International Journal of Analysis and Applications ISSN 2291-8639 Volume 13, Number 2 (2017), 198-205 http://www.etamaths.com BEST PROXIMITY POINTS FOR A NEW CLASS OF GENERALIZED PROXIMAL MAPPINGS TAYYAB KAMRAN1, MUHAMMAD USMAN ALI2 MIHAI POSTOLACHE3,4,∗, ADRIAN GHIURA4 AND MISBAH FARHEEN1 Abstract. The best proximity points are usually used to find the optimal approximate solution of the operator equation Tx = x, when T has no fixed point. In this paper, we prove some best proximity point theorems for nonself multivalued operators, following the foot steps of Basha and Shahzad [Best proximity point theorems for generalized proximal contractions, Fixed Point Theory Appl., 2012, 2012:42]. 1. Introduction Fixed point theory have an important role in many branches of mathematics such as differential and integral equations, optimization and variational analysis. This theory mainly concerns with the fixed point equation Tx = x, where T : A → B is some nonlinear operator. The solution of this equation is called a fixed point of the operator T . It is not necessary that the equation has a solution for every nonlinear operator T . For example this one has no solution when A ∩ B = ∅. In this case we may find a point x ∈ A which is closest to Tx, that is, the distance between Tx and x is least as compare to other elements of A. Such a point is called the best proximity point of T . The notion of best proximity point was initiated by Fan [1] for normed spaces. Eldred and Veeramani [2] generalized this notion in the context of metric spaces. In literature there are many important best proximity point theorems in different settings: Jleli et al. and Ali et al. [3, 4], for α-ψ-proximal mappings; Akbar and Gabeleh [5, 6], Derafshpour et al. [7], Di Bari et al. [8], Rezapour et al. [9], Vetro [10], for cyclic mappings; Alghamdi et al. [11] for mappings in geodesic metric spaces; Al-Thagafi and Shahzad [12], for Kakutani multimaps; Markin and Shahzad [13], for relatively u-continuous mappings; Nashine et al. [14], for rational proximal contractions; Akbar and Gabeleh [15], for multivalued non-self mappings; Choudhury et al. [16] for best proximity point and coupled best proximity point in partially ordered metric spaces; Shatanawi and Pitea [17], for best proximity points and best proximity coupled points in complete metric spaces with (P)-property; Jamali and Vaespour [18], for best proximity point for nonlinear contractions in Menger probabilistic metric spaces; Bejenaru and Pitea [19], for fixed point and best proximity point theorems in partial metric spaces. Motivated and inspired by the research introduced above, in this paper we introduce our best proximity point theorems for nonself multivalued operators, following the foot steps method of Basha and Shahzad [20]. 