Best proximity points for proximal contractive type mappings with C-class functions in S-metric spaces

Fixed Point Theory and Applications, Jun 2017

In this paper, we use the concept of C-class functions to establish the best proximity point results for a certain class of proximal contractive mappings in S-metric spaces. Our results extend and improve some known results in the literature. We give examples to analyze and support our main results.

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Best proximity points for proximal contractive type mappings with C-class functions in S-metric spaces

Ansari and Nantadilok Fixed Point Theory and Applications Best proximity points for proximal contractive type mappings with C -class functions in S-metric spaces Arslan Hojat Ansari 1 3 Jamnian Nantadilok 0 1 0 Department of Mathematics, Faculty of Science, Lampang Rajabhat University , Lampang, 52100 , Thailand 1 which implies φ(t) =  or 2 (t) = . That is, t =  , which is a contradiction. Hence, t = . That is 3 Department of Mathematics, Karaj Branch, Islamic Azad University , Karaj , Iran In this paper, we use the concept of C-class functions to establish the best proximity point results for a certain class of proximal contractive mappings in S-metric spaces. Our results extend and improve some known results in the literature. We give examples to analyze and support our main results. MSC: Primary 47H10; secondary 54H25 1 Introduction and preliminaries Consider the equation Tx = x. If the equation Tx = x does not possess a solution, then we attempt to resolve the problem of finding an element x such that x is in proximity to Tx. In fact, in the setting of a metric space (X, d), if T : A → X, then a best approximation theorem provides sufficient conditions that confirm the existence of an element x, known as the best approximant, such that d(x, Tx) = d(Tx, A), where d(A, B) := inf{d(x, y) : x ∈ A and y ∈ B} for any nonempty subsets A and B of X. Indeed, a classical best approximation theorem, due to Ky Fan [], states that if K is a nonempty compact convex subset of a Banach space X and T : K → X is a single-valued continuous mapping, then there exists an element x ∈ K such that d(x, Tx) = inf{d(y, Tx) : y ∈ K }, where d is a metric on X. This result has been generalized by many authors (see[-]). In other words, if A and B are two nonempty subsets of a metric space (X, d), then an element x ∈ A is said to be a fixed point of a given map T : A → B if Tx = x. Clearly, T (A) ∩ A = ∅ is a necessary (but not sufficient) condition for the existence of a fixed point of T . If T (A) ∩ A = ∅, then d(x, Tx) >  for all x ∈ A, that is, the set of fixed points of T is empty. In such a situation, one often attempts to find an element x which is in some sense closest to Tx. Best proximity point analysis has been developed in this direction. An element x∗ ∈ A is called a best proximity point of T if fixed points; proximity points; G-metric space; S-metric space; B-metric space; proximal contractive mapping; C-class function - d x∗, Tx∗ = d(A, B). Indeed, in view of the fact that d(x, Tx) ≥ d(A, B) for all x ∈ A, the global minimum of the mapping