Coincidence best proximity point theorems for proximal Berinde g-cyclic contractions in metric spaces

*Correspondence: inthira.c@msu.ac.th 1Department of Mathematics, Faculty of Science, Mahasarakham University, Mahasarakham 44150, Thailand Abstract In this paper, we introduce the notions of proximal Berinde g-cyclic contractions of two non-self-mappings and proximal Berinde g-contractions, called proximal Berinde g-cyclic contraction of the first and second kind. Coincidence best proximity point theorems for these types of mappings in a metric space are presented. Some examples illustrating our main results are also given. Our main results extend and generalize many existing results in the literature.


Introduction
Fixed point theory has an important role in the study of theory of nonlinear equations. Several problems can be formulated as equations of the form Jx = x, where J is a selfmapping in a metric space or in other spaces as appropriate framework. Fixed point theory concerning contraction and generalized contractions has been studied by many mathematicians. It is well-known that the first person who gave the idea of a contraction was Banach [4]. In 1922, he introduced the concept of a contraction of a self-mapping in a metric space and proved that every contraction mapping from a complete metric space X into itself has a unique fixed point. Later, in 2004, the Romanian mathematician Berinde [7] introduced the concept of a weak contraction mapping in a metric space, which is the generalization of the contraction mapping. He proved that every weak contraction mapping from a complete metric space X into itself has a fixed point. In addition to the references mentioned above, there are two interesting articles about the existence of a fixed point for a single-valued mapping which was studied by Hussain et al. [16] and Latif et al. [20]. Although fixed point theory has great importance in solving nonlinear equations of the form Jx = x, where J is a self-mapping, if J is a non-self-mapping, then it is possible that J has no fixed points. But we know that the distance between x and Jx is always greater than or equal to zero. where D(J) is the domain of J. From the above questions, this resulted in the study of a best proximity point theorem. Let (X, · ) be a normed vector space and C a nonempty compact convex subset in X. In 1969, Fan [12] was the first to study the best proximity point theorem, and he proved that if J : C → X is a contraction non-self-mapping, then there exists x * ∈ C such that x * -Jx * = d(Jx * , C), where d(Jx * , C) = min{ Jx *x : x ∈ C}. Clearly, if J(C) ⊆ C, then x * is a fixed point of J. Later, Kirk et al. [19] studied and introduced the concept of a cyclic mapping in the context of a metric space. Moreover, they proved that if J : Y ∪ Z → Y ∪ Z is a cyclic mapping on a complete metric space (X, d) and J satisfies the condition for some α ∈ [0, 1) and for all x ∈ Y , y ∈ Z, then Y ∩ Z = ∅ and J has a unique fixed point in Y ∪ Z. Three years later, Eldred and Veeramani [11] introduced the concept of a cyclic contraction and proved the existence and convergence results of a best proximity point for the cyclic contraction in a uniformly metric space and a convex Banach space, which is an extension of the results of Kirk et al. [19] to the case Y ∩ Z = ∅. Based on the concept of Eldred and Veeramani [11], many mathematicians got interested in studying best proximity point theorems in various directions. More details can be found in [1,9,13,15,17,21,22].
In 2011, Gabeleh and Abker [15] studied and discussed the existence and convergence of a best proximity point of a semicyclic contraction pair (J, T), where J and T are selfmappings on Y ∪ Z. In the same year, Basha [5] introduced the concept of a proximal cyclic contraction of two mappings J and T. Furthermore, the author gave the concept of a proximal contraction of the first and the second kind for non-self-mappings. Especially, the author obtained some interesting results on best proximity points for there mappings in a complete metric space.
The main aim of this paper is to study a coincidence best proximity point result for proximal Berinde g-cyclic contraction of two non-self-mappings, and a coincidence best proximity point result for proximal Berinde g-contraction of the first and second kind on a complete metric space.