2. Previous results Now, we recollect some basic notions, definitions and results which we require subsequently. Let (X,d) be a metric space. For A,B ⊆ X, dist(A,B) = inf{d(a,b) : a ∈ A, b ∈ B}, d(x,B) = inf{d(x,b) : b ∈ B}, A0 = {a ∈ A : d(a,b) = dist(A,B) for some b ∈ B}, B0 = {b ∈ B : d(a,b) = dist(A,B) for some a ∈ A}, while CB(B) is the set of all nonempty closed and bounded subsets of B. A point x∗ ∈ X is said to be a best proximity point of T : A → CB(B) if d(x∗,Tx∗) = dist(A,B). The set B is said to be approximatively compact with respect to the set A, if each {vn} in B with d(x,vn) → d(x,B) for some x in A, has a convergent subsequence [20]. 2010 Mathematics Subject Classification. 47H10, 54H25. Key words and phrases. F-contraction; F-contraction of Hardy Rogers and Ciric type; best proximity point. c©2017 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 198 BEST PROXIMITY POINTS FOR A NEW CLASS OF GENERALIZED PROXIMAL MAPPINGS 199 A class of all functions F : (0,∞) → R satisfying the conditions: (F1) F is strictly increasing, that is, for each a1,a2 ∈ (0,∞) with a1 < a2, we have F(a1) < F(a2), (F2) For each sequence {dn} of positive real numbers we have limn→∞ dn = 0 if and only if limn→∞F(dn) = −∞, (F3) For each sequence {dn} of positive real numbers with limn→∞ dn = 0, there exists k ∈ (0, 1) such that limn→∞ dn kF(dn) = 0, is called class F. A contraction involving a function F ∈ F is called an F-contraction. This class was introduced by Wardowski in [21]. In time, the functions from this class were used by various authors to generalize their contractive conditions: Cosentino and Vetro [22]; Minak et al. [23]; Sgroi and Vetro [24]; Paesano and Vetro [25]; Piri and Kumam [26]; Acar et al. [27]; Batra and Vashistha [28]. Recently, Basha and Shahzad [20] proved the following best proximity point theorem: Theorem 2.1. Let A and B be nonempty closed subsets of a complete metric space (X,d). Assume that A0 is nonempty and T : A → B is a mapping such that for each x1,x2,u1,u2 ∈ A with d(u1,Tx1) = dist(A,B) = d(u2,Tx2), we have d(u1,u2) ≤ a1d(x1,x2) + a2d(x1,u1) + a3d(x2,u2) + a4[d(x1,u2) + d(x2,u1)] (2.1) where a1,a2,a3,a4 ≥ 0 satisfying a1 + a2 + a3 + 2a4 < 1. Further assume that the following conditions hold: (i) T(A0) is contained in B0; (ii) B is approximatively compact with respect to A. Then T has a best proximity point. In this paper we introduce some new F type proximal contractions and prove some best proximity point theorems for such contractions. Our results generalize some existing best proximity point results. In particular Theorem 2.1 becomes a special case of one of our results (Theorem 3.1). 