ‖ x(t) 2 − y(t) 2 ‖ : x(t),y(t) ∈ K1 } , sup { ‖2x(t) − 2 − 2y(t) + 2‖ : x(t),y(t) ∈ K2 }} = max { sup {1 2 ‖x(t) −y(t)‖ : x(t),y(t) ∈ K1 } , sup { 2‖x(t) −y(t)‖ : x(t),y(t) ∈ K2 }} =2µ(T(K2)) = 2µ(K1 ∪K2) which shows that T is not noncyclic condensing mapping. But, T is noncyclic α-ψ-condensing, indeed if we define α : X ×X → X as α(x,Tx) = { 1, if x ∈ A; 0, otherwise. Then α(x,Tx)µ(T(K1) ∪T(K2)) ≤ ψ(µ(K1 ∪K2)), where ψ(t) = t2, t ≥ 0. Now we are ready to state and prove our first existence result for a best proximity point. Theorem 3.3. Let (A,B) be a nonempty, closed and convex pair in a reflexive and Busemann convex space (X,d) such that B is bounded and let closed convex hulls of finite sets be compact. Assume that µ is an MNC on X and T : A∪B → © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 410 Best proximity point (pair) results via MNC in Busemann convex metric spaces A∪B is a cyclic relatively nonexpansive α-admissible α-ψ-condensing operator. Then T has a best proximity point if there exists a point x0 in X such that α(x0,Tx0) ≥ 1. Proof. Let us define two sequences: (i) {(U2n,V2n)} consisting of nonempty, bounded, closed, convex, proximinal and T-invariant pairs as defined in Lemma 2.7. (ii) Let x0 ∈ X and {xn} such that xn = Txn−1 for every n ≥ 1. Since α(x0,Tx0) ≥ 1, α-admissibility of T implies that α(x1,x2) ≥ 1. Ap- plying recursion, we get α(xn,xn+1) ≥ 1, for every n ≥ 0. We note that if for some k ∈ N we have max{µ(U2k),µ(V2k)} = 0, then T : U2k∪V2k →U2k∪V2k is a compact and cyclic relatively nonexpansive mapping and so the result follows from Theorem 2.5. Assume that max{µ(U2n),µ(V2n)} > 0 for all n ∈ N. Since T is a α−ψ-condensing operator, µ(U2n+1 ∪V2n+1) ≤ α(xn,xn+1)µ(U2n+1 ∪V2n+1) = α(xn,Txn)µ(con(T(U2n)) ∪ con(T(V2n))) ≤ α(xn,Txn)µ(T(U2n) ∪T(V2n)) ≤ ψ(µ(U2n) ∪µ(V2n)) < µ(U2n) ∪µ(V2n). Repeating this pattern we get the following inequality µ(U2n+1 ∪V2n+1) ≤ ψn(µ(U0 ∪V0)), which yields us µ(U2n+1 ∪V2n+1) → 0, as n →∞. That is, lim n→∞ max{µ(U2n),µ(V2n)} = 0. Let U∞ = ∞⋂ n=0 U2n, & V∞ = ∞⋂ n=0 V2n. By the properties of the measure of noncompactness, the pair (U∞,V∞) ⊆ (A,B) is nonempty, compact and convex with dist(U∞,V∞) = dist(A,B). On the other hand, T : U∞ ∪V∞ → U∞ ∪V∞ is a cyclic relatively nonexpansive mapping. Now Theorem 2.3 ensures that T has a best proximity point. � We now state the noncyclic version of Theorem 3.3 in order to study the existence of best proximity pairs. Theorem 3.4. Let (A,B) be a nonempty, closed and convex pair in a reflexive and Busemann convex space (X,d) such that B is bounded and let closed convex hulls of finite sets be compact. Assume that µ is an MNC on X and T : A∪B → A∪B is a noncyclic relatively nonexpansive α-admissible and α-ψ-condensing operator. Then T admits a best proximity pair if there exists x0 ∈ X such that α(x0,Tx0) ≥ 1. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 411 M. Gabeleh and P. R. Patle Proof. Consider the sequence of pairs (Un,Vn) ⊆ (A,B) for n ∈ N ∪{0} as in Lemma 2.7. It follows from the proof of Theorem 3.11 of [13] that {(Un,Vn)} is a descending sequence of nonempty, bounded, closed, convex and proximinal pairs with dist(Un,Vn) = dist(A,B) which are T-invariant. If there exists some k ∈ N for which max{µ(Uk),µ(Vk)} = 0, then from Theorem 2.5 the result will be concluded. Now let x0 ∈ X. By α-admissibility of T and α(x0,Tx0) ≥ 1, we get α(xn,xn+1) ≥ 1. Suppose that max{µ(Un),µ(Vn)} > 0 for all n ∈ N∪{0}. By a discussion as in the proof of Theorem 3.3, we can see that limn→∞ max{µ(Un), µ(Vn)} = 0. Now, if we set U∞ = ∞⋂ n=0 Un, & V∞ = ∞⋂ n=0 Vn, then (U∞,V∞) is a nonempty, compact and convex pair with dist(U∞,V∞) = dist(A,B). Hence, the existence of a best proximity pair for the mapping T is deduced from Theorem 2.3. � 3.2. Results for β-ψ-condensing operator. Now we recall the concept of β-admissibility which is introduced by Rehman et al. in [19], with a slight modification as follows. Definition 3.5. Let β : 2X → [0, +∞). A mapping T : X → X is called β-admissible if for every M1,M2 ∈ 2X, we have β(M1 ∪ M2) ≥ 1 =⇒ β(con(T(M1) ∪T(M2))) ≥ 1. We now introduce a new class of cyclic (noncyclic) operators in the following definition. Definition 3.6. Let (A,B) be a nonempty and convex pair in a Banach space X and µ an MNC on X. A mapping T : A∪B → A∪B is said to be a cyclic (noncyclic) β-ψ-condensing operator if for any nonempty, bounded, closed, con- vex, proximinal and T-invariant pair (K1,K2) ⊆ (A,B) with dist(K1,K2) = dist(A,B), we have β(K1 ∪K2)µ(T(K1) ∪T(K2)) ≤ ψ(µ(K1 ∪K2)) where ψ ∈ Ψ is a nondecreasing function and β : 2X → [0, +∞). Example 3.7. Let X = BC(R+) be a Banach space and conisder A = {x ∈ BC(R+) : ||x|| ≤ 1} and B = {x ∈ BC(R+) : ||x|| > 1}. Let us define T : A∪B → A∪B as Tx =   x 3 , if x ∈ A; 2x− 3 2 , if x ∈ B. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 412 Best proximity point (pair) results via MNC in Busemann convex metric spaces Clearly T is noncyclic mapping. Let K1 = A and K2 = B. Then for µ(E) = diam(E), we have µ(T(K1) ∪T(K2)) = max{µ(T(K1)),µ(T(K2))} = max { sup { ‖Tx(t) −Ty(t)‖ : x(t),y(t) ∈ K1 } , sup { ‖Tx(t) −Ty(t)‖ : x(t),y(t) ∈ K2 }} = max { sup { ‖ x(t) 3 − y(t) 3 ‖ : x(t),y(t) ∈ K1 } , sup { ‖2x(t) − 3 2 − 2y(t) + 3 2 ‖ : x(t),y(t) ∈ K2 }} = max { sup {1 3 ‖x(t) −y(t)‖ : x(t),y(t) ∈ K1 } , sup { 2‖x(t) −y(t)‖ : x(t),y(t) ∈ K2 }} =2µ(T(K2)) = 2µ(K1 ∪K2), which shows that T is not noncyclic condensing mapping. But, T is noncyclic β-ψ-condensing. Indeed if we define β : 2X → [0, +∞) as β(K1∪K2) = 1, then β(K1 ∪K2)µ(T(K1) ∪T(K2)) ≤ ψ(µ(K1 ∪K2)), where ψ(t) = t2, t ≥ 0. Theorem 3.8. Let (A,B) be a nonempty, closed and convex pair in a reflexive and Busemann convex space (X,d) such that B is bounded and let closed convex hulls of finite sets be compact. Assume that µ is an MNC on X and T : A∪B → A∪B is a cyclic relatively nonexpansive β-admissible, β-ψ-condensing operator. Then T has a best proximity point if there exist nonempty, bounded, closed and convex X0,Y0 ⊆ X such that β(X0 ∪Y0) ≥ 1. Proof. Let us define sequences: (i) {(U2n,V2n)} consisting of nonempty, bounded, closed, convex, proximinal and T-invariant pairs as defined in the Lemma 2.7. We note that