EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 3, 2018, 869-875 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global A common best proximity point theorem for φ-dominated pair Mahdi Iranmanesh1, Ali Ganjbakhsh Sanatee1,∗ 1 Department of Pure Mathematics, Shahrood University of Technology, Shahrood, Iran Abstract. In the present research, an interesting common best proximity point theorem for pairs of non-self-mappings is presented. It satisfies a weakly contraction-like condition, thereby produc- ing common optimal approximate solutions of certain simultaneous fixed point equations. 2010 Mathematics Subject Classifications: 47H09, 47H10 Key Words and Phrases: Common best proximity point, φ-dominated pair, φ-contraction, fixed point 1. Introduction Fixed point theory is indispensable Tx = x for self-mappings T on subsets of metric space or normed space. Let A and B be non-empty subsets of metric space (X, d) and let T : A −→ B be non-self mapping. If the equation Tx = x does not possess solution, then d(x, Tx) > 0. In this case, it is important that we find an element x ∈ A such that d(x, Tx) is minimum in some sense. For example, the best approximation problem and best proximity problem are investigated in this regard (see [2] and [5]). An element x ∈ A is said to be a best proximity point of T if d(x, Tx) = d(A,B) where d(A,B) = inf{d(x, y) : x ∈ A, y ∈ B}. It is easy to check that if T is self-mapping the best proximity problem reduces to fixed point problem. There are several various of contractions that guarantee the existence of a best proximity point (see [2], [5], and [11]). Suppose that A and B be nonempty subsets of metric space (X, d). Let T : A −→ B and S : A −→ B be nonself mapping. Let considering the fact S and T are nonself- mappings, it is possible that the equations Tx = x and Sx = x have a common solution, considered as a common fixed point of the mappings T and S . When the equations have no common solution, one thinks to find an element x that is in near proximity to ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v11i3.3286 Email addresses: m.iranmanesh2012@gmail.com (M. Iranmanesh), alisanatee62@gmail.com (A. Ganjbakhsh Sanatee) http://www.ejpam.com 869 c© 2018 EJPAM All rights reserved. M. Iranmanesh, A. G. Sanatee / Eur. J. Pure Appl. Math, 11 (3) (2018), 869-875 870 Tx and Sx in the sense that d(x, Tx) and d(x, Sx) are minimal. In fact, one investigates the existence of such optimal approximate solutions, known as common best proximity points, to the equations Sx = x and Tx = x. Further, one can comprehend that the real valued functions x −→ d(x, Tx) and x −→ d(x, Sx) approximate the value of the error of proximate solution of the equations Tx = x and Sx = x. In view of the fact that d(A,B) ≤ d(x, Tx) and d(A,B) ≤ d(x, Sx), a common best proximity point theorem determines global minimum of both functions x −→ d(x, Tx) and x −→ d(x, Sx) by limiting a common approximate solution of the equations Tx = x and Sx = x to attain the requirement that d(x, Sx) = d(A,B) and d(x, Tx) = d(A,B). Common best proximity point problem was studied by many mathematicians (see [7], [8] and [11]). 