3. Main results We begin this section with the following definition. Definition 3.1. Let A and B be two nonempty subsets of a metric space (X,d). A mapping T : A → CB(B) is called αF -proximal contraction of Hardy Rogers type if there exist two functions α: A×A → [0,∞), F ∈ F and a constant τ > 0 such that for each x1,x2,u1,u2 ∈ A and v1 ∈ Tx1, v2 ∈ Tx2 with α(x1,x2) ≥ 1 and d(u1,v1) = dist(A,B) = d(u2,v2), we have α(u1,u2) ≥ 1 and τ + F(d(u1,u2)) ≤ F(N(x1,x2)) (3.1) whenever min{d(u1,u2),N(x1,x2)} > 0, where N(x1,x2) = a1d(x1,x2) + a2d(x1,u1) + a3d(x2,u2) + a4[d(x1,u2) + d(x2,u1)] with a1,a2,a3,a4 ≥ 0 satisfying a1 + a2 + a3 + 2a4 = 1 and a3 6= 1. Remark 3.1. By taking F(x) = ln x for each x ∈ (0,∞), one can see that (3.1) reduces to (2.1). Therefore, (3.1) is a proper generalization/extension of (2.1). Theorem 3.1. Let A and B be nonempty closed subsets of a complete metric space (X,d). Assume that A0 is nonempty and T : A → CB(B) is an αF -proximal contraction of Hardy Rogers type and satisfying the following conditions: (i) for each x ∈ A0, we have Tx ⊆ B0; (ii) there exist x1,x2 ∈ A0 and v1 ∈ Tx1 such that α(x1,x2) ≥ 1 and d(x2,v1) = dist(A,B); (iii) T is continuous, or, for any sequence {xn} ⊆ A such that xn → x as n → ∞ and α(xn,xn+1) ≥ 1 for each n ∈ N, we have α(xn,x) ≥ 1 for each n ∈ N; (iv) B is approximatively compact with respect to A. Then T has a best proximity point. 200 KAMRAN, ALI, POSTOLACHE, GHIURA AND FARHEEN Proof. By hypothesis (ii), we have x1,x2 ∈ A0 and v1 ∈ Tx1 for which α(x1,x2) ≥ 1 and d(x2,v1) = dist(A,B). As v2 ∈ Tx2 ⊆ B0, there is x3 ∈ A0 satisfying d(x3,v2) = dist(A,B). From (3.1), we get α(x2,x3) ≥ 1 and τ + F(d(x2,x3)) ≤ F(a1d(x1,x2) + a2d(x1,x2) + a3d(x2,x3) + a4[d(x1,x3) + d(x2,x2)]) ≤ F(a1d(x1,x2) + a2d(x1,x2) + a3d(x2,x3) + a4[d(x1,x2) + d(x2,x3)]) = F((a1 + a2 + a4)d(x1,x2) + (a3 + a4)d(x2,x3)). (3.2) As F is strictly increasing, from (3.2), we get d(x2,x3) < (a1 + a2 + a4)d(x1,x2) + (a3 + a4)d(x2,x3). That is, (1 −a3 −a4)d(x2,x3) < (a1 + a2 + a4)d(x1,x2). As a1 + a2 + a3 + 2a4 = 1, the above inequality implies that d(x2,x3) < d(x1,x2). Thus by (3.2), we have τ + F(d(x2,x3)) ≤ F(d(x1,x2)). (3.3) From above we have x2,x3 ∈ A0 and v2 ∈ Tx2 satisfying α(x2,x3) ≥ 1 and d(x3,v2) = dist(A,B). As v3 ∈ Tx3 ⊆ B0, there is x4 ∈ A0 such that d(x4,v3) = dist(A,B). From (3.1), we get α(x3,x4) ≥ 1 and τ + F(d(x3,x4)) ≤ F(a1d(x2,x3) + a2d(x2,x3) + a3d(x3,x4) + a4[d(x2,x4) + d(x3,x3)]) ≤ F(a1d(x2,x3) + a2d(x2,x3) + a3d(x3,x4) + a4[d(x2,x3) + d(x3,x4)]) = F((a1 + a2 + a4)d(x2,x3) + (a3 + a4)d(x3,x4)). After simplification we get τ + F(d(x3,x4)) ≤ F(d(x2,x3)). (3.4) From (3.4) and (3.3), we obtain F(d(x3,x4)) ≤ F(d(x1,x2)) − 2τ. Continuing the same process we get sequences {xn} in A0 and {vn} in B0 such that vn ∈ Txn, α(xn,xn+1) ≥ 1, d(xn+1,vn) = dist(A,B) and F(d(xn,xn+1)) ≤ F(d(x1,x2)) −nτ for each n ∈ N\{1}. (3.5) Letting n → ∞ in (3.5), we get limn→∞F(d(xn,xn+1)) = −∞. Thus, by property (F2), we have limn→∞d(xn,xn+1) = 0. Let dn = d(xn,xn+1) for each n ∈ N. From (F3) there exists k ∈ (0, 1) such that lim n→∞ dknF(dn) = 0. From (3.5) we have dknF(dn) −d k nF(d1) ≤−d k nnτ ≤ 0 for each n ∈ N. (3.6) Letting n →∞ in (3.6), we get lim n→∞ ndkn = 0. This implies that there