Coupled best proximity point theorem in metric Spaces

  • Wutiphol Sintunavarat1 and

    Affiliated with

    • Poom Kumam1Email author

      Affiliated with

      Fixed Point Theory and Applications20122012:93

      DOI: 10.1186/1687-1812-2012-93

      Received: 4 November 2011

      Accepted: 7 June 2012

      Published: 7 June 2012

      Abstract

      In this article the concept of coupled best proximity point and cyclic contraction pair are introduced and then we study the existence and convergence of these points in metric spaces. We also establish new results on the existence and convergence in a uniformly convex Banach spaces. Furthermore, we give new results of coupled fixed points in metric spaces and give some illustrative examples. An open problems are also given at the end for further investigation.

      1 Introduction

      The Banach contraction principle [1] states that if (X, d) is a complete metric space and T : XX is a contraction mapping (i.e., d(Tx, Ty) ≤ αd(x, y) for all x, yX, where α is a non-negative number such that α < 1), then T has a unique fixed point. This principle has been generalized in many ways over the years [215].

      One of the most interesting is the study of the extension of Banach contraction principle to the case of non-self mappings. In fact, given nonempty closed subsets A and B of a complete metric space (X, d), a contraction non-self-mapping T : AB does not necessarily has a fixed point.

      Eventually, it is quite natural to find an element x such that d(x, Tx) is minimum for a given problem which implies that x and Tx are in close proximity to each other.

      A point x in A for which d(x, Tx) = d(A, B) is call a best proximity point of T . Whenever a non-self-mapping T has no fixed point, a best proximity point represent an optimal approximate solution to the equation Tx = x. Since a best proximity point reduces to a fixed point if the underlying mapping is assumed to be self-mappings, the best proximity point theorems are natural generalizations of the Banach contraction principle.

      In 1969, Fan [16] introduced and established a classical best approximation theorem, that is, if A is a nonempty compact convex subset of a Hausdorff locally convex topological vector space B and T : AB is a continuous mapping, then there exists an element xA such that d(x, Tx) = d(Tx, A). Afterward, many authors have derived extensions of Fan's Theorem and the best approximation theorem in many directions such as Prolla [17], Reich [18], Sehgal and Singh [19, 20], Wlodarczyk and Plebaniak [2124], Vetrivel et al. [25], Eldred and Veeramani [26], Mongkolkeha and Kumam [27] and Sadiq Basha and Veeramani [2831].

      On the other hand, Bhaskar and Lakshmikantham [32] introduced the notions of a mixed monotone mapping and proved some coupled fixed point theorems for mappings satisfying the mixed monotone property. They have observation that their theorem can be used to investigate a large class of problems and discussed the existence and uniqueness of a solution for a periodic boundary value problem. For several improvements and generalizations see in [3336] and reference therein.

      The purpose of this article is to first introduce the notion of coupled best proximity point and cyclic contraction pair. We also establish the existence and convergence theorem of coupled best proximity points in metric spaces. Moreover, we apply this results in uniformly convex Banach space. We also study some results on the existence and convergence of coupled fixed point in metric spaces and give illustrative examples of our theorems. An open problem are also given at the end for further investigations.

      2 Preliminaries

      In this section, we give some basic definitions and concepts related to the main results of this article. Throughout this article we denote by ℕ the set of all positive integers and by ℝ the set of all real numbers. For nonempty subsets A and B of a metric space (X, d), we let
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equa_HTML.gif

      stands for the distance between A and B.

      A Banach space X is said to be

      (1) strictly convex if the following implication holds for all x, yX:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equb_HTML.gif
      (2) uniformly convex if for each ε with 0 < ε ≤ 2, there exists δ > 0 such that the following implication holds for all x, yX:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equc_HTML.gif

      It easily to see that a uniformly convex Banach space X is strictly convex but the converse is not true.

      Definition 2.1. [37] Let A and B be nonempty subsets of a metric space (X, d). The ordered pair (A, B) satisfies the property UC if the following holds:

      If {x n } and {z n } are sequences in A and {y n } is a sequence in B such that d(x n , y n ) → d(A, B) and d(z n , y n ) → d(A, B), then d(x n , z n ) → 0.

      Example 2.2. [37]The following are examples of a pair of nonempty subsets (A, B) satisfying the property UC.

      (1) Every pair of nonempty subsets A, B of a metric space (X, d) such that d(A, B) = 0.

      (2) Every pair of nonempty subsets A, B of a uniformly convex Banach space X such that A is convex.

      (3) Every pair of nonempty subsets A, B of a strictly convex Banach space which A is convex and relatively compact and the closure of B is weakly compact.

