@ Appl. Gen. Topol. 21, no. 2 (2020), 331-347 doi:10.4995/agt.2020.13926 c© AGT, UPV, 2020 Weak proximal normal structure and coincidence quasi-best proximity points Farhad Fouladi a, Ali Abkar a and Erdal Karapınar b,c a Department of Pure Mathemathics, Faculty of Science, Imam Khomeini International University, Qazvin 34149, Iran (fa folade@yahoo.com; abkar@sci.ikiu.ac.ir) b ETSI Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Viet- nam (erdalkarapinar@tdmu.edu.vn, erdalkarapinar@yahoo.com) c Department of Mathematics, Çankaya University, 06790, Etimesgut, Ankara, Turkey (erdal.karapinar@cankaya.edu.tr) Communicated by S. Romaguera Abstract We introduce the notion of pointwise cyclic-noncyclic relatively non- expansive pairs involving orbits. We study the best proximity point problem for this class of mappings. We also study the same problem for the class of pointwise noncyclic-noncyclic relatively nonexpansive pairs involving orbits. Finally, under the assumption of weak proximal normal structure, we prove a coincidence quasi-best proximity point theorem for pointwise cyclic-noncyclic relatively nonexpansive pairs in- volving orbits. Examples are provided to illustrate the observed results. 2010 MSC: 47H09; 46B20;46T99. Keywords: pointwise cyclic-noncyclic pairs; weak proximal normal struc- ture; coincidence quasi-best proximity point. 1. Introduction Let A, B be nonempty subsets of Banach space X. A mapping T : A ∪ B → A ∪ B is said to be cyclic provided that T(A) ⊆ B and T(B) ⊆ A. On the other hand, a mapping S : A ∪ B → A ∪ B is said to be noncyclic if S(A) ⊆ A and S(B) ⊆ B. Received 26 June 2020 – Accepted 15 July 2020 http://dx.doi.org/10.4995/agt.2020.13926 F. Fouladi, A. Abkar and E. Karapınar For a cyclic mapping T : A ∪ B → A ∪ B, a point p ∈ A ∪ B is said to be a best proximity point provided that d(p, Tp) = dist(A, B). Furthermore, we say that a pair (A, B) of subsets in a Banach space satisfies a property if each of the sets A and B has that property. Similarly, the pair (A, B) is called convex if both A and B are convex; moreover we write (A, B) ⊆ (E, F) ⇔ A ⊆ E, B ⊆ F. In addition, we will use the following notations: δ(A, B) = sup{‖x − y‖ : x ∈ A, y ∈ B}; δ(x, B) = sup{‖x − y‖ : y ∈ B}. For a nonempty, bounded and convex subset F of a Banach space X, we write rx(F) = sup{‖x − y‖ : y ∈ F}; r(F) = inf{rx(F) : x ∈ F}; Fc = {x ∈ F : rx(F) = r(F)}. In 2017, M. Gabeleh introduced the notion of a pointwise cyclic relatively nonexpansive mapping involving orbits, and proved a theorem on the existence of best proximity points. Definition 1.1 ([11]). Let (A, B) be a nonempty pair of subsets of a Banach space X. A mapping T : A∪B → A∪B is said to be pointwise cyclic relatively nonexpansive involving orbits if T is cyclic and for any (x, y) ∈ A × B, if ‖x − y‖ = dist(A, B), then ‖Tx − Ty‖ = dist(A, B), and otherwise, there exists a function α : A × B → [0, 1] such that ‖Tx − Ty‖ ≤ α(x, y)‖x − y‖ + (1 − α(x, y)) min{δx[O2(y; ∞)], δy[O2(x; ∞)]}, where, for any (x, y) ∈ A × B δx[O2(y; ∞)] = sup n∈N ‖x − T 2ny‖, δy[O2(x; ∞)] = sup n∈N ‖T 2nx − y‖. Note that, if A = B, then we say that T is a pointwise nonexpansive mapping involving orbits. In [12], M. Gabeleh, O. Olela Otafudu, and N. Shahzad considered a pair of mappings T and S. According to [12], for a nonempty pair of subsets (A, B) in a metric space (X, d), and a cyclic-noncyclic pair (T ; S) on A∪B (that is, T : A∪B → A∪B is cyclic and S : A∪B → A∪B is noncyclic); they called a point p ∈ A ∪ B a coincidence best proximity point for (T ; S) if d(Sp, Tp) = dist(A, B). Note that if S = I, the identity map on A ∪ B, then p ∈ A ∪ B is a best proximity point for T . c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 332 