exists n1 ∈ N such that ndkn ≤ 1 for each n ≥ n1. Thus, we have dn ≤ 1 n1/k , for each n ≥ n1. (3.7) BEST PROXIMITY POINTS FOR A NEW CLASS OF GENERALIZED PROXIMAL MAPPINGS 201 To prove that {xn} is a Cauchy sequence in A, consider m,n ∈ N with m > n > n1. By using the triangular inequality and (3.7), we have d(xn,xm) ≤ d(xn,xn+1) + d(xn+1,xn+2) + · · · + d(xm−1,xm) = m−1∑ i=n di ≤ ∞∑ i=n di ≤ ∞∑ i=n 1 i1/k . Since ∑∞ i=1 1 i1/k is convergent series, we get limn→∞d(xn,xm) = 0, which implies that {xn} is a Cauchy sequence in A. Since A is closed subset of a complete metric space, there exists x∗ in A such that xn → x∗ as n → ∞. As d(xn+1,vn) = dist(A,B), we have limn→∞d(x∗,vn) = dist(A,B). As B is approximatively compact with respect to A, we get a subsequence {vnk} of {vn} with vnk ∈ Txnk that converges to v∗. Thus, d(x∗,v∗) = lim k→∞ d(xnk,vnk ) = dist(A,B). By hypothesis (iii), when T is continuous, we get v∗ ∈ Tx∗. Hence dist(A,B) ≤ d(x∗,Tx∗) ≤ d(x∗,v∗) = dist(A,B). This implies that dist(A,B) = d(x∗,Tx∗). Now we prove the theorem for second assumption of hypothesis (iii), that is, α(xn,x ∗) ≥ 1 for each n ∈ N. Since x∗ ∈ A0, then Tx∗ ⊆ B0. This implies that for z∗ ∈ Tx∗, we have w∗ ∈ A0 such that d(w∗,z∗) = dist(A,B). Further note that d(xn+1,vn) = dist(A,B). We claim that d(x∗,w∗) = 0. Suppose on contrary that d(x∗,w∗) 6= 0 Now from (3.1), we get d(xn+1,w ∗) < a1d(xn,x ∗) + a2d(xn,xn+1) + a3d(x ∗,w∗) + a4[d(xn,w ∗) + d(x∗,xn+1)]. Letting n →∞, we get d(x∗,w∗) ≤ (a3 + a4)d(x∗,w∗), which is only possible when d(x∗,w∗) = 0. Thus we get dist(A,B) ≤ d(x∗,Tx∗) ≤ d(x∗,z∗) = dist(A,B), and this completes the proof. � Example 3.1. Let X = R × R be endowed with a metric d((x1,x2), (y1,y2)) = |x1 −y1| + |x2 −y2| for each x,y ∈ X. Take A = {(0,x) : −1 ≤ x ≤ 1} and B = {(1,x) : −1 ≤ x ≤ 1}. Define T : A → CB(B), T(0,x) =   {( 1, x + 1 2 )} if x ≥ 0 {(1,x), (1,x2)} otherwise, and α: A×A → [0,∞), α((0,x), (0,y)) = { 1 if x,y ∈ [0, 1] 0 otherwise. Take F(x) = ln x for each x ∈ (0,∞) and τ = 1 2 . It is easy to see that T is αF -proximal contraction of Hardy Rogers type with a0 = 1 and a2 = a3 = a4 = 0. For each x ∈ A0, we have Tx ⊆ B0. Also for x1 = (0, 1 2 ) ∈ A0 and v1 = (1, 34 ) ∈ Tx1, we have x2 = (0, 3 4 ) such that α(x1,x2) = 1 and d(x2,v1) = dist(A,B). Moreover, for any sequence {xn} ⊆ A such that xn → x as n → ∞ and α(xn,xn+1) = 1 for each n ∈ N, we have α(xn,x) = 1 for each n ∈ N. Further note that B is approximatively compact with respect to A, therefore, by Theorem 3.1, T has a best proximity point. Remark 3.2. Note that Theorem 2.1 is not applicable in the above example. Therefore, our theorem properly generalizes/extends Theorem 2.1. Definition 3.2. Let A and B be two nonempty subsets of a metric space (X,d). A mapping T : A → CB(B) is called αF -proximal contraction of Ciric type if there exist two functions α: A×A → [0,∞), continuous F in F and a constant