      Definition 2.3. Let A and B be nonempty subsets of a metric space (X, d) and T : AB be a mapping. A point xA is said to be a best proximity point of T if it satisfies the condition that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equd_HTML.gif

      It can be observed that a best proximity point reduces to a fixed point if the underlying mapping is a self-mapping.

      Definition 2.4. [32] Let A be a nonempty subset of a metric space X and F : A X AA. A point (x, x') ∈ A × A is called a coupled fixed point of F if
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Eque_HTML.gif

      3 Coupled best proximity point theorem

      In this section, we study the existence and convergence of coupled best proximity points for cyclic contraction pairs. We begin by introducing the notion of property UC* and a coupled best proximity point.

      Definition 3.1. Let A and B be nonempty subsets of a metric space (X, d). The ordered pair (A, B) satisfies the property UC* if (A, B) has property UC and the following condition holds:

      If {x n } and {z n } are sequences in A and {y n } is a sequence in B satisfying:

      (1) d(z n , y n ) → d(A, B).

      (2) For every ε > 0 there exists N ∈ ℕ such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equf_HTML.gif

      for all m > nN,

      then, for every ε > 0 there exists N 1 ∈ ℕ such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equg_HTML.gif

      for all m > nN 1.

      Example 3.2. The following are examples of a pair of nonempty subsets (A, B) satisfying the property UC*.

      (1) Every pair of nonempty subsets A, B of a metric space (X, d) such that d(A, B) = 0.

      (2) Every pair of nonempty closed subsets A, B of a uniformly convex Banach space X such that A is convex [[38], Lemma 3.7].

      Definition 3.3. Let A and B be nonempty subsets of a metric space X and F : A × AB. A point (x, x') ∈ A × A is called a coupled best proximity point of F if
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equh_HTML.gif

      It is easy to see that if A = B in Definition 3.3, then a coupled best proximity point reduces to a coupled fixed point.

      Next, we introduce the notion of a cyclic contraction for a pair of two binary mappings.

      Definition 3.4. Let A and B be nonempty subsets of a metric space X, F : A × AB and G : B × BA. The ordered pair (F, G) is said to be a cyclic contraction if there exists a non-negative number α < 1 such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equi_HTML.gif

      for all (x, x') ∈ A × A and (y, y') ∈ B × B.

      Note that if (F, G) is a cyclic contraction, then (G, F ) is also a cyclic contraction.

      Example 3.5. Let X = ℝ with the usual metric d(x, y) = |x - y| and let A = [2,4] and B = [-4, -2]. It easy to see that d(A, B) = 4. Define F : A × AB and G : B × BA by
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equj_HTML.gif
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equk_HTML.gif
      For arbitrary (x, x') ∈ A × A and (y, y') ∈ B × B and fixed http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq1_HTML.gif , we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equl_HTML.gif

      This implies that (F, G) is a cyclic contraction with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq2_HTML.gif .

      Example 3.6. Let X = ℝ2 with the metric d((x, y), (x', y')) = max{|x - x'|, |y - y'|} and let A = {(x, 0): 0 ≤ x ≤ 1} and B = {(x, 1): 0 ≤ x ≤ 1}. It easy to prove that d(A, B) = 1. Define F : A × AB and G : B × BA by
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equm_HTML.gif
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equn_HTML.gif
      We obtain that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equo_HTML.gif
      Also for all α > 0, we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equp_HTML.gif

      This implies that (F, G) is cyclic contraction.

      The following lemma plays an important role in our main results.

      Lemma 3.7. Let A and B be nonempty subsets of a metric space X, F : A × AB, G : B × BA and (F, G) be a cyclic contraction. If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq3_HTML.gif and we define
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equq_HTML.gif
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equr_HTML.gif

      for all n ∈ ℕ ∪ {0}, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq4_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq5_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq6_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq7_HTML.gif .

      Proof. For each n ∈ ℕ ∪ {0}, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equs_HTML.gif
      By induction, we see that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equt_HTML.gif
      Taking n, we obtain
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equ1_HTML.gif
      (3.1)
      For each n ∈ ℕ ∪ {0}, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equu_HTML.gif
      By induction, we see that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equv_HTML.gif
      Setting n, we obtain
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equ2_HTML.gif
      (3.2)

      By similar argument, we also have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq8_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq9_HTML.gif for all n ∈ ℕ ∪ {0}.    □