Weak proximal normal structure In 2019, A. Abkar and M. Norouzian introduced the concept of coincidence quasi-best proximity point and proved the existence of such points for quasi- cyclic-noncyclic contraction pairs. We remark that the coincidence quasi-best proximity point theory is more general and includes both the best proximity point theory and the coincidence best proximity point theory. Definition 1.2 ([2]). Let (A, B) be a nonempty pair of subsets of a metric space (X, d) and T, S : X → X be a quasi-cyclic-noncyclic pair on A ∪ B; that is, T(A) ⊆ S(B) and T(B) ⊆ S(A). A point p ∈ A ∪ B is said to be a coincidence quasi-best proximity point for (T ; S) if d(Sp, Tp) = dist(S(A), S(B)). In case that S = I, the point p reduces to a best proximity point for T . In this article, we will focus on the coincidence quasi-best proximity point problem for pointwise cyclic-noncyclic and noncyclic-noncyclic relatively non- expansive pairs. To do this, we need to recall some definitions and theorems. We begin with the following definition which is a modification of a concept in [8]. Definition 1.3. Let (A, B) be a nonempty pair of subsets of a Banach space X and S : A ∪ B → A ∪ B be a noncyclic mapping on A ∪ B. A convex pair (S(A), S(B)) is called a proximal pair if for each (a1, b1) ∈ A × B, there exists (a2, b2) ∈ A × B such that for each i, j ∈ {1, 2} with i ∕= j we have ‖Sai − Sbj‖ = dist(S(A), S(B)). Given (A, B) a pair of nonempty subsets of a Banach space X, the associated proximal pair of (S(A), S(B)) is the pair (S(As0), S(B s 0)) given by As0 := {a ∈ A : ‖Sa − Sb‖ = dist(S(A), S(B)) for some b ∈ B}, Bs0 := {b ∈ B : ‖Sa − Sb‖ = dist(S(A), S(B)) for some a ∈ A}, In fact, if the pair (S(A), S(B)) is nonempty, weakly compact and convex, then its associated pair (S(As0), S(B s 0)) is also nonempty, weakly compact and convex. Furthermore, we have dist(S(As0), S(B s 0)) = dist(S(A), S(B)). The proof of the above statements goes in the same lines as in the case for the pair (A, B); see for instance [21]. Here’s a definition we derive from [8] and we’ve made some changes to meet our needs. Definition 1.4. Let (K1, K2) be a nonempty pair of subsets of a Banach space X and S : K1 ∪K2 → K1 ∪K2 be a noncyclic mapping on K1 ∪K2. We say that a convex pair (S(K1), S(K2)) has proximal normal structure (PNS) if for any closed, bounded, convex and proximal pair (S(H1), S(H2)) ⊆ (S(K1), S(K2)) which dist(S(H1), S(H2)) = dist(S(K1), S(K2)), δ(S(H1), S(H2)) > dist(S(H1), S(H2)), c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 333 F. Fouladi, A. Abkar and E. Karapınar there exists (x, y) ∈ H1 × H2 such that δ(Sx, S(H2)) < δ(S(H1), S(H2)), δ(Sy, S(H1)) < δ(S(H1), S(H2)). Note that the pair (K, K) has proximal normal structure if and only if K has normal structure in the sense of Brodskii and Milman (see [4] and [20]). Theorem 1.5 ([8]). Every bounded, closed and convex pair in a uniformly convex Banach space X has proximal normal structure. The following definition is a modification of what already appeared in [11]. Definition 1.6. Let (K1, K2) be a nonempty pair of subsets of a Banach space X and S : K1 ∪ K2 → K1 ∪ K2 be a noncyclic mapping on K1 ∪ K2. We say that a convex pair (S(K1), S(K2)) has weak proximal normal structure (WPNS) if for each nonempty, weakly compact and convex proximal pair (S(H1), S(H2)) ⊆ (S(K1), S(K2)) for which dist(S(H1), S(H2)) = dist(S(K1), S(K2)), δ(S(H1), S(H2)) > dist(S(H1), S(H2)), there exists (x, y) ∈ H1 × H2 such that δ(Sx, S(H2)) < δ(S(H1), S(H2)), δ(Sy, S(H1)) < δ(S(H1), S(H2)). In this article, we intend to generalize some results of [8] and [11]. Our results have the following advantages: First, we introduce the class of the pointwise cyclic-noncyclic