τ > 0 such that for each x1,x2,u1,u2 ∈ A and v1 ∈ Tx1, v2 ∈ Tx2 with α(x1,x2) ≥ 1 and d(u1,v1) = dist(A,B) = d(u2,v2), we have α(u1,u2) ≥ 1 and τ + F(d(u1,u2)) ≤ F(M(x1,x2)) (3.8) 202 KAMRAN, ALI, POSTOLACHE, GHIURA AND FARHEEN whenever min{d(u1,u2),M(x1,x2)} > 0, where M(x1,x2) = max { d(x1,x2),d(x1,u1),d(x2,u2), d(x1,u2) + d(x2,u1) 2 } . Theorem 3.2. Let A and B be nonempty closed subsets of a complete metric space (X,d). Assume that A0 is nonempty and T : A → CB(B) is an αF -proximal contraction of Ciric type satisfying the following conditions: (i) for each x ∈ A0, we have Tx ⊆ B0; (ii) there exist x1,x2 ∈ A0 and v1 ∈ Tx1 such that α(x1,x2) ≥ 1 and d(x2,v1) = dist(A,B); (iii) T is continuous, or, for any sequence {xn} ⊆ A such that xn → x as n → ∞ and α(xn,xn+1) ≥ 1 for each n ∈ N, we have α(xn,x) ≥ 1 for each n ∈ N; (iv) B is approximatively compact with respect to A. Then T has a best proximity point. Proof. By hypothesis (ii), we have x1,x2 ∈ A0 and v1 ∈ Tx1 for which α(x1,x2) ≥ 1 and d(x2,v1) = dist(A,B). As v2 ∈ Tx2 ⊆ B0, there is x3 ∈ A0 satisfying d(x3,v2) = dist(A,B). From (3.8), we get α(x2,x3) ≥ 1 and τ + F(d(x2,x3)) ≤ F ( max { d(x1,x2),d(x1,x2),d(x2,x3), d(x1,x3) + d(x2,x2) 2 }) = F ( max{d(x1,x2),d(x2,x3)} ) = F(d(x1,x2)), (3.9) otherwise we have a contradiction. From above we have x2,x3 ∈ A0 and v2 ∈ Tx2 satisfying α(x2,x3) ≥ 1 and d(x3,v2) = dist(A,B). As v3 ∈ Tx3 ⊆ B0, there is x4 ∈ A0 such that d(x4,v3) = dist(A,B). From (3.8), we get α(x3,x4) ≥ 1 and τ + F(d(x3,x4)) ≤ F ( max { d(x2,x3),d(x2,x3),d(x3,x4), d(x2,x4) + d(x3,x3) 2 }) = F ( max{d(x2,x3),d(x3,x4)} ) = F(d(x2,x3)), (3.10) otherwise we have a contradiction. From (3.9) and (3.10), we have F(d(x3,x4)) ≤ F(d(x1,x2)) − 2τ. Continuing the same process we get sequences {xn} in A0 and {vn} in B0 such that vn ∈ Txn, α(xn,xn+1) ≥ 1, d(xn+1,vn) = dist(A,B) and F(d(xn,xn+1)) ≤ F(d(x1,x2)) −nτ for each n ∈ N−{1}. Working on the same lines as the proof of Theorem 3.1 is done. We prove that {xn} is a Cauchy sequence in A. Since A is closed subset of a complete metric space, there exists x∗ in A such that xn → x∗ as n → ∞. As d(xn+1,vn) = dist(A,B). Thus, we have limn→∞d(x∗,vn) = dist(A,B). Since B is approximatively compact with respect to A, we get a subsequence {vnk} of {vn} with vnk ∈ Txnk that converges to v∗. Thus, d(x∗,v∗) = lim k→∞ d(xnk+1,vnk ) = dist(A,B). By hypothesis (iii), when T is continuous, we get v∗ ∈ Tx∗. Hence dist(A,B) ≤ d(x∗,Tx∗) ≤ d(x∗,v∗) = dist(A,B). Now assume that we have α(xn,x ∗) ≥ 1 for each n ∈ N. Since x∗ ∈ A0, then BEST PROXIMITY POINTS FOR A NEW CLASS OF GENERALIZED PROXIMAL MAPPINGS 203 Tx∗ ⊆ B0. This implies that for z∗ ∈ Tx∗, we have w∗ ∈ A0 such that d(w∗,z∗) = dist(A,B). Further note that d(xn+1,vn) = dist(A,B). We claim that d(x∗,w∗) = 0. On contrary assume that d(x∗,w∗) 6= 0. Now, from (3.8), we get τ + F(d(xn+1,w ∗)) < F ( max { d(xn,x ∗),d(xn,xn+1),d(x ∗,w∗), d(xn,w ∗) + d(xn+1,x ∗) 2 ) . Letting n →∞, we obtain τ + F(d(x∗,w∗)) ≤ F(d(x∗,w∗)), which is not possible. Hence, we have d(x∗,w∗) = 0. Thus we get dist(A,B) ≤ d(x∗,Tx∗) ≤ d(x∗,z∗) = dist(A,B), and this completes the proof. � Example 3.2. Let X = R × R be endowed with a metric d((x1,x2), (y1,y2)) = |x1 −y1| + |x2 −y2| for each x,y ∈ X. Take A = {(0,x) : −1 ≤ x ≤ 1} and B = {(1,x) : −1 ≤ x ≤ 1}. Define T : A → CB(B), T(0,x) = { {(1, x 3 ), (1, x 2 )} if x ≥ 0 {(1,x), (1,x2)} otherwise, and α: A×A → [0,∞) α((0,x), (0,y)) = { 1 if x,y ∈ [0, 1] 0 otherwise. Take F(x) = ln x for each x ∈ (0,∞) and τ = 1 2 . It is easy to see that T is αF -proximal contraction of Ciric type. For each x ∈ A0, we have Tx ⊆ B0. Also for x1 = (0, 13 ) ∈ A0 and v1 = (1, 1 6 ) ∈ Tx1, we have x2 = (0, 1 6 ) such that α(x1,x2) = 1 and d(x2,v1) = dist(A,B). Moreover, for any sequence {xn} ⊆ A such that xn → x as n → ∞ and α(xn,xn+1) = 1 for each n ∈ N, we have α(xn,x) = 1 for each n ∈ N. Further, note that B is approximatively compact with respect to A. Therefore, by Theorem 3.2, T has a best proximity point. 4. Consequences By taking α(x,y) = 1 for each x,y ∈ A, the following two theorems immediately follow from our results. Theorem 4.1. Let A and B be nonempty closed subsets of a complete metric space (X,d). Assume that A0 is nonempty and T : A → CB(B) is a mapping for which there exist a function F ∈ F and a constant τ > 0 such that for each x1,x2,u1,u2 ∈ A and v1 ∈ Tx1, v2 ∈ Tx2 with d(u1,v1) = dist(A,B) = d(u2,v2), we have τ + F(d(u1,u2)) ≤ F(N(x1,x2)) whenever min{d(u1,u2),N(x1,x2)} > 0, where N(x1,x2) = a1d(x1,x2) + a2d(x1,u1) + a3d(x2,u2) + a4[d(x1,u2) + d(x2,u1)] with a1,a2,a3,a4 ≥ 0 satisfying a1 + a2 + a3 + 2a4 = 1 and a3 6= 1. Further assume that the following conditions hold: (i) for each x ∈ A0, we have Tx ⊆ B0; (ii) B is approximatively compact with respect to A. Then T has a best proximity point. Theorem 4.2. Let A and B be nonempty closed subsets of a complete metric space (X,d). Assume that A0 is nonempty and T : A → CB(B) is a mapping for which there exist a continuous function F ∈ F and a constant τ > 0 such that for each x1,x2,u1,u2 ∈ A and v1 ∈ Tx1, v2 ∈ Tx2 with d(u1,v1) = dist(A,B) = d(u2,v2), we have τ + F(d(u1,u2)) ≤ F(M(x1,x2)) 204 KAMRAN, ALI, POSTOLACHE, GHIURA AND FARHEEN whenever min{d(u1,u2),M(x1,x2)} > 0, where M(x1,x2) = max { d(x1,x2),d(x1,u1),d(x2,u2), d(x1,u2) + d(x2,u1) 2 } . Further assume that the following conditions hold: (i) for each x ∈ A0, we have Tx ⊆ B0; (ii) B is approximatively compact with respect to A. Then T has a best proximity point. When we take X = A = B, we get the following fixed point theorems from our results: Theorem 4.3. Let (X,d) be a complete metric space. Assume T : X → CB(X) is a mapping for which there are two functions α: A × A → [0,∞), F ∈ F and a constant τ > 0 such that for each x1,x2 ∈ X and u1 ∈ Tx1, u2 ∈ Tx2 with α(x1,x2) ≥ 1, we have α(u1,u2) ≥ 1 and τ + F(d(u1,u2)) ≤ F(N(x1,x2)) whenever min{d(u1,u2),N(x1,x2)} > 0, where N(x1,x2) = a1d(x1,x2) + a2d(x1,u1) + a3d(x2,u2) + a4[d(x1,u2) + d(x2,u1)] with a1,a2,a3,a4 ≥ 0 satisfying a1 +a2 +a3 +2a4 = 1 and a3 6= 1. Further assume that T is continuous, or, for any sequence {xn}⊆ X such that xn → x as n →∞ and α(xn,xn+1) ≥ 1 for each n ∈ N, we have α(xn,x) ≥ 1 for each n ∈ N. Then T has a fixed point. Theorem 4.4. Let (X,d) be a complete metric space. Assume T : X → CB(X) is a mapping for which there is α: A×A → [0,∞), continuous function, F in F and τ > 0 such that for each x1,x2 ∈ X and u1 ∈ Tx1, u2 ∈ Tx2 with α(x1,x2) ≥ 1, we have α(u1,u2) ≥ 1 and τ + F(d(u1,u2)) ≤ F(M(x1,x2)) whenever min{d(u1,u2),M(x1,x2)} > 0, where M(x1,x2) = max { d(x1,x2),d(x1,u1),d(x2,u2), d(x1,u2) + d(x2,u1) 2 } . Further assume that T is continuous, or, for any sequence {xn}⊆ X such that xn → x as n →∞ and α(xn,xn+1) ≥ 1 for each n ∈ N, we have α(xn,x) ≥ 1 for each n ∈ N. Then T has a fixed point. References [1] K. Fan, Extensions of two fixed point theorems of F. E. Browder. Math. Z., 112 (1969), 234-240. [2] A. Eldred, P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl., 323 (2006), 1001-1006. [3] M. Jleli, B. Samet, Best proximity point for α-ψ-proximal contraction type mappings and applications, Bull. Sci. Math., 137 (2013), 977-995. [4] M. U. Ali, T. Kamran, N. Shahzad, Best proximity point for α-ψ-proximal contractive multimaps, Abstr. Appl. Anal., 2014 (2014), Art. ID 181598. [5] A. Abkar, M. Gabeleh, Best proximity points for asymptotic cyclic contraction mappings, Nonlinear Anal., 74 (2011), 7261-7268. [6] A. Abkar, M. Gabeleh, Best proximity points for cyclic mappings in ordered metric spaces, J. Optim. Theory Appl., 151 (2011), 418-424. [7] M. Derafshpour, S. Rezapour, N. Shahzad, Best proximity points of cyclic ϕ-contractions in ordered metric spaces, Topol. Meth. Nonlin. Anal., 37 (2011), 193-202. [8] C. Di Bari, T. Suzuki, C. Vetro, Best proximity point for cyclic Meir-Keeler contraction, Nonlinear Anal., 69 (2008), 3790-3794. [9] S. Rezapour, M. Derafshpour, N. Shahzad, Best proximity points of cyclic φ-contractions on reflexive Banach spaces, Fixed Point Theory Appl., 2010 (2010), Art. ID 946178. [10] C. Vetro, Best proximity points: convergence and existence theorems for p-cyclic mappings, Nonlinear Anal. 73 (7) (2010), 2283-2291. [11] M. A. Alghamdi, M. A. Alghamdi, N. Shahzad, Best proximity point results in geodesic metric spaces, Fixed Point Theory Appl., 2012 (2012), Art. ID 234. [12] M. A. Al-Thagafi, N. Shahzad, Best proximity pairs and equilibrium pairs for Kakutani multimaps, Nonlinear Anal., 70 (3) (2009), 1209-1216. [13] J. Markin, N. Shahzad, Best proximity points for relatively u-continuous mappings in Banach and hyperconvex spaces, Abstr. Appl. Anal. 2013 (2013), Art. ID 680186. BEST PROXIMITY POINTS