x → d(x, Tx) is attained at a best proximity point. Clearly, if the underlying © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. mapping is self-mapping, then a best proximity point reduces to a fixed point. The goal of best proximity point theory is to furnish sufficient conditions that assure the existence of such points. For more details on this approach, we refer the reader to [–] and the references therein. Mustafa and Sims [] introduced the notion of G-metric and obtained some wellknown fixed point results in the setting of G-metric spaces. Many authors have obtained fixed point results in the context of G-metric spaces [–]. In , Sedghi et al. [, ] introduced a D∗-metric space, which is a modification of D-metric spaces introduced by Dhage [], and established common fixed point theorems in D∗-metric spaces. We note that every G-metric is a D∗-metric, but in general the converse is not true (see []). In , Sedghi et al. [] introduced the concept of an S-metric space, a modification of D∗metric and G-metric spaces, and gave a generalization of fixed point theorems in S-metric spaces, but the best proximity point results in S-metric spaces still remain open. Recently, Ansari [] introduced the concept of C-class functions which can be used to generalize many fixed point theorems in the literature (see, for example, []). Later, Nantadilok [] obtained best proximity point results for a certain class of proximal contractive mappings in complete S-metric spaces. Inspired and motivated by Ansari [] and Nantadilok [], in this paper, we establish best proximity point results for proximal contractive type mappings with C-class functions in the setting of S-metric spaces. We also give examples to support our results. Now we collect some necessary definitions and results in this direction. The notion of S-metric spaces is defined as follows. Definition . (see []) Let X be a nonempty set. An S-metric on X is a function S : X → [, ∞) that satisfies the following conditions, for each x, y, z, a ∈ X. (i) S(x, y, z) ≥ ; (ii) S(x, y, z) =  if and only if x = y = z; (iii) S(x, y, z) ≤ S(x, x, a) + S(y, y, a) + S(z, z, a). The function S is called an S-metric on X, and the pair (X, S) is called an S-metric space. Remark . This notion is a modification of a G-metric space [] and a D∗-metric space []. Lemma . (see []) Let (X, S) be an S-metric space. Then S(x, x, y) = S(y, y, x) for all x, y ∈ X. Lemma . (see []) Let (X, S) be an S-metric space. Then for all x, y, z ∈ X. Definition . (see []) Let (X, S) be an S-metric space. (i) A sequence {xn} ⊂ X is said to converge to x ∈ X if S(xn, xn, x) →  as n → ∞. That is, for each ε > , there exists n ∈ N such that for all n ≥ n we have S(xn, xn, x) < ε. We write xn → x for brevity. (ii) A sequence {xn} ⊂ X is called a Cauchy sequence if S(xn, xn, xm) →  as n, m → ∞. That is, for each ε > , there exists n ∈ N such that for all n, m ≥ n we have S(xn, xn, xm) < ε. (iii) The S-metric space (X, S) is said to be complete if every Cauchy sequence is a convergent sequence. Some geometric examples for S-metric spaces can be seen in []. Definition . (see []) Let X be a nonempty set. A B-metric on X is a