if for some k ∈ N we have max{µ(U2k),µ(V2k)} = 0, then T : U2k∪V2k →U2k∪V2k is a compact and cyclic relatively nonexpansive mapping and so the result follows from Theorem 2.5. Assume that max{µ(U2n),µ(V2n)} > 0 for all n ∈ N. Since β(U0 ∪ V0) ≥ 1, β-admissibility of T implies that β(U1 ∪V1) ≥ 1. Applying recursion, we get β(U2n ∪U2n) ≥ 1, for every n ≥ 0. Since T is a β −ψ-condensing operator, µ(U2n+1 ∪V2n+1) ≤ β(U2n ∪U2n)µ(U2n+1 ∪V2n+1) = β(U2n ∪U2n)µ(con(T(U2n)) ∪ con(T(V2n))) ≤ β(U2n ∪U2n)µ(T(U2n) ∪T(V2n)) ≤ ψ(µ(U2n) ∪µ(V2n)). Repeating this pattern we get the following inequality µ(U2n+1 ∪V2n+1) ≤ ψn(µ(U0 ∪V0)), which yields us µ(U2n+1 ∪V2n+1) → 0, as n →∞. That is, lim n→∞ max{µ(U2n),µ(V2n)} = 0. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 413 M. Gabeleh and P. R. Patle Let U∞ = ∞⋂ n=0 U2n, & V∞ = ∞⋂ n=0 V2n. By the properties of the measure of noncompactness, the pair (U∞,V∞) ⊆ (A,B) is nonempty, compact and convex with dist(U∞,V∞) = dist(A,B). On the other hand, T : U∞ ∪V∞ → U∞ ∪V∞ is a cyclic relatively nonexpansive mapping. Now Theorem 2.3 ensures that T has a best proximity point. � We now state the noncyclic version of Theorem 3.8 in order to study the existence of best proximity pairs. Theorem 3.9. Let (A,B) be a nonempty, closed and convex pair in a reflexive and Busemann convex space (X,d) such that B is bounded and let closed convex hulls of finite sets be compact. Assume that µ is an MNC on X and T : A∪B → A∪B is a noncyclic relatively nonexpansive β-admissible and β−ψ-condensing operator. Then T admits a best proximity pair if there exist nonempty, bounded, closed and convex X0,Y0 ⊆ X such that β(X0 ∪Y0) ≥ 1. Proof. Consider the sequence of pairs (Un,Vn) ⊆ (A,B) for n ∈ N ∪{0} as in Lemma 2.7. It follows from the proof of Theorem 3.11 of [13] that {(Un,Vn)} is a descending sequence of nonempty, bounded, closed, convex and proximinal pairs with dist(Un,Vn) = dist(A,B) which are T-invariant. If there exists some k ∈ N for which max{µ(Uk),µ(Vk)} = 0, then from Theorem 2.5 the result will be concluded. Now by β-admissibility of T and β(U0,V0) ≥ 1, we get β(Un,Vn) ≥ 1. Suppose that max{µ(Un),µ(Vn)} > 0, ∀n ∈ N∪{0}. By a discussion as in the proof of Theorem 3.8, we can see that limn→∞ max{µ(Un), µ(Vn)} = 0. Now, if we set U∞ = ∞⋂ n=0 Un, & V∞ = ∞⋂ n=0 Vn, then (U∞,V∞) is a nonempty, compact and convex pair with dist(U∞,V∞) = dist(A,B). Hence, the existence of a best proximity pair for the mapping T is deduced from Theorem 2.3. � 4. Coupled best proximity point results In this section, we study the existence of coupled best proximity points in the setting of Busemann convex spaces. To this end, we recall the following concepts which first appeared in the Ph.D Thesis of the first author ([15]). Let (A,B) be a nonempty pair in a metric space (X,d) and S : (A×A) ∪ (B×B) → A∪B be a cyclic mapping, that is, S(A×A) ⊆ B and S(B×B) ⊆ A. A point (u,v) ∈ (A×A)∪(B×B) is called a coupled best proximity point for the mapping S provided that d(u,S(u,v)) = d(v,S(v,u)) = dist(A,B). © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 414 Best