FOR A NEW CLASS OF GENERALIZED PROXIMAL MAPPINGS 205 [14] H. K. Nashine, P. Kumam, C. Vetro, Best proximity point theorems for rational proximal contractions, Fixed Point Theory Appl., 2013 (2013), Art. ID 95. [15] A. Abkar, M. Gabeleh, The existence of best proximity points for multivalued non-self mappings, RACSAM, 107 (2) (2013), 319-325. [16] B. S. Choudhury, N. Metiya, M. Postolache, P. Konar, A discussion on best proximity point and coupled best proximity point in partially ordered metric spaces. Fixed Point Theory Appl., 2015 (2015), Art. ID 170. [17] W. Shatanawi, A. Pitea, Best proximity point and best proximity coupled point in a complete metric space with (P)-property, Filomat, 29 (1) (2015), 63-74. [18] M. Jamali, S.M. Vaezpour, Best proximity point for certain nonlinear contractions in Menger probabilistic metric spaces, J. Adv. Math. Stud., 9 (2) (2016), 338-347. [19] A. Bejenaru, A. Pitea, Fixed point and best proximity point theorems in partial metric spaces, J. Math. Anal., 7 (4) (2016), 25-44. [20] S. S. Basha, N. Shahzad, Best proximity point theorems for generalized proximal contractions, Fixed Point Theory Appl., 2012 (2012), Art. ID 42. [21] D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl., 2012 (2012), Art. ID 94. [22] M. Cosentino, P. Vetro, Fixed point results for F-contractive mappings of Hardy-Rogers-type, Filomat 28 (4) (2014), 715-722. [23] G. Minak, A. Helvac, I. Altun, Ciric type generalized F-contractions on complete metric spaces and fixed point results, Filomat, 28 (6) (2014), 1143-1151. [24] M. Sgroi, C. Vetro, Multi-valued F-contractions and the solution of certain functional and integral equations, Filomat 27 (7) (2013), 1259-1268. [25] D. Paesano, C. Vetro, Multi-valued F-contractions in 0-complete partial metric spaces with application to Volterra type integral equation,Rev. R. Acad. Cienc. Exactas Fs. Nat., Ser. A Mat., 108 (2) (2014), 1005-1020. [26] H. Piri, P. Kumam, Some fixed point theorems concerning F-contraction in complete metric spaces, Fixed Point Theory Appl., 2014 (2014), Art. ID 210. [27] O. Acar, I. Altun, A fixed point theorem for multivalued mappings with δ-distance, Abstr. Appl. Anal., 2014 (2014), Art. ID 497092. [28] R. Batra, S. Vashistha, Fixed points of an F-contraction on metric spaces with a graph, Inter. J. Comput. Math., 91 (12) (2014), 2483-2490. 1Department of Mathematics, Quaid-i-Azam University, Islamabad-Pakistan 2Department of Mathematics, School of Natural Sciences, National University of Sciences and Tech- nology Islamabad-Pakistan 3Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan 4Department of Mathematics & Computer Science, University Politehnica of Bucharest, 313 Splaiul Independenţei, 060042 Bucharest, Romania ∗Corresponding author: mi.postolache@mail.cmuh.org.tw, mihai@mathem.pub.ro 1. Introduction 2. Previous results 3. Main results 4. Consequences References