function d : X → [, ∞) if there exists a real umber b ≥  such that the following conditions hold for all x, y, z ∈ X. (B) d(x, y) =  if and only if x = y. (B) d(x, y) = d(y, x). (B) d(x, y) ≤ b[d(x, z) + d(y, z)]. The function d is called a B-metric on X, and the pair (X, d) is called a B-metric space. Theorem . (see []) Let (X, S) be an S-metric space, and let d(x, y) = S(x, x, y) for all x, y ∈ X. Then we have (i) d is a B-metric on X; (ii) xn → x in (X, S) if and only if xn → x in (X, d); (iii) {xn} is a Cauchy sequence in (X, S) if and only if {xn} is a Cauchy sequence in (X, d). Now we recall the notion of C-class functions introduced in [] as follows. Definition . (see []) A mapping f : [, ∞) → R is called a C-class function if it is continuous and satisfies the following properties: () f (s, t) ≤ s; () f (s, t) = s implies that either s = , or t =  for all s, t ∈ [, ∞). We will denote the family of C-class functions as C. Note that for some F ∈ C, we have F(, ) = . Example . (see []) The following functions F : [, ∞) → R are elements of C, for all s, t ∈ [, ∞): () F(s, t) = s – t, F(s, t) = s ⇒ t = ; () F(s, t) = ms,  < m < , F(s, t) = s ⇒ s = ; () F(s, t) = (+st)r , r ∈ (, ∞), F(s, t) = s ⇒ s =  or t = ; () F(s, t) = log(t + as)/( + t), a > , F(s, t) = s ⇒ s =  or t = ; () F(s, t) = ln( + as)/, a > e, F(s, ) = s ⇒ s = ; () F(s, t) = (s + l)(/(+t)r) – l, l > , r ∈ (, ∞), F(s, t) = s ⇒ t = ; () F(s, t) = s logt+a a, a > , F(s, t) = s ⇒ s =  or t = ; () F(s, t) = s – ( ++ss )( +tt ), F(s, t) = s ⇒ t = ; () F(s, t) = sβ(s), β : [, ∞) → [, ), and is continuous, F(s, t) = s ⇒ s = ; () F(s, t) = s – k+tt , F(s, t) = s ⇒ t = ; () F(s, t) = s – ϕ(s), F(s, t) = s ⇒ s = , where ϕ : [, ∞) → [, ∞) is a continuous function such that ϕ(t) =  ⇔ t = ; () F(s, t) = sh(s, t), F(s, t) = s ⇒ s = , where h : [, ∞) × [, ∞) → [, ∞) is a continuous function such that h(t, s) <  for all t, s > ; () F(s, t) = s – ( ++tt )t, F(s, t) = s ⇒ t = ; () F(s, t) = s – ( ++ss )( +tt ), F(s, t) = s ⇒ t = ; () F(s, t) = √nln( + sn), F(s, t) = s ⇒ s = ; () F(s, t) = (+ss)r ; r ∈ (, ∞), F(s, t) = s ⇒ s = ; () F(s, t) = ϑ (s); ϑ : R+ × R+ → R is a generalized Mizoguchi-Takahashi type function, F(s, t) = s ⇒ s = . Definition . (see []) A function ψ : [, ∞) → [, ∞) is called an altering distance function if the following properties are satisfied: (i) ψ is nondecreasing and continuous, (ii) ψ (t) =  if and only if t = . denote the class of altering distance functions. Definition . (see []) An ultra altering distance function is a continuous, nondecreasing mapping ψ : [, ∞) → [, ∞) such that ψ (t) > , t >  and ψ () ≥ . We let u denote the set of all ultra altering distance functions. We note that every S-metric on X induces a metric dS on X defined by dS(x, y) = S(x, x, y) + S(y, y, x), for all x, y ∈ X. We show that a metric dS on X defined by () is a B-metric on X. Conditions (B) and (B) are easy to check. It follows from the definition of S-metric and Lemma . that dS(x, y) = S(x, x, y) + S(y, y, x) ≤ S(x, x, z) + S(x, x, z) + S(y, y, z) + S(y, y, z) + S(y, y, z) + S(x, x, z) = dS(x, z) + dS(y, z) + S(x, x, z) + S(y, y, z) ≤ dS(x, z) + dS(y, z) + S(x, x, z) + S(y, y, z) =  