2. Preliminary Concepts Definition 1. An element x ∈ X is said to be common best proximity point of the non- self-mappings S : A −→ B and T : A −→ B if it satisfies the condition that d(x, Sx) = d(x, Tx) = d(A,B). Definition 2. [4] A function φ : [0,+∞) −→ [0,+∞) is called a comparison if it satisfies the following conditions: • φ is increasing, • the sequence (φn(t))n∈N converges to 0 as n→ +∞, for all t ∈ [0,+∞). We recall that a self-mapping T on a metric space (X, d) is said to be φ−contraction if d(T (x), T (y)) ≤ φ(d(x, y)) for any x, y ∈ X; where φ is comparison function. Remark 1. If φ is comparison function then • φ(t) < t for any t ∈ (0,+∞), • φ(t) = 0 if and only if t = 0. Lemma 1. [4] Let (X, d) be a metric space and let (xn)n∈N be a sequence in X such that d(xn+1, xn) → 0. If (xn)n∈N is not Cauchy sequence then there exists ε > 0 and sequences (n(k)) and (m(k)) of positive integers such that the following sequences tend to ε as k → +∞: d(xm(k), xn(k)), d(xm(k), xn(k)+1), d(xm(k)−1, xn(k)), d(xm(k)−1, xn(k)+1), d(xm(k)+1, xn(k)+1), d(xm(k)+1, xn(k)). Definition 3. Let S : A −→ B, T : A −→ B and F : B −→ A be given. F is said to commute with the pair (S, T ) if SFT = TFS. M. Iranmanesh, A. G. Sanatee / Eur. J. Pure Appl. Math, 11 (3) (2018), 869-875 871 Definition 4. Let S : A −→ B, T : A −→ B, F : B −→ A and G : B −→ A be mappings. The pair (F, S) is said to be φ−dominated by the pair (G,T ) if for any x ∈ A and y ∈ B it satisfies the condition that d(FSx, FSy) ≤ φ(d(GTx,GTy)) where φ is comparison function. 3. Main Results From here throughout this paper, X denotes a complete metric space and A and B are its nonempty subsets. Now, we are ready to present our main result. Theorem 1. Let A and B be closed. Moreover, assume that S : A −→ B, T : A −→ B, F : B −→ A and G : B −→ A are continuous functions satisfying the following conditions: (1) FS commutes with GT and SF commutes with TG. (2) (F, S) is φ−dominated by (G,T ) and (S, F ) is ψ−dominated by (T,G), where φ and ψ are comparison functions. (3) FS(A) ⊂ GT (A) and SF (B) ⊂ TG(B). (4) S and T commute with the pair (F,G), and F and G commute with the pair (S, T ). (5) There is a non-negative number α < 1 such that for all x ∈ A d(Sx, FSx) ≤ αd(Tx,GTx) + (1− α)d(A,B). Then, there exists u ∈ A and v ∈ B such that d(u, Su) = d(u, Tu) = d(A,B) d(v, Fv) = d(v,Gv) = d(A,B) d(u, v) = d(A,B). If (I, S) is φ−dominated by (I, T ), where I is the identity mapping on B, then 2d(A,B) + φ(2d(A,B) + d(a, a′))− d(a, a′) ≥ 0 whevever a′ is another common best proximity point of S and T . Proof. Let x0 be an element in A. Since FS(A) ⊂ GT (A), there exists an element x1 ∈ A such that FS(x0) = GT (x1). Again by FS(A) ⊂ GT (A), we can choose an element x2 ∈ A such that FS(x1) = GT (x2). By continuing this process, we can construct a sequence (xn) such that FS(xn) = GT (xn+1). By condition (2) there exists continuous non-decreasing