proximity point (pair) results via MNC in Busemann convex metric spaces We also need the following lemma. Lemma 4.1 ([1]). Suppose that µ1,µ2, · · · ,µn are measures of noncompactness on the metric spaces X1,X2, · · · ,Xn, respectively. Moreover, assume that the function Θ : [0,∞)n → [0,∞) is convex and Θ(x1,x2, · · · ,xn) = 0 if and only if xj = 0 for all j = 1, 2, · · · ,n. Then µ(E) = Θ ( µ1(E1),µ2(E2), · · · ,µn(En) ) , defines a measure of noncompactness on X1×X2×···×Xn, where Ej denotes the natural projection of E into Ej for j = 1, 2, · · · ,n. The next lemma was given in [16]. For the convenience of the reader, we include a proof of this fact. Lemma 4.2 (see [16, Lemma 4.2]). Let (A,B) be a nonempty pair in a metric space (X,d). Consider the product space X ×X with the metric d∞ ( (x1,y1), (x2,y2) ) = max{d(x1,x2),d(y1,y2)}, ∀(x1,y1), (x2,y2) ∈ X×X. Then the pair (A,B) is proximinal in X if and only if (A × A,B × B) is proximinal in X ×X. Proof. Note that dist(A × A,B × B) = dist(A,B). In fact, for any (a,a′) ∈ A×A, (b,b′) ∈ B ×B we have dist(A×A,B ×B) = inf( (a,a′),(b,b′) ) ∈(A×A)×(B×B) d∞ ( (a,a′), (b,b′) ) = inf( (a,a′),(b,b′) ) ∈(A×A)×(B×B) max{d(a,b),d(a′,b′)} = dist(A,B). Suppose (A,B) is proximinal and (a,a′) ∈ A × A, then we can find b,b′ ∈ B such that d(a,b) = d(a′,b′) = dist(A,B). Thus for an element (b,b′) ∈ B × B we have d∞ ( (a,a′), (b,b′) ) = dist(A,B) ( = dist(A × A,B × B) ) , that is, (A×A)0 = A×A. By a similar manner, (B ×B)0 = B ×B and so the pair( (A×A), (B×B) ) is proximinal in X×X. Now assume that ( (A×A), (B×B) ) is proximinal and a ∈ A, then (a,a) ∈ A × A and so there exists a point (b,b′) ∈ B × B for which d∞ ( (a,a), (b,b′) ) = dist(A,B) which deduces that d(a,b) = d(a,b′) = dist(A,B), that is, A0 = A. Equivalently, B0 = B which implies that (A,B) is proximinal and this completes the proof of lemma. � We now state our first coupled best proximity point result. Theorem 4.3. Let (A,B) be a nonempty, closed and convex pair in a reflexive and Busemann convex space (X,d) such that B is bounded and let closed convex hulls of finite sets be compact. Assume that µ is an MNC on X and S : (A×A)∪(B×B) → A∪B is a cyclic mapping satisfying following conditions. (i) Let (K1,K2) ⊆ (A,B) and (K′1,K′2) ⊆ (A,B) be nonempty, bounded, closed, convex, proximinal and S-invariant pairs with dist(K1,K2) = © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 415 M. Gabeleh and P. R. Patle dist(A,B) = dist(K′1,K ′ 2), γ : X 2 × X2 → [0, +∞) such that for any (x1,y1) ∈ X ×X we have γ((x1,y1), (x2,y2))µ((S(K1×K′1)∪S(K2×K ′ 2))) ≤ 1 2 ψ ( max{µ(K1∪K′1),µ(K2∪K ′ 2)} ) , where ψ ∈ Ψ is nondecreasing. (ii) For all (x,y), (u,v) ∈ X ×X and γ ( (x,y), (u,v) ) ≥ 1 we have γ ( (S(x,y),S(y,x)), (S(u,v),S(v,u) ) ≥ 1. (iii) Moreover, d(S(x1,x2),S(y1,y2)) ≤ d∞ ( (x1,y1), (x2,y2) ) , ∀(x1,x2) ∈ A×A, ∀(y1,y2) ∈ B ×B. (iv) Furthermore, there exists (x0,y0) ∈ X ×X such that γ ( (x0,y0), (S(x0,y0),S(y0,x0)) ) ≥ 1, γ ( (y0,x0), (S(y0,x0),S(x0,y0)) ) ≥ 1. Then S admits a coupled best proximity point. Proof. Set µ̃(E) := max{µ(E1),µ(E2)}, where Ej denotes the natural