dS(x, z) + dS(y, z) . This shows that dS is a B-metric. Definition . (see []) Let (X, S) be an S-metric space, and let A and B be two nonempty subsets of X. Then B is said to be approximately compact with respect to A if every sequence {yn} in B, satisfying the condition dS(x, yn) → dS(x, B) for some x in A, has a convergent subsequence. Let Φ denote the class of all functions ϕ : [, ∞) → [, ∞) which satisfy . ϕ continuous and nondecreasing, . ϕ(t) =  if and only if t = , . ϕ(t + s) ≤ ϕ(t) + ϕ(s), ∀t, s ∈ [, ∞). Definition . (see []) Let A and B be two nonempty subsets of an S-metric space (X, S). Let T : A → B be a nonself-mapping. We say that T is an S-ϕ-ψ -proximal contractive mapping, if for all x, y, u, v ∈ A, dS(u, Tx) = dS(A, B) dS(v, Ty) = dS(A, B) holds, where ϕ ∈ Definition . Let A and B be two nonempty subsets of an S-metric space (X, S). An element x∗ ∈ A is said to be a best proximity point of a nonself-mapping T if dS(x∗, Tx∗) = dS(A, B). The main result obtained in [] is the following best proximity point theorem. Theorem . (see []) Let A, B be two nonempty subsets of an S-metric space (X, S) such that (A, S) is a complete S-metric space, A is nonempty, and B is approximately compact with respect to A. Assume that T : A → B is an S-ϕ-ψ -proximal contractive mapping such that T (A) ⊆ B. Then T has a unique best proximity point; that is, there exists a unique element z ∈ A such that dS(z, Tz) = dS(A, B). 2 Main results Let (X, S) be an S-metric space. Suppose that A and B are nonempty subsets of an S-metric space (X, S). We will use the following notations: A = a ∈ A : dS(a, b) = dS(A, B) for some b ∈ B B = b ∈ B : dS(a, b) = dS(A, B) for some a ∈ A , where dS(A, B) = inf{dS(x, y) : x ∈ A, y ∈ B}. We introduce the following definitions. Definition . Let A and B be two nonempty subsets of an S-metric space (X, S). Let T : A → B be a nonself-mapping. We say that T is an S-(F, ϕ, ψ )-proximal contractive mapping, if for all x, y, u, v ∈ A, dS(u, Tx) = dS(A, B) dS(v, Ty) = dS(A, B) holds, where F ∈ C, ϕ ∈ Definition . Let A and B be two nonempty subsets of an S-metric space (X, S). Let T : A → B be a nonself-mapping. We say that T is an S-(F, ϕ, ψ )-sum-proximal contractive mapping, if for all x, y, u, v ∈ A, ϕ S(u, u, v) ≤ F ϕ(m(u,u∗,v,x,y)), ψ (m(u,u∗,v,x,y)) holds, where F ∈ C, ϕ ∈ , ψ ∈ u and with a, b, c, d ≥  and a + b + c + d > . We note that these kind of generalizations make sense, since they extend and cover those corresponding classes of proximal contractive mappings defined in []. We state and prove our main results. Theorem . Let A, B be two nonempty subsets of an S-metric space (X, S) such that (A, S) is a complete S-metric space, A is nonempty, and B is approximately compact with respect to A. Assume that T : A → B is an S-(F, ϕ, ψ )-proximal contractive mapping such that T (A) ⊆ B. Then T has a unique best proximity point; that is, there exists a unique element z ∈ A such that dS(z, Tz) = dS(A, B). Proof Since the subset A is not empty, we take x in A. Taking Tx ∈ T (A) ⊆ B into account, we can find x ∈ A such that dS(x, Tx) = dS(A, B). Further, since Tx ∈ T (A) ⊆ B, it follows that there