      Lemma 3.8. Let A and B be nonempty subsets of a metric space X such that (A, B) and (B, A) have a property UC, F : A × AB, G : B × BA and let the ordered pair (F, G) is a cyclic contraction. If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq10_HTML.gif and define
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equw_HTML.gif
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equx_HTML.gif
      for all n ∈ ℕ ∪ {0}, then for ε > 0, there exists a positive integer N 0 such that for all m > nN 0,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equ3_HTML.gif
      (3.3)
      Proof. By Lemma 3.7, we have d(x 2n , x 2n+1) → d(A, B) and d(x 2n+1, x 2n+2) → d(A, B). Since (A, B) has a property UC, we get d(x 2n , x 2n+2) → 0. A similar argument shows that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq11_HTML.gif . As (B, A) has a property UC, we also have d(x 2n+1, x 2n+3) → 0 and . Suppose that (3.3) does not hold. Then there exists ε' > 0 such that for all k ∈ ℕ, there is m k > n k k satisfying
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equy_HTML.gif
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equz_HTML.gif
      Therefore, we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equaa_HTML.gif
      Letting k, we obtain to see that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equ4_HTML.gif
      (3.4)
      By using the triangle inequality we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equab_HTML.gif
      Taking k, we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equac_HTML.gif

      which contradicts. Therefore, we can conclude that (3.3) holds.    □

      Lemma 3.9. Let A and B be nonempty subsets of a metric space X, (A, B) and (B, A) satisfy the property UC*. Let F : A × AB, G : B × BA and (F, G) be a cyclic contraction. If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq13_HTML.gif and define
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equad_HTML.gif
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equae_HTML.gif

      for all n ∈ ℕ ∪ {0}, then {x 2n }, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq14_HTML.gif , {x 2n+1} and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq15_HTML.gif are Cauchy sequences.

      Proof. By Lemma 3.7, we have d(x 2n , x 2n+1) → d(A, B) and d(x 2n+1, x 2n+2) → d(A, B). Since (A, B) has a property UC*, we get d(x 2n , x 2n+2) → 0. As (B, A) has a property UC*, we also have d(x 2n+1, x 2n+3) → 0.

      We now show that for every ε > 0 there exists N such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equ5_HTML.gif
      (3.5)

      for all m > nN.

      Suppose (3.5) not, then there exists ε > 0 such that for all k ∈ ℕ there exists m k > n k k such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equ6_HTML.gif
      (3.6)
      Now we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equaf_HTML.gif

      Taking k, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq16_HTML.gif .

      By Lemma 3.8, there exists N ∈ ℕ such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equ7_HTML.gif
      (3.7)
      for all m > nN. By using the triangle inequality we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equag_HTML.gif
      Taking k, we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equah_HTML.gif

      which contradicts. Therefore, condition (3.5) holds. Since (3.5) holds and d(x 2n , x 2n+1) → d(A, B), by using property UC* of (A, B), we have {x 2n } is a Cauchy sequence. In similar way, we can prove that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq17_HTML.gif , {x 2n+1} and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq18_HTML.gif are Cauchy sequences.    □

      Here we state the main results of this article on the existence and convergence of coupled best proximity points for cyclic contraction pairs on nonempty subsets of metric spaces satisfying the property UC*.

      Theorem 3.10. Let A and B be nonempty closed subsets of a complete metric space X such that (A, B) and (B, A) satisfy the property UC*. Let F : A × AB, G : B × BA and (F, G) be a cyclic contraction. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq19_HTML.gif and define
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equai_HTML.gif
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equaj_HTML.gif
      for all n ∈ ℕ ∪ {0}. Then F has a coupled best proximity point (p, q) ∈ A × A and G has a coupled best proximity point (p', q') ∈ B × B such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equak_HTML.gif

      Moreover, we have x 2n p, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq20_HTML.gif , x 2n+1 p' and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq21_HTML.gif .

      Proof. By Lemma 3.7, we get d(x 2n , x 2n+1) → d(A, B). Using Lemma 3.9, we have {x 2n } and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq22_HTML.gif are Cauchy sequences. Thus, there exists p, qA such that x 2n p and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq23_HTML.gif . We obtain that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equ8_HTML.gif
      (3.8)
      Letting n in (3.8), we have d(p, x 2n-1) → d(A, B). By a similar argument we also have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq24_HTML.gif . It follows that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equal_HTML.gif

      Taking n, we get d(p, F (p, q)) = d(A, B). Similarly, we can prove that d(q, F (q, p)) = d(A, B). Therefore, we have (p, q) is a coupled best proximity point of F.

      In similar way, we can prove that there exists p', q' ∈ B such that x 2n+1 p' and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq25_HTML.gif . Moreover, we also have d(p', G(p', q')) = d(A, B) and d(q', G(q', p')) = d(A, B) and so (p', q') is a coupled best proximity point of G.