and noncyclic-noncyclic relatively nonexpansive pairs involving orbits, that in particular, includes the class of pointwise cyclic-noncyclic and noncyclic-noncyclic relatively nonexpansive mappings involving orbits. Second, we consider a pair of mappings while the previous articles are concerned with one single mapping, and finally, we study the coincidence quasi-best proximity point problem, which in particular, includes the best proximity point problem as a special case. 2. Cyclic-noncyclic pairs We begin this section by introducing the new concept of a pointwise cyclic-noncyclic relatively nonexpansive pair involving orbits. Definition 2.1. Assume that (A, B) is a nonempty pair of subsets of a Banach space X and T, S : A ∪ B → A ∪ B are two mappings. A pair (T ; S) is said to be a pointwise cyclic-noncyclic relatively nonexpansive pair involving orbits if (T ; S) is a cyclic-noncyclic pair and for any (x, y) ∈ A × B, if ‖x − y‖ = dist(S(A), S(B)), then ‖Tx − Ty‖ = dist(S(A), S(B)), ‖Sx − Sy‖ = dist(S(A), S(B)) and otherwise, there exists a function α : A × B → [0, 1] such that ‖Tx−Ty‖ ≤ α(x, y)‖Sx−Sy‖+(1−α(x, y)) max{δx[O2(y; ∞)], δy[O2(x; ∞)]}, where, for any (x, y) ∈ A × B δx[O2(y; ∞)] = sup n∈N ‖x − T 2ny‖, δy[O2(x; ∞)] = sup n∈N ‖T 2nx − y‖. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 334 Weak proximal normal structure We note that if S = I, then the class of pointwise cyclic-noncyclic relatively nonexpansive pairs involving orbits reduces to the class of pointwise cyclic relatively nonexpansive mappings involving orbits introduced in [11]. Definition 2.2 ([20]). We say that a Banach space X has the property (C) if every bounded decreasing sequence of nonempty, closed and convex subsets of X have a nonempty intersection. For C ⊆ X, we denote the diameter of C by δ(C). A point x ∈ C is a diametral point of C provided that sup{‖x − y‖ : y ∈ C} = δ(C). A convex set K ⊆ X is said to have normal structure if for each bounded convex subset H of K which contains at least two points, there is some point x ∈ H which is not a diametral point of H. Lemma 2.3 ([20]). Assume that X is a Banach space with the property (C), then Fc is nonempty, closed and convex. Lemma 2.4 ([20]). Assume that F is a closed and convex subset of a Banach space X which contains at least two points. If F has normal structure, then δ(Fc) < δ(F). Theorem 2.5. Assume that K is a nonempty, bounded, closed and convex subset of a Banach space X with property (C). Suppose that K has normal structure. Let (T, S) be a pointwise cyclic-noncyclic relatively nonexpansive pair involving orbits on K. Then there exists a point p ∈ K such that ‖Tp−Sp‖ = 0. Proof. Suppose Γ denotes the collection of all nonempty, closed and convex subsets of K such that (T, S) is a pointwise cyclic-noncyclic relatively non- expansive pair involving orbits on K. By Zorn’s Lemma, Γ has a minimal member, say F . We complete the proof by verifying that F consists of a single point. Assume that x ∈ Fc. In this case, for any y ∈ Fc we have ‖Sx − y‖ ≤ sup{‖z − y‖ : z ∈ F} = ry(F) = r(F), therefore, sup{‖Sx − y‖ : x ∈ Fc} ≤ r(F). Then, rSx(F) = sup{‖Sx − y‖ : y ∈ F} ≤ sup{‖Sx − y‖ : x ∈ Fc, y ∈ F} ≤ sup{r(F), y ∈ F} = r(F). c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 335 F. Fouladi, A. Abkar and E. Karapınar Then, for any x ∈ Fc we have rSx(F) = r(F); that is, S : Fc → Fc. Moreover, for any x, y ∈ Fc we have ‖Sx−Sy‖ ≤ r(F). On other hand, for any x, y ∈ Fc, δx[O2(y; ∞)] = sup n∈N ‖x − T 2ny‖ ≤ sup{‖x − z‖ : z ∈ F} = rx(F) = r(F). Similarly, for any x, y ∈ Fc we have δy[O2(x; ∞)] ≤ r(F). In particular, for each x, y ∈ Fc, ‖Tx − Ty‖ ≤ α(x, y)‖Sx − Sy‖ + (1 − α(x, y)) max{δx[O2(y; ∞)], δy[O2(x; ∞)]} ≤ α(x, y)r(F) + (1 − α(x, y))r(F) = r(F); that is, rT x(F) = r(F). Then, T : Fc → Fc. By Lemma 2.3, we have Fc ∈ Γ. If δ(F) > 0, then by Lemma 2.4, Fc is properly contained in F which contradicts the minimality of F . Hence δ(F) = 0 and F consists of a single point; this is, there exists a point p ∈ K such that Tp = p and Sp = p. So, there exists a p ∈ K such that ‖Tp − p‖ = 0. □ Theorem 2.6. Assume that (A, B) is a nonempty pair of subsets in a Banach space X with PNS. Let T, S : A ∪ B → A ∪ B be a pointwise cyclic-noncyclic relatively nonexpansive pair involving orbits, and such that T(A) ⊆ S(B) and T(B) ⊆ S(A). Suppose that (S(A), S(B)) is a weakly compact and convex pair of subsets in X. Then there exists (x, y) ∈ A × B such that for p ∈ {x, y} we have ‖Tp − Sp‖ = dist(S(A), S(B)). Proof. The result follows from Theorem 2.5 if dist(S(A), S(B)) = 0, so we assume that dist(S(A), S(B)) > 0. Let (S(As0), S(B s 0)) be the associated prox- imal pair of (S(A), S(B)). We have already observed that S(As0) and S(B s 0) are nonempty, weakly compact and convex, moreover dist(S(As0), S(B s 0)) = dist(S(A), S(B)). Assume that x ∈ As0, then there exists y ∈ Bs0 such that ‖Sx − Sy‖ = dist(S(A), S(B)). On other hand, (T ; S) is a pointwise cyclic-noncyclic rel- atively nonexpansive pair involving orbits. Thus, ‖T(Sx) − T(Sy)‖ = dist(S(A), S(B)), ‖S(Sx) − S(Sy)‖ = dist(S(A), S(B)). This implies that ‖S(Sx) − S(Sy)‖ = dist(S(As0), S(B s 0)), and ‖T(Sx) − T(Sy)‖ = dist(S(As0), S(B s 0)). Therefore, we have T(Sx) ∈ S(Bs0), T(Sy) ∈ S(A s 0); c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 336 Weak proximal normal structure that is, T(S(As0)) ⊆ S(B s 0), T(S(B s 0)) ⊆ S(A s 0). Similarly, S(S(As0)) ⊆ S(A s 0), S(S(B s 0)) ⊆ S(B s 0). So, for each x ∈ As0 and y ∈ Bs0 we have ‖T(Sx) − T(Sy)‖ = dist(S(As0), S(B s 0)), and ‖S(Sx) − S(Sy)‖ = dist(S(As0), S(B s 0)). Clearly (S(As0), S(B s 0)) also has proximal normal structure. Now, assume that Ω denotes the collection of all nonempty subsets S(F) of S(As0) ∪ S(Bs0) for which S(F) ∩ S(As0) and S(F) ∩ S(Bs0) are nonempty, closed, convex, and such that T(S(F) ∩ S(As0)) ⊆ S(F) ∩ S(B s 0), T(S(F) ∩ S(B s 0)) ⊆ S(F) ∩ S(A s 0), and S(S(F) ∩ S(As0)) ⊆ S(F) ∩ S(A s 0), S(S(F) ∩ S(B s 0)) ⊆ S(F) ∩ S(B s 0), and so dist(S(F) ∩ S(As0), S(F) ∩ S(B s 0)) = dist(S(A), S(B)). Since, S(As0)∪S(Bs0) ∈ Ω and Ω is nonempty, we may assume that {S(Fα)}α∈Ω is a decreasing chain in Ω such that S(F0) = ∩α∈ΩS(Fα). Then S(F0) ∩ S(As0) = ∩α∈Ω(S(Fα) ∩ S(As0)), so S(F0) ∩ S(As0) is nonempty, closed and convex. Similarly, S(F0) ∩ S(Bs0) is nonempty, closed and convex. Also, T(S(F0) ∩ S(As0)) ⊆ S(F0) ∩ S(B s 0), T(S(F0) ∩ S(B s 0)) ⊆ S(F0) ∩ S(A s 0) and S(S(F0) ∩ S(As0)) ⊆ S(F0) ∩ S(A s 0), S(S(F0) ∩ S(B s 0)) ⊆ S(F0) ∩ S(B s 0). To show that S(F0) ∈ Ω we only need to verify that dist(S(F0) ∩ S(As0), S(F0) ∩ S(B s 0)) = dist(S(A), S(B)). Note that for each α ∈ J it is possible to select Sxα ∈ S(Fα) ∩ S(As0), Syα ∈ S(Fα) ∩ S(B s 0) such that ‖Sxα − Syα‖ = dist(S(A), S(B)). It is also possible to choose convergent subnets {Sxα′} and {Syα′} (with the same indices), say lim α′ Sxα′ = Sx, lim α′ Syα′ = Sy. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 337 F. Fouladi, A. Abkar and E. Karapınar Then clearly Sx ∈ S(F0) ∩ S(As0) and Sy ∈ S(F0) ∩ S(Bs0). By weak lower semicontinuity of the norm, we have ‖Sx − Sy‖ ≤ dist(S(A), S(B)); hence, dist(S(A), S(B)) ≤ dist(S(F0) ∩ S(As0), S(F0) ∩ S(B s 0)) ≤ ‖Sx − Sy‖ ≤ dist(S(A), S(B)). Therefore, dist(S(F0) ∩ S(As0), S(F0) ∩ S(B s 0)) = dist(S(A), S(B)). Since, every chain in Ω is bounded below by a member of Ω, Zorn’s Lemma implies that Ω has a minimal element, say S(K). Assume that S(K1) = S(K)∩ S(As0) and S(K2) = S(K) ∩ S(Bs0). Observe that if