projec- tion of E into Ej for j = 1, 2. Thus by Lemma 4.1 µ̃ is an MNC on X × X. Let T : (A×A) ∪ (B ×B) → (A×A) ∪ (B ×B) defined by T(u,v) = (S(u,v),S(v,u)), ∀(u,v) ∈ (A×A) ∪ (B ×B). Note that if (u,v) ∈ A×A, then by the fact that S is cyclic, (S(u,v),S(v,u)) ∈ B×B, that is, T(A×A) ⊆ B×B. Equivalently, T(B×B) ⊆ A×A. Therefore, T is cyclic on (A×A) ∪ (B ×B). Besides, for any ((x1,x2), (y1,y2)) ∈ (A×A) × (B ×B) we have d∞ ( T(x1,x2),T(y1,y2) ) = d∞ (( S(x1,x2),S(x2,x1) ) , ( S(y1,y2),S(y2,y1) )) = max{d ( S(x1,x2),S(y1,y2) ) ,d ( S(x2,x1),S(y2,y1) ) } ≤ max{d∞ ( (x1,y1), (x2,y2) ) ,d∞ ( (x2,y2),d(x1,y1) ) } = d∞ ( (x1,x2), (y1,y2) ) . Hence, T is relatively nonexpansive. Let us now define α : X2 ×X2 → [0, +∞) as follows: α((x1,y1), (x2,y2)) = min{γ((x1,y1), (x2,y2)),γ((y1,x1), (y2,x2))}. By our hypothesis (ii), it is clear that whenever α((x1,y1), (x2,y2)) ≥ 1, we have α(T(x1,y1),T(x2,y2)) ≥ 1, which shows that T is α-admissible. Also by hy- pothesis (iv) it is clear that there exists (x0,y0) ∈ X×X such that α((x0,y0), (y0,x0)) ≥ 1. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 416 Best proximity point (pair) results via MNC in Busemann convex metric spaces Moreover, we have α((x1,y1), (x2,y2))µ̃(T(K1 ×K′1) ∪T(K2 ×K ′ 2)) = α((x1,y1), (x2,y2)) max{µ̃ ( T(K1 ×K′1) ) , µ̃ ( T(K2 ×K′2) ) } = α((x1,y1), (x2,y2)) max{µ̃ ( S(K1 ×K′1) ×S(K ′ 1 ×K1) ) , µ̃ ( S(K2 ×K′2) ×S(K ′ 2 ×K2) ) } = α((x1,y1), (x2,y2)) max { max{µ ( S(K1 ×K′1) ) ,µ ( S(K′1 ×K1) ) }, max{µ ( S(K2 ×K′2) ) ,µ ( S(K′2 ×K2) ) } } = α((x1,y1), (x2,y2)) max { max{µ ( S(K1 ×K′1) ) ,µ ( S(K2 ×K′2) ) }, max{µ ( S(K′1 ×K1) ) ,µ ( S(K′2 ×K2) ) } } = α((x1,y1), (x2,y2)) max { µ ( S(K1 ×K′1) ∪S(K2 ×K ′ 2) ) ,µ ( S(K′1 ×K1) ∪S(K ′ 2 ×K2) )} ≤ α((x1,y1), (x2,y2)) [ µ ( S(K1 ×K′1) ∪S(K2 ×K ′ 2) ) + µ ( S(K′1 ×K1) ∪S(K ′ 2 ×K2) )] ≤ 1 2 ψ ( max{µ(K1 ∪K′1),µ(K2 ∪K ′ 2)} ) + 1 2 ψ ( max{µ(K′1 ∪K1),µ(K ′ 2 ∪K2)} ) by hypothesis(i) = ψ ( max{µ(K1 ∪K′1),µ(K2 ∪K ′ 2)} ) = ψ ( max { max{µ(K1),µ(K′1)}, max{µ(K2),µ(K ′ 2)} }) = ψ ( max { µ̃(K1 ×K′1), µ̃(K2 ×K ′ 2) }) = ψ ( µ̃ ( (K1 ×K′1) ∪ (K2 ×K ′ 2) )) . This implies that T is a α-ψ-condensing operator. Now, Theorem 3.3 ensures that T has a best proximity point, called (p,q) ∈ (A×A) ∪ (B ×B). That is, dist(A,B) = d∞ ( (p,q),T(p,q) ) = d∞ ( (p,q), (S(p,q),S(q,p)) ) = max{d(p,S(p,q)),d(q,S(q,p))}. Thus (p,q) is a coupled best proximity point of S. � Theorem 4.4. Let (A,B) be a nonempty, closed and convex pair in a reflexive and Busemann convex space (X,d) such that B is bounded and let closed convex hulls of finite sets be compact. Assume that µ is an MNC on X and S : (A×A)∪(B×B) → A∪B is a cyclic mapping satisfying following conditions. (i) For all nonempty, bounded, closed, convex, proximinal and S-invariant pairs (K1,K2) ⊆ (A,B) and (K′1,K′2) ⊆ (A,B) with dist(K1,K2) = dist(A,B) = dist(K′1,K ′ 2) and γ : 2 X×X → [0, +∞) such that γ(K1×K2)µ((S(K1×K′1)∪S(K2×K ′ 2))) ≤ 1 2 ψ ( max{µ(K1∪K′1),µ(K2∪K ′ 2)} ) , where ψ ∈ Ψ is nondecreasing. (ii) For any (U,V ) ∈ X ×X and γ(U,V ) ≥ 1 we have γ ( con(S(U ×V ) ×S(V ×U)) ) ≥ 1. (iii) Moreover, d(S(x1,x2),S(y1,y2)) ≤ d∞ ( (x1,y1), (x2,y2) ) , ∀(x1,x2) ∈ A×A, ∀(y1,y2) ∈ B ×B, (iv) Furthermore, there exists closed and convex X0,Y0 ⊆ X such that γ ( (X0 ×Y0)) ) ≥ 1 and γ ( Y0 ×X0 ) ≥ 1. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 417 M. Gabeleh and P. R. Patle Then S admits a coupled best proximity point. Proof. Let us set µ̃(E) := max{µ(E1),µ(E2)}, where Ej denotes the natural projection of E into Ej for j = 1, 2. Thus by Lemma 4.1 µ̃ is an MNC on X ×X. Then following proof of Theorem 4.3 it is easy to show T is cyclic on (A×A) ∪ (B ×B) and T is relatively nonexpansive. Let us now define β : 2X×X → [0, +∞) as follows: β(E1 ×E2) = min{γ(E1 ×E2),γ(E2 ×E1)}. By our hypothesis (ii), it is clear that whenever β(E1 × E2) ≥ 1, we have β(con(T(E1 ×E2))) ≥ 1, which shows that T is β-admissible. Also by hypoth- esis (iv) it is clear that there exist X0,Y0 ⊆ X such that β(X0 ×Y0) ≥ 1. Moreover, we have β(K1 ×K2)µ̃(T(K1 ×K′1) ∪T(K2 ×K ′ 2)) = β(K1 ×K2) max{µ̃ ( T(K1 ×K′1) ) , µ̃ ( T(K2 ×K′2) ) } = β(K1 ×K2) max{µ̃ ( S(K1 ×K′1) ×S(K ′ 1 ×K1) ) , µ̃ ( S(K2 ×K′2) ×S(K ′ 2 ×K2) ) } = β(K1 ×K2) max { max{µ ( S(K1 ×K′1) ) ,µ ( S(K′1 ×K1) ) }, max{µ ( S(K2 ×K′2) ) ,µ ( S(K′2 ×K2) ) } } = β(K1 ×K2) max { max{µ ( S(K1 ×K′1) ) ,µ ( S(K2 ×K′2) ) }, max{µ ( S(K′1 ×K1) ) ,µ ( S(K′2 ×K2) ) } } = β(K1 ×K2) max { µ ( S(K1 ×K′1) ∪S(K2 ×K ′ 2) ) ,µ ( S(K′1 ×K1) ∪S(K ′ 2 ×K2) )} ≤ β(K1 ×K2) [ µ ( S(K1 ×K′1) ∪S(K2 ×K ′ 2) ) + µ ( S(K′1 ×K1) ∪S(K ′ 2 ×K2) )] ≤ 1 2 ψ ( max{µ(K1 ∪K′1),µ(K2 ∪K ′ 2)} ) + 1 2 ψ ( max{µ(K′1 ∪K1),µ(K ′ 2 ∪K2)} ) by hypothesis(i) = ψ ( max{µ(K1 ∪K′1),µ(K2 ∪K ′ 2)} ) = ψ ( max { max{µ(K1),µ(K′1)}, max{µ(K2),µ(K ′ 2)} }) = ψ ( max { µ̃(K1 ×K′1), µ̃(K2 ×K ′ 2) }) = ψ ( µ̃ ( (K1 ×K′1) ∪ (K2 ×K ′ 2) )) . This implies that T is a β-ψ-condensing operator. Now, Theorem 3.8 ensures that T has a best proximity point, called (p,q) ∈ (A × A) ∪ (B × B). Thus (p,q) is a coupled best proximity point of S. � Remark 4.5. It is remarkable to note that by a classical theorem due to S. Mazur, the convex hull of a compact set in a Banach space is again relatively compact. But this fact may not be true in geodesic metric spaces in general (see [4] for more information about the geodesic spaces which satisfies the Mazur’s theorem). Therefore, we should consider the compactness assumption of closed convex hulls of finite sets in our main existence theorems. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 418 Best proximity point (pair) results via MNC in Busemann convex metric spaces Remark 4.6. It is worth noticing that the class of reflexive and Busemann convex spaces contains the class of reflexive and strictly convex Banach spaces as a subclass. For example consider X = R2 with radial metric defined with d ( (x1,y1), (x2,y2) ) =  ρ ( (x1,y1), (x2,y2) ) ; if (0, 0), (x1,y1), (x2,y2) are colinear, ρ ( (x1,y1), (0, 0) ) + ρ ( (x2,y2), (0, 0) ) ; otherwise, where ρ denotes the usual Euclidean metric on R2. Then (X,d) is a complete R- tree and so is a reflexive and Busemann convex space (see [8] for more details). Note that the radial metric does not induced with any norm. 5. Application This section is dedicated to prove a result which shows the existence of optimum solutions of a system of second order differential equation with two initial conditions. Let τ,γ ∈ R+, I = [0,τ] and (E,‖.