function φ : [0,+∞) −→ [0,+∞), with limn⇒∞ φn(t) = 0 for all t ∈ [0,+∞), such that d(FSxn, FSxn+1) ≤ φ(d(GTxn, GTxn+1)). M. Iranmanesh, A. G. Sanatee / Eur. J. Pure Appl. Math, 11 (3) (2018), 869-875 872 Since FS(xn) = GT (xn+1) so, d(FSxn, FSxn+1) ≤ φ(d(GTxn, GTxn+1)) = φ(d(FSxn−1, FSxn)) ≤ φ2(d(GTxn−1, GTxn)) = φ2(d(FSxn−2, GTxn−1)) ≤ φ3(d(FSxn−3, FSxn−2)) ≤ . . . ≤ φn(d(FSx0, FSx1)). Hence, d(FSxn, FSxn+1) ≤ φn(d(FSx0, FSx1)) taking n −→ +∞ we have lim n→+∞ d(FSxn, FSxn+1) = 0. Clime: (FSxn) is a cauchy sequence. Let (FSxn) is not Cauchy. Then there exists ε > 0 and two sequences ( n(k) ) and ( m(k) ) of positive integers such that n(k) > m(k) > k, d(FSxm(k), FSxn(k)−1) < ε and d(FSxm(k), FSxn(k)) ≥ ε. (1) Using condition (2) and the fact that φ is a comparison function we obtain that d(FSxm(k)+1, FSxn(k)) ≤ φ ( d(GTxm(k)+1, GTxn(k)) ) = φ ( d(FSxm(k), FSxn(k)−1) ) ≤ φ(ε) < ε. (2) Taking n −→ +∞ in (2) we get ε = lim k→+∞ d(FSxm(k)+1, FSxn(k)) ≤ φ(ε) < ε. This is a contradiction. Hence (FSxn) is Cauchy sequence. Obviosly, (GTxn) is also Cauchy. Because of the completeness of the space, there exists an element x ∈ A such that FSxn −→ x. By continuity of FS and GT ; (GT )(FS)xn −→ GTx (FS)(GT )xn −→ FSx. By condition (1), it follows that GTx = FSx. Put a = GTx = FSx. Then, by condition (1) and (2), d(FSa, a) = d(FSa, FSx) ≤ φ(d(GTa,GTx)) = φ(d(GT (FSx), a)) = φ(d(FS(GTx), a)) = φ(d(FSa, a)). Since φ is comparison by Remark 1 we get d(FSa, a) = 0 =⇒ FSa = a. M. Iranmanesh, A. G. Sanatee / Eur. J. Pure Appl. Math, 11 (3) (2018), 869-875 873 Further, GTa = (GT )(FSx) = (FS)(GTx) = FSa = a. A similar argument can be given to assert that there exists an element b ∈ B such that SFb = TGb = b. Also, since T commutes with the pair (F,G), GTFb = FTGb = Fb. So, d(a, Fb) = d(FSa, FS(Fb)) ≤ φ(d(GTa,GT (Fb))) = φ(d(a, Fb)). Since φ is a comparison by Remark 1, it follows that Fb = a. By the same argument we can show that Gb = a, Sa = b and Ta = b. Consequently, by condition (5) there exists α ∈ [0, 1) such that d(a, b) = d(Sa, FSa) ≤ αd(Ta,GTa) + (1− α)d(A,B) = αd(a, b) + (1− α)d(a, b). So d(a, b) ≤ d(A,B) and hence d(a, b) = d(A,B). Therefore, d(a, Sa) = d(a, Ta) = d(a, b) = d(A,B) d(b, Fb) = d(b,Gb) = d(a, b) = d(A,B). If (I, S) is φ−dominated by (I, T ) and a′ is another common best proximity point of S and T , then d(a, a′) ≤ d(a, Sa) + d(Sa, Sa′) + d(a′, Sa′) ≤ 2d(A,B) + φ(d(Ta, Ta′)) ≤ 2d(A,B) + φ(2d(A,B) + d(a, a′)) and hence 2d(A,B) + φ(2d(A,B) + d(a, a′))− d(a, a′) ≥ 0. Example 1. Consider the space of real numbers with the Euclidean meteric. Let A = [3,+∞) and B = (−∞,−3] Suppose that S, T : A → B, F,G : B → A and φ, ψ : [0,+∞)→ [0,+∞) are defined by S(x) = −3; T (x) = −x; F (y) = { 3 y ∈ Z 4 y ∈ R \ Z ;G(y) = −y and φ(x) = ψ(x) = x 1 + x . It is easy to check that d(A,B) = 6 and the mapping S, T, F and G are satisfied the conditions in Theorem 1 and d(3, S(3)) = d(3, T (3)) = d(A,B) d(−3, F (−3)) = d(−3, G(−3)) = d(A,B) d(3,−3) = d(A,B). 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