δ(S(K1), S(K2)) = dist(S(K1), S(K2)), then for any x ∈ S(K1), we have ‖Tx − Sx‖ = dist(S(K1), S(K2)) = dist(S(A), S(B)). Similarly, for any y ∈ S(K2), we have ‖Ty − Sy‖ = dist(S(K1), S(K2)) = dist(S(A), S(B)). Now, we assume that δ(S(K1), S(K2)) > dist(S(K1), S(K2)). We complete the proof by showing that this leads to a contradiction. Since S(K) is minimal, it follows that (S(K1), S(K2)) is a proximal pair in (S(A s 0), S(B s 0)). By the PNS property of X, there exist (x1, y1) ∈ K1 × K2 and β ∈ (0, 1) such that δ(Sx1, S(K2)) ≤ βδ(S(K1), S(K2)) and δ(Sy1, S(K1)) ≤ βδ(S(K1), S(K2)). Since, (S(K1), S(K2)) is a proximal pair, there exists (x2, y2) ∈ K1 × K2 such that for each distinct i, j ∈ {1, 2}, we have ‖Sxi − Syj‖ = dist(S(K1), S(K2)). So, for each u ∈ S(K2) we have ‖ Sx1 + Sx2 2 − u‖ ≤ ‖ Sx1 − u 2 ‖ + ‖ Sx2 − u 2 ‖ ≤ βδ(S(K1), S(K2)) 2 + δ(S(K1), S(K2)) 2 = αδ(S(K1), S(K2)), where α = 1+β 2 ∈ (0, 1). Assume that Sw1 = (Sx1+Sx2) 2 and Sw2 = (Sy1+Sy2) 2 . Then δ(Sw1, S(K2)) ≤ αδ(S(K1), S(K2)) and δ(Sw2, S(K1)) ≤ αδ(S(K1), S(K2)). c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 338 Weak proximal normal structure Since, dist(S(K1), S(K2)) ≤ ‖Sw1 − Sw2‖ = ‖ (Sx1 + Sx2) 2 − (Sy1 + Sy2) 2 ‖ ≤ 1 2 [‖Sx1 − Sy2‖ + ‖Sx2 − Sy1‖] = dist(S(K1), S(K2)), we have ‖Sw1 − Sw2‖ = dist(S(K1), S(K2)). Put S(L1) = {Sx ∈ S(K1) : δ(Sx, S(K2)) ≤ αδ(S(K1), S(K2))}, S(L2) = {Sy ∈ S(K2) : δ(Sy, S(K1)) ≤ αδ(S(K1), S(K2))}. Then for each i = 1, 2, S(Li) is a nonempty, closed and convex subset of S(Ki) and since Sw1 ∈ S(L1) and Sw2 ∈ S(L2), we have dist(S(K1), S(K2)) ≤ dist(S(L1), S(L2)) ≤ ‖Sw1 − Sw2‖ = dist(S(K1), S(K2)). Therefore, dist(S(L1), S(L2)) = dist(S(K1), S(K2)) = dist(S(A), S(B)). Now, assume that Sx ∈ S(L1) and Sy ∈ S(K2). Then Sx ∈ S(As0) and Sy ∈ S(Bs0); that is, x ∈ As0 and y ∈ Bs0. Thus, ‖T(Sx) − T(Sy)‖ = dist(S(A), S(B)) ≤ δ(Sx, S(K2)) ≤ αδ(S(K1), S(K2)). So, T(Sy) ∈ B(T(Sx); αδ(S(K1), S(K2))) ∩ S(K1); that is, T(S(K2)) ⊆ B(T(Sx); αδ(S(K1), S(K2))) ∩ S(K1) := S(K′1). Clearly S(K′1) is closed and convex. Also, if Sy ∈ S(K2) satisfies ‖Sx − Sy‖ = dist(S(A), S(B)), then ‖T(Sx) − T(Sy)‖ = dist(S(K1), S(K2)). Since, T(Sy) ∈ S(K′1), we conclude that dist(S(K′1), S(K2)) = dist(S(A), S(B)). Therefore, S(K′1) ∪ S(K2) ∈ Ω and by the minimality of S(K) we must have S(K′1) = S(K1). Hence, S(K1) ⊆ B(T(Sx); αδ(S(K1), S(K2))); that is, δ(T(Sx), S(K1)) ≤ αδ(S(K1), S(K2)) and since Sx ∈ S(L1) was arbi- trary, we obtain T(S(L1)) ⊆ S(L2). Similarly, T(S(L2)) ⊆ S(L1), S(S(L1)) ⊆ S(L1) and S(S(L2)) ⊆ S(L2). Thus, S(L1)∪S(L2) ∈ Ω, but δ(S(L1), S(L2)) ≤ αδ(S(K1), S(K2)), contradicting the minimality of S(K). □ Corollary 2.7. Assume that (A, B) is a nonempty pair of subsets in a uni- formly convex Banach space X. Let T, S : A ∪ B → A ∪ B be a pointwise cyclic-noncyclic relatively nonexpansive pair involving orbits, and such that T(A) ⊆ S(B) and T(B) ⊆ S(A). Suppose that (S(A), S(B)) is a bounded, c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 339 F. Fouladi, A. Abkar and E. Karapınar closed and convex pair of subsets in X. Then there exists (x, y) ∈ A × B such that for p ∈ {x, y} we have ‖Tp − Sp‖ = dist(S(A), S(B)). 3. Noncyclic-noncyclic pairs In this section we study the case in which both mappings are noncyclic. In- deed, we first introduce a pointwise noncyclic-noncyclic relatively nonexpansive pair involving orbits, and proceed to study its best proximity points. Definition 3.1. Assume that (A, B) is a nonempty pair of subsets of a Banach space X and T, S : A ∪ B → A ∪ B are two mappings. A pair (T ; S) is said to be a pointwise noncyclic-noncyclic relatively nonexpansive pair involving orbits if (T ; S) is a noncyclic-noncyclic pair and for any (x, y) ∈ A × B, if ‖x − y‖ = dist(S(A), S(B)), then ‖Tx − Ty‖ = dist(S(A), S(B)), ‖Sx − Sy‖ = dist(S(A), S(B)) and otherwise, there exists a function α : A × B → [0, 1] such that ‖Tx − Ty‖ ≤ α(x, y)‖Sx − Sy‖ + (1 − α(x, y)) max{δx[O(y; ∞)], δy[O(x; ∞)]}, where, for any (x, y) ∈ A × B δx[O(y; ∞)] = sup n∈N ‖x − T ny‖, δy[O(x; ∞)] = sup n∈N ‖T nx − y‖. Theorem 3.2. Assume that (A, B) is a nonempty pair of subsets in a strictly convex Banach space X with PNS, and T, S : A ∪ B → A ∪ B is a point- wise noncyclic-noncyclic relatively nonexpansive pair involving orbits such that T(A) ⊆ S(A) and T(B) ⊆ S(B). Suppose that (S(A), S(B)) is a weakly com- pact and convex pair of subsets in X. Then, there exists x0 ∈ A and y0 ∈ B such that Tx0 = x0, Ty0 = y0 and ‖x0 − y0‖ = dist(S(A), S(B)). Proof. Suppose that (S(As0), S(B s 0)) is the associated proximal pair of (S(A), S(B)), and choose x ∈ As0. Then there exists y ∈ Bs0 such that ‖Sx − Sy‖ = dist(S(A), S(B)), and furthermore ‖T(Sx) − T(Sy)‖ = dist(S(A), S(B)) = dist(S(As0), S(B s 0)). Thus, T : S(As0) → S(As0) and similarly, T : S(Bs0) → S(Bs0). Now let Ω denote the collection of nonempty subsets S(F) of S(As0) ∪ S(Bs0) for which S(F) ∩ S(As0) and S(F) ∩ S(Bs0) are nonempty, closed and convex, T(S(F) ∩ S(As0)) ⊆ S(F) ∩ S(A s 0), T(S(F) ∩ S(B s 0)) ⊆ S(F) ∩ S(B s 0), S(S(F) ∩ S(As0)) ⊆ S(F) ∩ S(A s 0), S(S(F) ∩ S(B s 0)) ⊆ S(F) ∩ S(B s 0) c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 340 Weak proximal normal structure and dist(S(F) ∩ S(As0), S(F) ∩ S(B s 0)) = dist(S(A), S(B)). Since, S(As0) ∪ S(Bs0) ∈ Ω, Ω is nonempty. We proceed as in the proof of Theorem 2.6 to show that Ω has a minimal element S(K). Assume that S(K1) = S(K) ∩ S(As0), and S(K2) = S(K) ∩ S(Bs0). First, assume that one of the sets is a singleton, say S(K1) = {x}. Then Tx = x and if y is the unique point of S(K2) for which ‖x − y‖ = dist(S(K1), S(K2)), it must be the case that Ty = y. Since, ‖y − x‖ = dist(S(A), S(B)), we are finished. So, we may assume that S(K1) and S(K2) have positive diameter and because the space is strictly convex, this in turn implies that δ(S(K1), S(K2)) > dist(S(K1), S(K2)). We shall see that this leads to a contradiction. Since (S(As0), S(B s 0)) has prox- imal normal structure, we may define S(L1) and S(L2) as in the proof of The- orem 2.6. Choose Sx ∈ S(L1). For any Sy ∈ S(K2), we have Sx ∈ S(As0) and Sy ∈ S(Bs0); that is, x ∈ As0 and y ∈ Bs0. Thus, ‖Sx − Sy‖ = dist(S(A), S(B)) and so, ‖T(Sx) − T(Sy)‖ = dist(S(A), S(B)) ≤ δ(Sx, S(K2)) ≤ αδ(S(K1), S(K2)). This implies that T(Sy) ∈ B(T(Sx); αδ(S(K1), S(K2))) ∩ S(K2), thus, T(S(K2)) ⊆ B(T(Sx); αδ(S(K1), S(K2))) ∩ S(K2). It follows from the minimality of S(K) that S(K2) ⊆ B(T(Sx); αδ(S(K1), S(K2))) and this in turn implies that δ(T(Sx), S(K2)) ≤ αδ(S(K1), S(K2)). Therefore, T(Sx) ∈ S(L1); in fact T(S(L1)) ⊆ S(L1). Similarly, T(S(L2)) ⊆ S(L2), S(S(L1)) ⊆ S(L1) and S(S(L2)) ⊆ S(L2). Since, S(L1) and S(L2) are, respectively, nonempty, closed and convex subsets of S(K1) and S(K2) and since for α < 1 we have δ(S(L1), S(L2)) ≤ αδ(S(K1), S(K2)), which contradicts the minimality of S(K). □ Corollary 3.3. Assume that (A, B) is a nonempty pair of subsets in a uni- formly convex Banach space X and T, S : A ∪ B → A ∪ B is a pointwise noncyclic-noncyclic relatively nonexpansive pair involving orbits such that T(A) ⊆ S(A) and T(B) ⊆ S(B). Suppose that (S(A), S(B)) is a bounded, closed and convex pair of subsets in X. Then, there exists x0 ∈ A and y0 ∈ B such that Tx0 = x0, Ty0 = y0 and ‖x0 − y0‖ = dist(S(A), S(B)). c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 341 F. Fouladi, A. Abkar and E. Karapınar 4. WPNS and cyclic-noncyclic pairs In this section, and under weak proximal normal structure, we discuss the coincidence quasi-best proximity point problem for