‖) be a Banach space. Let B1 = B(α0,γ), B2 = B(β0,γ) where α0,β0 ∈ E. We consider the following sys- tem of second order differential equation with two initial conditions (5.1) x ′′ (s) = f(s,x(s)), x(0) = α0, x ′ (0) = α1, y ′′ (s) = g(s,y(s)), y(0) = β0, y ′ (0) = β1, where, f : I × B1 → R, g : I × B2 → R are continuous functions such that ‖f(s,x)‖≤ A1, ‖g(s,y)‖≤ A2, s ∈I and α1,β1 ∈ E. Twice integrating (5.1) and usage of initial conditions yields us (5.2) x(s) = α0 + ∫ s 0 (α1 + (s−r)f(r,x(r))dr, y(s) = β0 + ∫ s 0 (β1 + (s−r)g(r,x(r))dr. It is clear that the systems (5.1) and (5.2) are equivalent to each other. Let J ⊆ I, S = C(J ,E) be a Banach space of continuous mappings from J into E endowed with supremum norm and consider S1 = C(J ,B1) = {x : J → B1 : x ∈S, x(0) = α0, x′(0) = α1}, S2 = C(J ,B2) = {y : J → B2 : y ∈S, y(0) = β0, y′(0) = β1}. So, (S1,S2) is NBCC pair in S. Now, for every x ∈ S1 and every y ∈ S2, we have ‖x−y‖ = sup s∈J ‖x(s) −y(s)| ≥ ‖α0 −β0‖. So, dist(S1,S2) = ‖α0−β0‖. Let us define operator T : S1∪S2 →S as follows: Tx(s) =   β0 + ∫ s 0 ( β1 + (s−r)g(r,x(r)) ) dr, x ∈S1, α0 + ∫ s 0 ( α1 + (s−r)f(r,x(r)) ) dr, x ∈S2. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 419 M. Gabeleh and P. R. Patle It is clear that T is cyclic operator. It is known that w ∈ S1 ∪ S2 is an optimum solution of the system (5.2) if ‖w −Tw‖ = dist(S1 ∪S2) is satisfied. Equivalently, w is the best proximity point of the operator T . Before proving the existence of an optimum solution of system (5.2) we recall an extension of the mean values theorem for integrals, which is presented according to our notations. Theorem 5.1 ([12]). For I,J ,B1,B2,f and g as given in the discussion above with s ∈ J we have α0 + ∫ s 0 (α1 + (s−r)f(r,x(r)))dr ∈ α0 + s con ( {α1 + (s−r)f(r,x(r)) : r ∈ [0,s]} ) and β0 + ∫ s 0 (β1 + (s−r)g(r,x(r)))dr ∈ β0 + s con ( {β1 + (s−r)g(r,x(r)) : r ∈ [0,s]} ) . The following theorem shows the existence of optimum solutions for the system (5.1). Theorem 5.2. Let µ be an arbitrary MNC on S, τ(τA2 +‖β1‖) ≤ γ, τ(τA1 + ‖α1‖) ≤ γ and τ ≤ 1. The system (5.1) has an optimal solution if the following condition holds true: (1) For any bounded pair (K1,K2) ⊆ (B1,B2), there is a nondecreasing function ψ ∈ Ψ such that µ(f(J ×K1) ∪g(J ×K2)) < 1 s ψ(µ(N1 ∪N2)). (2) For each x ∈S1 and for all y ∈S2, ‖g(r,x(r)) −f(r,y(r))‖≤ 1 s2 (‖x(r) −y(r)‖−‖β0 −α0‖ + ‖β1 −α1‖s). Proof. As the system (5.1) and (5.2) are equivalent to each other, in order to show (5.1) has an optimal solution it is sufficient to show (5.2) has an optimal solution. From the above discussion it is clear that the operator T is cyclic. Our first task is to show that T(S1) is a bounded and equicontinuous subset of S2. For each x ∈S1, ‖Tx(t)‖ = ‖β0 + ∫ s 0 (β1 + (s−r)g(r,x(r))dr‖ ≤‖β0‖ + ∫ s 0 ‖β1 + (s−r)g(r,x(r)‖dr ≤‖β0‖ + τ(‖β1‖ + τA2) ≤‖β0‖ + γ. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 420 Best proximity point (pair) results via MNC in Busemann convex metric spaces Thus T(S1) is bounded. Now for s,s′ ∈ J and x ∈S1, ‖Tx(s) −Tx(s′)‖ = wwww ∫ s 0 (β1 + (s−r)g(r,x(r))dr − ∫ s′ 0 (β1 + (s−r)g(r,x(r))dr wwww ≤ ∣∣∣∣ ∫ s′ s wwβ1 + (s−r)g(r,x(r)wwdr∣∣∣∣ ≤(τA2 + ‖β1‖) ‖s−s′‖ ≤M|s−s′|, where M = τA2 + ‖β1‖, which means that T(S1) is equicontinuous. By a similar argument T(S2) is a bounded and equicontinuous subset of S1. Thus an application of the Arzela-Ascoli theorem concludes that (S1,S2) is relatively compact. Now our aim is to show that T is a relatively nonexpansive cyclic β-ψ- condensing operator. For each (x,y) ∈ S1 ×S2 with the help of assumption (2), we have ‖Tx(s) −Ty(s)‖ = wwwwβ0 + ∫ s 0 (β1 + (s−r)g(r,x(r))dr − [α0 + ∫ s 0 (α1 + (s−r)f(r,x(r))dr] wwww ≤‖β0 −α0‖ + wwww ∫ s 0 [ (β1 −α1) + (s−r)(g(r,x(r)) −f(r,x(r)) ] dr wwww ≤‖β0 −α0‖ + ‖β1 −α1‖ s + wwwws ∫ s 0 ( g(r,x(r)) −f(r,x(r) ) dr wwww ≤‖β0 −α0‖ + ‖β1 −α1‖ s + (‖x(s) −y(s)‖−‖β0 −α0‖−‖β1 −α1‖ s) =‖x(s) −y(s)‖. This means that T is relatively nonexpansive. In order to show that T is cyclic α-ψ-condensing, suppose that the pair (K1,K2) ⊆ ( S1,S2 ) is NBCC, proximinal, T-invariant and dist(K1,K2) = dist(S1,S2)(= ‖α0 − β0‖). Now © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 421 M. Gabeleh and P. R. Patle using Theorem 5.1 and assumption (1) we have µ(T(K1) ∪T(K2)) = max{µ(T(K1)),µ(T(K2))} = max { sup s∈J { µ ( {Tx(s) : x ∈ K1} )} , sup s∈J { µ ( {Ty(s) : y ∈ K2} )}} = max { sup s∈J { µ ({ β0 + ∫ s 0 (β1 + (s−r)g(r,x(r)))dr : x ∈ K1 })} , sup s∈J { µ ({ α0 + ∫ s 0 (α1 + (s−r)f(r,x(r)))dr : x ∈ K1 })}} = max { sup s∈J { µ ({ β0 + s con ( {β1 + (s−r)g(r,x(r)) : r ∈ [0,s]} )} , sup s∈J { µ ({ α0 + s con ( {α1 + (s−r)f(r,x(r)) : r ∈ [0,s]} )}} = max { µ ({ β0 + s ( β1 + con ( {g(r,x(r)) : r ∈ [0,s]} )}) , µ ({ α0 + s ( α1 + con ( {f(r,x(r)) : r ∈ [0,s]} )})} ≤max { sµ ( {g(J ×K1)} ) ,sµ ( {f(J ×K2)} )} =sµ ( {f(J ×K1) ∪g(J ×K2)} ) ≤ s ψ(µ(K1 ∪K2)) s . Thus we get µ(T(K1 ∪T(K2))) ≤ ψ(µ(K1 ∪K2). Furthermore, we define the function β : 2E → [0, +∞) as follows: β(K1∪K2) = 1. The latter implies that β(K1 ∪K2)µ(T(K1 ∪T(K2))) ≤ ψ(µ(K1 ∪K2)). Thus necessary requirements of Theorem 3.4 are satisfied. So the operator T has a best proximity point and hence the system (5.1) has an optimal solution. � Remark 5.3. We know that, the Banach space C(J,X) in Theorem 5.2 is not reflexive and the reflexivity condition in Theorem 3.4 is essential. In fact, the reflexivity condition in Theorem 3.4 was used to prove that the proximal pair (A0,B0) is nonempty. But in Theorem 5.2 the proximal pair (S1,S2) is nonempty, automatically because of the fact that (α0,β0) ∈S1 ×S2. As an application of Theorem 5.2, we conclude the following existence result of a solution for a system of second order differential equations satisfying the same two initial conditions. Corollary 5.4. Under the notations defined above in this section and the as- sumptions of Theorem 5.2 if α0 = β0 = x0 and α1 = β1 = x1 , then the system © AGT, UPV, 2022 Appl. Gen. 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