      Finally, we show that d(p, p') + d(q, q') = 2d(A, B). For n ∈ ℕ ∪ {0}, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equam_HTML.gif
      Letting n, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equ9_HTML.gif
      (3.9)
      For n ∈ ℕ ∪ {0}, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equan_HTML.gif
      Letting n, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equ10_HTML.gif
      (3.10)
      It follows from (3.9) and (3.10) that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equao_HTML.gif
      which implies that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equ11_HTML.gif
      (3.11)
      Since d(A, B) ≤ d(p, p') and d(A, B) ≤ d(q, q'), we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equ12_HTML.gif
      (3.12)
      From (3.11) and (3.12), we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equap_HTML.gif

      This complete the proof.    □

      Note that every pair of nonempty closed subsets A, B of a uniformly convex Banach space X such that A is convex satisfies the property UC*. Therefore, we obtain the following corollary.

      Corollary 3.11. Let A and B be nonempty closed convex subsets of a uniformly convex Banach space X, F : A × AB, G : B × BA and (F, G) be a cyclic contraction. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq26_HTML.gif and define
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equaq_HTML.gif
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equar_HTML.gif
      for all n ∈ ℕ ∪ {0}. Then F has a coupled best proximity point (p, q) ∈ A × A and G has a coupled best proximity point (p', q') ∈ B × B such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equas_HTML.gif

      Moreover, we have x 2n p, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq27_HTML.gif , x 2n+1 p' and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq28_HTML.gif .

      Next, we give some illustrative example of Corollary 3.11.

      Example 3.12. Consider uniformly convex Banach space X = ℝ with the usual norm. Let A = [1,2] and B = [-2, -1]. Thus d(A, B) = 2. Define F : A × AB and G : B × BA by
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equat_HTML.gif
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equau_HTML.gif
      For arbitrary (x, x') ∈ A × A and (y, y') ∈ B × B and fixed http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq29_HTML.gif , we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equav_HTML.gif
      This implies that (F, G) is a cyclic contraction with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq30_HTML.gif . Since A and B are convex, we have (A, B) and (B, A) satisfy the property UC*. Therefore, all hypothesis of Corollary 3.11 hold. So F has a coupled best proximity point and G has a coupled best proximity point. We note that a point (1, 1) ∈ A × A is a unique coupled best proximity point of F and a point (-1, -1) ∈ B × B is a unique coupled best proximity point of G. Furthermore, we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equaw_HTML.gif
      Theorem 3.13. Let A and B be nonempty compact subsets of a metric space X, F : A×AB, G : B × BA and (F, G) be a cyclic contraction pair. If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq31_HTML.gif and define
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equax_HTML.gif
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equay_HTML.gif
      for all n ∈ ℕ ∪ {0}, then F has a coupled best proximity point (p, q) ∈ A × A and G has a coupled best proximity point (p', q') ∈ B × B such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equaz_HTML.gif
      Proof. Since x 0, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq32_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equba_HTML.gif
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equbb_HTML.gif
      for all n ∈ ℕ ∪ {0}, we have x 2n , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq33_HTML.gif and x 2n+1, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq34_HTML.gif for all n ∈ ℕ ∪ {0}. As A is compact, the sequence {x 2n } and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq35_HTML.gif have convergent subsequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq36_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq37_HTML.gif , respectively, such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equbc_HTML.gif
      Now, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equ13_HTML.gif
      (3.13)
      By Lemma 3.7, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq38_HTML.gif . Taking k in (3.13), we get http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq39_HTML.gif . By a similar argument we observe that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq40_HTML.gif . Note that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equbd_HTML.gif

      Taking k, we get d(p, F (p, q)) = d(A, B). Similarly, we can prove that d(q, F(q, p)) = d(A, B). Thus F has a coupled best proximity (p, q) ∈ A × A. In similar way, since B is compact, we can also prove that G has a coupled best proximity point in (p', q') ∈ B × B. For d(p, p') + d(q, q') = 2d(A, B) similar to the final step of the proof of Theorem 3.10. This complete the proof.    □

      4 Coupled fixed point theorem

      In this section, we give the new coupled fixed point theorem for a cyclic contraction pair.

      Theorem 4.1. Let A and B be nonempty closed subsets of a complete metric space X, F : A × AB, G : B × BA and (F, G) be a cyclic contraction. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq41_HTML.gif and define
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Eqube_HTML.gif
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equbf_HTML.gif

      for all n ∈ ℕ ∪ {0}. If d(A, B) = 0, then F and G have a unique common coupled fixed point (p, q) ∈ A ∩ B × A ∩ B. Moreover, we have x 2n p, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq42_HTML.gif , x 2n+1 p and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq43_HTML.gif .