pointwise cyclic-noncyclic relatively nonexpansive pairs involving orbits. Lemma 4.1. Assume that (A, B) is a nonempty pair of subsets in a Banach space X, and T, S : A ∪ B → A ∪ B is a pointwise cyclic-noncyclic relatively nonexpansive pair involving orbits such that T(A) ⊆ S(B) and T(B) ⊆ S(A). Suppose that (S(A), S(B)) is a weakly compact and convex pair of subsets in X. Then, there exists (S(K1), S(K2)) ⊆ (S(As0), S(Bs0)) ⊆ (S(A), S(B)) which is minimal with respect to being nonempty, closed, convex and T and S-invariant pair of subsets of (S(A), S(B)), such that dist(S(K1), S(K2)) = dist(S(A), S(B)). Moreover, the pair (S(K1), S(K2)) is proximal. Proof. The proof essentially goes in the same lines as in the proof of Theorem 2.6. We omit the details. □ Theorem 4.2. Assume that (A, B) is a nonempty pair of subsets in a Ba- nach space X with WPNS, and T, S : A ∪ B → A ∪ B is a pointwise cyclic- noncyclic relatively nonexpansive pair involving orbits such that T(A) ⊆ S(B) and T(B) ⊆ S(A). Suppose that (S(A), S(B)) is a weakly compact and convex pair of subsets in X. Then (T ; S) has a coincidence quasi-best proximity point. Proof. By Lemma 4.1, assume that (S(K1), S(K2)) is a minimal, weakly com- pact, convex and proximal pair which is T and S-invariant, and such that dist(S(K1), S(K2)) = dist(S(A), S(B)). Notice that con(T(S(K1))) ⊆ S(K2) and so, T(con(T(S(K1)))) ⊆ T(S(K2)) ⊆ con(T(S(K2))). Similarly, T(con(T(S(K2)))) ⊆ con(T(S(K1))); that is, T is cyclic on con(T(S(K1))) ∪ con(T(S(K2))). On other hand, S is noncyclic on con(S(S(K1))) ∪ con(S(S(K2))). The minimality of (S(K1), S(K2)) implies that con(T(S(K1))) = S(K2) and con(T(S(K2))) = S(K1). Besides, con(S(S(K1))) = S(K1) and con(S(S(K2))) = S(K2). We note that if δ(S(K1), S(K2)) = dist(S(K1), S(K2)) = dist(S(A), S(B)), then every point of S(K1) ∪ S(K2) is a coincidence quasi-best proximity point c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 342 Weak proximal normal structure of (T ; S) and we are finished. Otherwise, since (S(A), S(B)) has WPNS, there exists a point (x1, y1) ∈ K1 × K2 and c ∈ (0, 1), so that δ(Sx1, S(K2)) ≤ c δ(S(K1), S(K2)), δ(Sy1, S(K1)) ≤ c δ(S(K1), S(K2)). Since (S(K1), S(K2)) is a proximal pair, there exists (x2, y2) ∈ K1 × K2 such that ‖Sx1 − Sy2‖ = ‖Sx2 − Sy1‖ = dist(S(A), S(B)). Put Su := Sx1+Sx2 2 and Sv := Sy1+Sy2 2 . Then, (Su, Sv) ∈ S(K1) × S(K2) and ‖Su − Sv‖ = dist(S(K1), S(K2)). Moreover, for each z ∈ K2, we have ‖Su − Sz‖ = ‖ Sx1 + Sx2 2 − Sz‖ ≤ 1 2 [‖Sx1 − Sz‖ + ‖Sx2 − Sz‖] ≤ c + 1 2 δ(S(K1), S(K2)). Now, if r := c+1 2 , then r ∈ (0, 1) and δ(Su, (S(K2)) ≤ rδ(S(K1), S(K2)). Similarly, we can see that δ(Sv, (S(K1)) ≤ rδ(S(K1), S(K2)). Assume that S(L1) = {Sx ∈ S(K1) : δ(Sx, S(K2)) ≤ rδ(S(K1), S(K2))}, S(L2) = {Sy ∈ S(K2) : δ(Sy, S(K1)) ≤ rδ(S(K1), S(K2))}. Thus, (Su, Sv) ∈ S(L1)×S(L2) and so, dist(S(L1), S(L2)) = dist(S(K1), S(K2)). Moreover, (S(L1), S(L2)) is a weakly compact and convex pair in X. We show that T is cyclic on S(L1)∪S(L2). Suppose Sx ∈ S(L1) and Sy ∈ S(K2). Then, similar to proof of Theorem 2.6, Sx ∈ S(As0) and Sy ∈ S(Bs0); that is, x ∈ As0 and y ∈ Bs0. Thus, ‖T(Sx) − T(Sy)‖ = dist(S(A), S(B)) ≤ δ(Sx, S(K2)) ≤ rδ(S(K1), S(K2)). So, T(Sy) ∈ B(T(Sx); rδ(S(K1), S(K2))); that is, T(S(K2)) ⊆ B(T(Sx); rδ(S(K1), S(K2))) and S(K1) = conT(S(K2)) ⊆ B(T(Sx); rδ(S(K1), S(K2))). Therefore, δ(T(Sx), S(K1)) ≤ rδ(S(K1), S(K2)); that is, T(Sx) ∈ S(L2). Thus, T(S(L1)) ⊆ S(L2). Similarly, T(S(L2)) ⊆ S(L1), S(S(L1)) ⊆ S(L1) and S(S(L2)) ⊆ S(L2). Hence, T is cyclic and S is noncyclic on S(L1)∪S(L2). The minimality of (S(K1), S(K2)) now implies that S(L1) = S(K1) and S(L2) = S(K2). Now, we have δ(S(K1), S(K2)) = sup x∈K1 δ(Sx, S(K2)) ≤ rδ(S(K1), S(K2)), which is a contradiction. □ c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 343 F. Fouladi, A. Abkar and E. Karapınar 5. Examples We clarify the above results with some examples. Example 5.1. Let A = [−4, 0] and B = [0, 4] be subsets of the uniformly convex Banach space (R, |.