Preliminaries
In this section, we will review the notations and definitions to provide basic knowledge for creating the main results of this article. Definition 2.1 ([7]) Let (X, d) be a metric space. A mapping J : X → X is called a weak contraction if there exist k ∈ (0, 1) and L ≥ 0 such that d(Jx, Jy) ≤ ad(x, y) + Ld(x, Jy), for all x, y ∈ X. (2.1) Remark 2.2 In some articles, the mapping J satisfying the condition (2.1) is called an almost contraction (see, [8]).
Let (X, d) be a metric space and Y , Z be nonempty subsets of X. Let J : We give the meaning of the sets Y 0 and Z 0 as follows: (i) J is a cyclic mapping; (ii) There exists k ∈ (0, 1) such that d(Jx, Jy) ≤ kd(x, y) Definition 2.4 ([15]) Let Y , Z be two nonempty closed subsets of a complete metric space (X, d), and let J, T be two self-maps on Y ∪ Z. We call (J, T) a semicyclic contraction pair if the following conditions hold: (i) J is a cyclic mapping; (ii) There exists α ∈ (0, 1) such that d(Jx, Ty) ≤ αd(x, y) Obviously, in the case that J = T, a semicyclic contraction pair reduces to a cyclic contraction.
Note that, if J is a self-mapping, then, using Definition 2.6, we get that J is a contraction. So, the pair (J, J) forms a proximal cyclic contraction.
It is remarked that in the case that J is a non-self-mapping, a proximal contraction of the first kind reduces to a contraction and every contraction is a proximal contraction of the second kind. for all x 1 and x 2 in Y . Also, Basha [5] proved the existence of the following best proximity point in a complete metric space. Theorem 2.9 (Basha [5], Theorem 3.1) Let Y and Z be nonempty closed subsets of a complete metric space such that Y 0 and Z 0 are nonempty. Let J : Y → Z, T : Z → Y , and g : Y ∪ Z → Y ∪ Z satisfy the following conditions: (a) J and T are proximal contractions of the first kind; Motivated by Eldred and Veeramani [11], Gabeleh and Abker [15], and Basha [5], and the idea of Berinde [7], we introduce the new classes of proximal Berinde g-contractions of the first and second kind, and proximal Berinde g-cyclic contractions which are more general than the class of non-self-mappings in [5]. Moreover, we obtain a coincidence best proximity point theorem. We also give some examples to illustrate our results.

Main results
In this section, we shall first introduce proximal Berinde g-contractions of the first and second kind, as well as proximal Berinde g-cyclic contractions. Then, we prove the existence of coincidence best proximity points for these non-self-mappings in a metric space. Throughout this section, Y and Z are two nonempty subsets of a metric space (X, d).
From Definition 3.2, if we take L 1 = 0 and g as the identity mapping, then a proximal Berinde g-contraction of the first kind reduces to a proximal contraction of the first kind which was introduced by Basha [5].
Then Jx 1 = (0, 1), Jx 2 = ( It follows that where L 1 ≥ √ 2. From each case, we can conclude that J is a proximal Berinde gcontraction of the first kind with α = 1 2 and L 1 = √ 2. From Case 4, we noticed that J is not a proximal contraction of the first kind when L 1 = 0 and g is the identity mapping.

Definition 3.4
Let J : Y → Z and g : Y → Y be mappings. A mapping J is said to be a proximal Berinde g-contraction of the second kind if there exist β ∈ [0, 1) and L 2 ≥ 0 such that for all x 1 , From Definition 3.4, if we take L 2 = 0 and g as the identity mapping, then a proximal Berinde g-contraction of the second kind reduces to a proximal contraction of the second kind which was introduced by Basha [5].
From Definition 3.5, if we take L = 0 and g as the identity mapping, then a proximal Berinde g-cyclic contraction reduces to a proximal cyclic contraction which was introduced by Basha [5].
Before giving the coincidence best proximity point theorems, we give the following lemmas, which are important tools for proving the existence of coincidence best proximity points in a metric space.