      Proof. Since d(A, B) = 0, we get (A, B) and (B, A) have the property UC*. Therefore, by Theorem 3.10 claim that F has a coupled best proximity point (p, q) ∈ A × A that is
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equ14_HTML.gif
      (4.1)
      and G has a coupled best proximity point (p', q') ∈ B × B that is
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equ15_HTML.gif
      (4.2)
      Moreover, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equ16_HTML.gif
      (4.3)
      From (4.1) and d(A, B) = 0, we conclude that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equbg_HTML.gif
      that is (p, q) is a coupled fixed point of F . It follows from (4.2) and d(A, B) = 0, we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equbh_HTML.gif
      that is (p', q') is a coupled fixed point of G. Using (4.3) and the fact that d(A, B) = 0, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equbi_HTML.gif

      which implies that p = p' and q = q'. Therefore, we conclude that (p, q) ∈ A ∩ B × A ∩ B is a common coupled fixed point of F and G.

      Finally, we show the uniqueness of common coupled fixed point of F and G. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq44_HTML.gif be another common coupled fixed point of F and G. So http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq45_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq46_HTML.gif . Now, we obtain that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equ17_HTML.gif
      (4.4)
      and also
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equ18_HTML.gif
      (4.5)
      It follows from (4.4) and (4.5) that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equbj_HTML.gif

      which implies that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq47_HTML.gif and so http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq48_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq49_HTML.gif . Therefore, (p, q) is a unique common coupled fixed point in A ∩ B × A ∩ B.    □

      Example 4.2. Consider X = ℝ with the usual metric, A = [-1, 0] and B = [0,1] . Define F : A × AB by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq50_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq51_HTML.gif . Then d(A, B) = 0 and (F, G) is a cyclic contraction with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq52_HTML.gif . Indeed, for arbitrary (x, x') ∈ A × A and (y, y') ∈ B × B, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equbk_HTML.gif

      Therefore, all hypothesis of Theorem 4.1 hold. So F and G have a unique common coupled fixed point and this point is (0, 0) ∈ A ∩ B × A ∩ B.

      If we take A = B in Theorem 4.1, then we get the following results.

      Corollary 4.3. Let A be nonempty closed subsets of a complete metric space X, F : A×AA and G : A×AA and let the order pair (F, G) is a cyclic contraction. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq53_HTML.gif and define
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equbl_HTML.gif
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equbm_HTML.gif

      for all n ∈ ℕ ∪ {0}. Then F and G have a unique common coupled fixed point (p, q) ∈ A×A. Moreover, we have x 2n p, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq54_HTML.gif , x 2n+1 p and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq55_HTML.gif

      We take F = G in Corollary 4.3, then we get the following results.

      Corollary 4.4. Let A be nonempty closed subsets of a complete metric space X, F : A×AA and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equ19_HTML.gif
      (4.6)

      for all (x, x'), (y, y') ∈ A × A. Then F has a unique coupled fixed point (p, q) ∈ A × A.

      Example 4.5. Consider X = ℝ with the usual metric and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq56_HTML.gif . Define F : A×AA by
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equbn_HTML.gif

      We show that F satisfies (4.6) with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq57_HTML.gif . Let (x, x'), (y, y') ∈ A × A.

      Case 1: If x < x' and y < y', then
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equbo_HTML.gif
      Case 2: If x < x' and yy', then
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equbp_HTML.gif

      Case 3: If xx' and y < y'. In this case we can prove by a similar argument as in case 2.

      Case 4: If xx' and yy', then
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_Equbq_HTML.gif

      Thus condition (4.6) holds with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-93/MediaObjects/13663_2011_212_IEq58_HTML.gif . Therefore, by Corollary 4.4 F has the unique coupled fixed point in A that is a point (0, 0).

      Open problems:

      • In Theorem 3.10, can be replaced the property UC* by a more general condition ?

      • In Theorem 3.10, can be drop the property UC* ?

      • Can be extend the result in this article to another spaces ?

      Declarations

      Acknowledgements

      The first author would like to thank the Research Professional Development Project Under the Science Achievement Scholarship of Thailand (SAST) for some financial support. Furthermore, the second author was supported by the Commission on Higher Education, the Thailand Research Fund and the King Mongkut's University of Technology Thonburi for financial support during the preparation of this manuscript.

      Authors’ Affiliations

      (1)
      Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT)

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      Copyright

      © Sintunavarat and Kumam; licensee Springer. 2012