|). For any x ∈ A ∪ B we define Tx = − 1 4 x, Sx = 1 2 x. Then, T(A) = [0, 1] ⊆ [0, 2] = S(B), T(B) = [−1, 0] ⊆ [−2, 0] = S(A). Moreover, for any (x, y) ∈ A × B, we define α(x, y) = ! 0, if x = y 1, if x ∕= y. If (x, y) ∈ A × B such that ‖x − y‖ = dist(S(A), S(B)) = 0, then x = y and ‖Tx − Ty‖ = dist(S(A), S(B)), ‖Sx − Sy‖ = dist(S(A), S(B)). Otherwise, ‖Tx − Ty‖ = ‖ 1 4 y − 1 4 x‖ = 1 2 ‖ 1 2 y − 1 2 x‖ = 1 2 ‖Sy − Sx‖ = 1 2 ‖Sx − Sy‖ ≤ ‖Sx − Sy‖ = α(x, y)‖Sx − Sy‖ + (1 − α(x, y)) max{δx[O2(y; ∞)], δy[O2(x; ∞)]}. Thus, (T ; S) is a pointwise cyclic-noncyclic relatively nonexpansive pair involv- ing orbits, and by Corollary 2.7, there exists (x, y) ∈ A × B such that ‖Tx − Sx‖ = dist(S(A), S(B)), ‖Ty − Sy‖ = dist(S(A), S(B)). Example 5.2. Let A = [−4, −1] and B = [1, 4] be subsets in (R, |.|). Let K1 = [−4, −2], K2 = [2, 4] and Sx = " ###$ ###% − √ −x − 2, if x ∈ A \ K1√ x + 2, if x ∈ B \ K2 −3, if x ∈ K1 3, if x ∈ K2. Therefore, S is a noncyclic mapping. Moreover, S(A) = [−4, −3] ⊆ A, S(B) = [3, 4] ⊆ B. So, (S(A), S(B)) is a closed, convex and bounded pair and we have dist(S(A), S(B)) = 6. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 344 Weak proximal normal structure Suppose that Tx = " ###$ ###% √ −x + 2, if x ∈ A \ K1 − √ x − 2, if x ∈ B \ K2 3, if x ∈ K1 −3, if x ∈ K2. Therefore, T is a cyclic mapping. Besides, T(A) = [3, 4] = S(B) ⊆ B, T(B) = [−4, −3] = S(A) ⊆ A. Moreover, we suppose that for any (x, y) ∈ A × B, α(x, y) = ! 1, if (x, y) ∈ (A \ K1) × (B \ K2) 0, otherwise. If ‖x − y‖ = dist(S(A), S(B)), then (x, y) ∈ K1 × K2 and we have ‖Sx − Sy‖ = ‖ − 3 − 3‖ = 6 = dist(S(A), S(B)) and ‖Tx − Ty‖ = ‖3 − (−3)‖ = 6 = dist(S(A), S(B)). Onherwise, for any (x, y) ∈ (A \ K1) × (B \ K2), we have ‖Tx − Ty‖ = ‖ √ −x + 2 − (− √ y − 2)‖ = ‖ √ −x + √ y + 4‖ = ‖ √ y + 2 − (− √ −x − 2)‖ = ‖Sy − Sx‖ = ‖Sx − Sy‖ ≤ α(x, y)‖Sx − Sy‖ + (1 − α(x, y)) max{δx[O(y; ∞)], δy[O(x; ∞)]}. Thus, (T ; S) is a pointwise cyclic-noncyclic relatively nonexpansive pair involv- ing orbits, and by Corollary 2.7, there exists (x, y) ∈ A × B such that ‖Tx − Sx‖ = dist(S(A), S(B)), ‖Ty − Sy‖ = dist(S(A), S(B)). In fact, for any (x, y) ∈ K1 × K2, we have ‖Tx − Sx‖ = 6 = dist(S(A), S(B)), ‖Ty − Sy‖ = 6 = dist(S(A), S(B)). We clarify the above result with an example. Example 5.3. Assume that A = [−4, 0] and B = [0, 4] are subsets of (R, |.|). For any x ∈ A ∪ B, we set Tx = 1 4 x, Sx = 1 2 x. Then, T(A) = [−1, 0] ⊆ [−2, 0] = S(A), T(B) = [0, 1] ⊆ [0, 2] = S(B). Moreover, we suppose that for any (x, y) ∈ A × B, α(x, y) = ! 0, if x = y 1, if x ∕= y. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 345 F. Fouladi, A. Abkar and E. Karapınar If (x, y) ∈ A × B such that ‖x − y‖ = dist(S(A), S(B)) = 0, then x = y and ‖Tx − Ty‖ = dist(S(A), S(B)), ‖Sx − Sy‖ = dist(S(A), S(B)). Otherwise, ‖Tx − Ty‖ = ‖ 1 4 x − 1 4 y‖ = 1 2 ‖ 1 2 x − 1 2 y‖ = 1 2 ‖Sx − Sy‖ ≤ ‖Sx − Sy‖ = α(x, y)‖Sx − Sy‖ + (1 − α(x, y)) max{δx[O(y; ∞)], δy[O(x; ∞)]}. Thus, (T ; S) is a pointwise noncyclic-noncyclic relatively nonexpansive pair involving orbits, and by Corollary 3.3, there exists (x0, y0) ∈ A × B such that ‖x0 − y0‖ = dist(S(A), S(B)). In fact, for x0 = 0 and y0 = 0, we have Tx0 = x0, Ty0 = y0 and ‖x0 − y0‖ = dist(S(A), S(B)). Acknowledgements. The authors express their gratitude to colleagues at China Medical University for their sincere hospitality during the visit of China Medical University, Taichung, Taiwan. References [1] A. Abkar and M. 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