is an element x in A such that dS(x, Tx) = dS(A, B). Recursively, we obtain a sequence {xn} in A satisfying dS(xn+, Txn) = dS(A, B), ∀n ∈ N ∪ {}. dS(u, Tx) = dS(A, B), dS(v, Ty) = dS(A, B), ϕ S(xn, xn, xn+) ≤ F ϕ S(xn–, xn–, xn) , ψ S(xn–, xn–, xn) ≤ ϕ S(xn–, xn–, xn) , We will show that {xn}n∞= is an S-Cauchy sequence. Suppose, on the contrary, that there exist ε >  and a subsequence {xnk } of {xn} such that with nk ≥ mk > k. Further, corresponding to mk , we can choose nk in such a way that it is the smallest integer with nk > mk and it satisfies (). Hence, So, the sequence {S(xn, xn, xn+)} is a decreasing sequence in R+ and thus it is convergent to t ∈ R+. We claim that t = . Suppose, on the contrary, that t > . Taking the limit as n → ∞ in (), we get ≤ S(xnk , xnk , xnk–) + S(xmk , xmk , xnk–) Letting k → ∞ in (), we derive that Again, by Lemmas . and ., we obtain the following inequalities: S(xmk , xmk , xnk ) ≤ S(xmk , xmk , xmk–) + S(xnk , xnk , xmk–) ≤ S(xmk , xmk , xmk–) + S(xnk , xnk , xnk–) + S(xmk–, xmk–, xnk–) S(xmk–, xmk–, xnk–) ≤ S(xmk–, xmk–, xmk ) + S(xnk–, xnk–, xmk ) ≤ S(xmk–, xmk–, xmk ) + S(xnk–, xnk–, xnk ) + S(xmk , xmk , xnk ) Letting k → ∞ in () and applying (), we find that From () with u = xmk , x = xmk–, v = xnk and y = xnk–, we have ϕ S(xmk , xmk , xnk ) ≤ F ϕ S(xmk–, xmk–, xnk–) , Taking the limit as k → ∞ in the above inequality, we obtain That is, {xn}n∞= is a Cauchy sequence. Since (A, S) is a complete S-metric space, there exists z ∈ A such that xn → z as n → ∞. On the other hand, for all n ∈ N, we can write dS(z, B) ≤ dS(z, Txn) ≤ dS(z, xn+) + dS(xn+, Txn) = dS(z, xn+) + dS(A, B). nl→im∞ dS(z, Txn) = dS(z, B) = dS(A, B). dS z, y∗ = lim dS(xnk+, Txnk ) = dS(A, B), k→∞ Taking the limit as n → ∞ in the above inequality, we obtain Since B is approximately compact with respect to A, the sequence {Txn} has a subsequence {Txnk } that converges to some y∗ ∈ B. Hence, and so z ∈ A. Now, since Tz ∈ T (A) ⊆ B, there exists w ∈ A such that dS(w, Tz) = dS(A, B). From () with u = xn+, x = xn, v = w and y = z, we have ϕ S(xn+, xn+, w) ≤ F ϕ S(xn, xn, z) , ψ S(xn, xn, z) . Taking the limit as n → ∞, we get ϕ S(p, p, q) ≤ F ϕ S(p, p, q) , ψ S(p, p, q) , Example . Let X = [, ∞). Define an S-metric on X by This implies S(z, z, w) = . That is, w = z. Thus dS(z, Tz) = dS(A, B). Therefore T has a best proximity point. To prove uniqueness, suppose that p = q such that dS(p, Tp) = dS(A, B) and dS(q, Tq) = dS(A, B). Now, by () with u = x = p and v = y = q, we get Let F(s, t) = s – t for all s, t ∈ [, ∞). Also define ϕ, ψ : [, ∞) → [, ∞) by ϕ(t) = t and ψ (t) =  t. Clearly, dS(A, B) = , A = {}, B = {} and T (A) ⊆ B. Let dS(u, Tx) = dS(A, B) and dS(v, Ty) = dS(A, B), then u = v = , x = , , and y = , . Now since u = v = , ϕ(S(u, u, v)) = . Hence, Therefore, we have dS(u, Tx) = dS(A, B) dS(v, Ty) = dS(A, B) ϕ S(u, u, v) ≤ F ϕ S(x, x, y) , ψ S(x, x, y) . ϕ S(u, u, v) ≤ F ϕ S(x, x, y) , ψ S(x, x, y) . Thus T is an S-(F, ϕ, ψ )-proximal contractive mapping. All the conditions of Theorem . hold true, and T has a unique best proximity point. Here, z =  is the unique best proximity point of T . Remark . If we take F(s, t) = s – t in Theorem ., then our result reduces to Theorem . in []. Theorem . Let A, B be two nonempty subsets of an S-metric space (X, S) such that (A, S) is a complete S-metric space, A is nonempty, and B is approximately compact with respect to A. Assume that T : A → B is an S-(F, ϕ, ψ )-sum-proximal contractive mapping such that T (A) ⊆ B. Then T has a unique best proximity point; that is, there exists a unique element z ∈ A such that dS(z, Tz) = dS(A, B). Proof Since the subset A is not empty, we take x in A. Taking Tx ∈ T (A) ⊆ B into account, we can find x ∈ A such that dS(x, Tx) = dS(A, B). Further, since Tx ∈ T (A) ⊆ B, it follows that there is an element x in A such that dS(x, Tx) = dS(A, B). Recursively, we obtain a sequence {xn} in A satisfying dS(xn+, Txn) = dS(A, B), ∀n ∈ N ∪ {}. ϕ S(xn+, xn+, xn+) ≤ F ϕ + cS(xn+, xn+, xn+) + dS(xn+, xn+, xn+) , + cS(xn+, xn+, xn+) + dS(xn+, xn+, xn+) ≤ ϕ + (c + d)S(xn+, xn+, xn+) , S(xn, xn, xn+) ≤ S(xn–, xn–, xn). So, the sequence {S(xn, xn, xn+)} is a decreasing sequence in R+ and thus it is convergent to t ∈ R+. We claim that t = . Suppose, on the contrary, that t > . Taking the limit as n → ∞ in (), we get which implies ϕ(t) =  or ψ (t) = . That is, t = , which is a contradiction. Hence, t = . That is, We will show that {xn}n∞= is an S-Cauchy sequence. Suppose, on the contrary, that there exist ε >  and a subsequence {xnk } of {xn} such that with nk ≥ mk > k. Further, corresponding to mk , we can choose nk in such a way that it is the smallest integer with nk > mk satisfying (). Hence, ≤ S(xnk , xnk , xnk–) + S(xmk , xmk , xnk–) Letting k → ∞, we derive Again, by using Lemmas . and ., we obtain the following inequalities: S(xmk , xmk , xnk ) ≤ S(xmk , xmk , xmk–) + S(xnk , xnk , xmk–) ≤ S(xmk , xmk , xmk–) + S(xnk , xnk , xnk–) + S(xmk–, xmk–, xnk–) S(xmk–, xmk–, xnk–) ≤ S(xmk–, xmk–, xmk ) + S(xnk–, xnk–, xmk ) ≤ S(xmk–, xmk–, xmk ) + S(xnk–, xnk–, xnk ) + S(xmk , xmk , xnk ) Letting k → ∞ in () and () and applying (), we find that + cS(xnk–, xnk–, xmk–) + dS(xnk–, xnk–, xnk ) , + cS(xnk–, xnk–, xmk–) + dS(xnk–, xnk–, xnk ) Taking the limit as k → ∞ in the above inequality, we obtain This proves that {xn} is a Cauchy sequence in an S-metric space (X, S). Since (A, S) is a complete metric space, there exists z ∈ A such that {xn} converges to z. As in the proof of Theorem ., we have dS(w, Tz) = dS(A, B) for some w ∈ A. From () with x = xn–, u = xn, u∗ = xn+, y = z and v = w, we have ≤ F ϕ + cS(z, z, xn+) + dS(z, z, w) , + cS(z, z, xn+) + dS(z, z, w) Taking the limit as n → ∞ in the above inequality, we get ≤ F ϕ ≤ ϕ ≤ F ϕ ≤ ϕ which implies ϕ( a+b+c+d S(p, p, q)) =  or ψ( a+bb++cc+d S(p, p, q)) = , so S(p, p, q) = . Hence b+c p = q, that is, T has the unique best proximity point. Let F(s, t) = s– t+tk for all s, t ∈ [, ∞). This is a C-class function. Also define ϕ, ψ : [, ∞) → [, ∞) by ϕ(t) = t and ψ(t) =  t. Clearly, dS(A, B) = , A = {}, B = {} and T(A) ⊆ B. Let dS(u, Tx) = dS(A, B), dS(u∗, Tu) = dS(A, B) and dS(v, Ty) = dS(A, B), then we get u = u∗ = v = , x = , , and y = , . Now since u = u∗ = v = , ϕ(S(u, u, v)) = . Hence, = ϕ(m(u,u∗,v,x,y)) – with a, b, c, d ≥  and a + b + c + d > . Therefore, we have ϕ S(u, u, v) ≤ F ϕ(m(u,u∗,v,x,y)), ψ(m(u,u∗,v,x,y)) . Thus T is an S-(F, ϕ, ψ)-sum-proximal contractive mapping. All the conditions of Theorem . hold true and T has a unique best proximity point. Here, z =  is the unique best proximity point of T . Remark . If we take F(s, t) = (a + b + c + d)s,  < a + b + c + d <  and ϕ(t) = t in Theorem ., then our result reduces to Theorem . in []. ϕ S(u, u, v) ≤ F ϕ(m(u,u∗,v,x,y)), ψ(m(u,u∗,v,x,y)) . ϕ S(u, u, v) ≤ F ϕ(m(u,v,x,y)), ψ (m(u,,v,x,y)) , dS(u, Tx) = dS(A, B) dS(v, Ty) = dS(A, B) × aS(x, x, u) + b S(x,+x,Sy()uS,(ux,, xv),u) + cS(x, x, y) + dS(x, x, u) Proof Following the same lines as those in the proof of Theorem ., we can construct a sequence {xn} in A satisfying dS(xn+, Txn) = dS(A, B), ∀n ∈ N ∪ {}. From () with x = xn–, u = xn, y = xn and v = xn+, we have Theorem . Let A, B be two nonempty subsets of an S-metric space (X, S) such that (A, S) is a complete S-metric space, A is nonempty, and B is approximately compact with respect to A. Assume that T : A → B is a nonself-mapping such that T (A) ⊆ B and, for x, y, u, v ∈ A, ϕ S(xn, xn, xn+) ≤ F ϕ aS(xn–, xn–, xn) S(xn–, xn–, xn)S(xn–, xn–, xn)  + S(xn, xn, xn+) + cS(xn–, xn–, xn) + dS(xn–, xn–, xn) , aS(xn–, xn–, xn) S(xn–, xn–, xn)S(xn–, xn–, xn)  + S(xn, xn, xn+) + cS(xn–, xn–, xn) + dS(xn–, xn–, xn) ≤ ϕ S(xn–, xn–, xn) for all n ∈ N ∪ {}. This implies S(xn, xn, xn+) ≤ S(xn–, xn–, xn). So, the sequence {S(xn, xn, xn+)} is a decreasing sequence in R+ and thus it is convergent to t ∈ R+. We claim that t = . Suppose, on the contrary, that t > . Taking limit as n → ∞ in (), we get which implies ϕ(t) =  or ψ (t) = . That is, t =  which is a contradiction. Hence, t = . That is, Similarly, one can see that {xn} is a Cauchy sequence in an S-metric space (X, S). Due to the completeness of (A, S), there exists z ∈ A such that {xn} converges to z. As in the proof of Theorem ., we have dS(w, Tz) = dS(A, B) for some w ∈ A. Now, from () with x = xn–, u = xn, y = z and v = w, we deduce ≤ F ϕ aS(xn–, xn–, xn) S(xn–, xn–, z)S(xn–, xn–, xn)  + S(xn, xn, w) + cS(xn–, xn–, z) + dS(xn–, xn–, xn) , aS(xn–, xn–, xn) S(xn–, xn–, z)S(xn–, xn–, xn)  + S(xn, xn, w) + cS(xn–, xn–, z) + dS(xn–, xn–, xn) S(p, p, q) ≤ aS(p, p, p) + b By taking the limit as n → ∞ in the above inequality, we get S(z, z, w) = ; that is, z = w. Hence, dS(z, Tz) = dS(w, Tz) = dS(A, B); that is, T has a best proximity point. To prove uniqueness, assume that p = q, such that dS(p, Tp) = dS(A, B) and dS(q, Tq) = dS(A, B). Now, by () with x = u = p and y = v = q, we have Remark . By taking F(s, t) = (a + b + c + d)s,  < a + b + c + d <  and ϕ(t) = t in Theorem ., our result reduces to Theorem . in []. Acknowledgements The second author is grateful to Lampang Rajabhat University for financial support during the preparation of this manuscript and to the referees for useful suggestions. 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Arslan Hojat Ansari, Jamnian Nantadilok. Best proximity points for proximal contractive type mappings with C-class functions in S-metric spaces, Fixed Point Theory and Applications, 2017, 12, DOI: 10.1186/s13663-017-0606-z