Lemma 3.6 Let Y and Z be nonempty subsets of a metric space
, there exists y ∈ Y 0 such that z = gy, and then d(gy, Jx) = d(Y , Z).
(ii) Let x 0 ∈ Y 0 be given. By using (i), we get that there exists x 1 ∈ Y 0 such that Again, from x 1 ∈ Y 0 and using (i), we have that there exists x 2 ∈ Y 0 such that Continuing in the same way, we can establish a sequence {x n } in Y 0 such that The proof is completed.
Since α ∈ (0, 1), {x n } is a Cauchy sequence in X, and hence converges to some point x in Y since Y is closed. (ii) g is an isometry with Y 0 ⊆ g(Y 0 ). Then, there exists a point x ∈ Y such that x n → x, as n → ∞.
Proof Let x 0 ∈ Y 0 be given. By Lemma 3.6, we can find a sequence {x n } in Y 0 defined by Since J is a proximal Berinde g-contraction of the first kind and d(gx n , Jx n-1 ) = d(Y , Z) = d(gx n+1 , Jx n ), for all n ∈ N, we obtain that for each n ∈ N, d(gx n , gx n+1 ) ≤ αd(gx n-1 , gx n ) + L min d(gx n-1 , gx n+1 ), d(gx n , gx n ) = αd(gx n-1 , gx n ).
It implies that because g is an isometry. Since α ∈ [0, 1) and using Lemma 3.7, we obtain that there exists x ∈ Y such that x n → x as n → ∞.  In addition, if α + L 1 < 1 and β + L 2 < 1, then there exists a unique element x and there exists a unique element y which satisfy the equation (3.5).
Proof From J satisfies the conditions (i)-(iii) and using Lemma 3.8, we get that for x 0 ∈ Y 0 , we can find a sequence {x n } in Y 0 such that Z), for all n ∈ N * , and which converges to some element x ∈ Y . Similarly, for y 0 ∈ Z 0 , we can find a sequence {y n } in Z 0 such that d(gy n+1 , Jy n ) = d(Y , Z), for all n ∈ N * , and which converges to some element y ∈ Z. Since the pair (J, T) is a proximal Berinde g-cyclic contraction and d(gx n+1 , Jx n ) = d(Y , Z) = d(gy n+1 , Ty n ), for all n ∈ N * , there exist γ ∈ [0, 1) and L ≥ 0 such that d(gx n+1 , gy n+1 ) ≤ γ d(gx n , gy n ) + (1γ )d(Y , Z) + Ld(gx n , gx n+1 ).

It implies that
Taking limit as n → ∞, we have d(x, y) ≤ γ d(x, y) since Y 0 ⊆ g(Y 0 ) and Z 0 ⊆ g(Z 0 ). Since J is a proximal Berinde g-contraction of the first kind and Taking n → ∞ in above inequality, by the continuity of g, we get d(gw, gx) = 0, and so, gx = gw. It implies that Similarly, it is easy to verify that d(gy, Ty) = d(Y , Z). Thus, we can conclude that Therefore, (x, y) is a coincidence best proximity point of the triple (g, J, T). Next, we will show that (x, y) is unique. Suppose that α + L 1 < 1, β + L 2 < 1 and there exists gw ∈ Y 0 such that Since J is a proximal Berinde g-contraction of the first kind, we have that d(gx, gw) ≤ αd(gx, gw) + L 1 min d(gx, gw), d(gw, gx) ≤ (α + L 1 )d(gx, gw).
Since α + L 1 < 1, d(gx, gw) = 0. It follows that gx = gw, which implies that there exists a unique x ∈ Y such that d(gx, Jx) = d (Y , Z). Similarly, we can show that there exists a unique y ∈ Z such that d(gy, Ty) = d (Y , Z). Therefore, the pair (x, y) is a unique coincidence best proximity point of the triple (g, J, T). Example 3.10 Consider the complete metric space R 2 with the usual metric defined by d(x, y) = (x 1y 1 ) 2 + (x 2y 2 ) 2 , for all x = (x 1 , x 2 ), y = (y 1 , y 2 ) ∈ R 2 .
Obviously, g is an isometry, J and T are proximal Berinde g-contractions of the first kind, and the pair (J, T) is a proximal Berinde g-cyclic contraction. Clearly, the mappings J, T, and g satisfy the conditions (ii) and (iv). Therefore, all hypothesis of Theorem 3.9 are satisfied. Moreover, (0, 0) ∈ Y is a coincidence best proximity point of the pair (g, J) and (0, 1) ∈ Z is a coincidence best proximity point of the pair (g, T), i.e., ((0, 0), (0, 1)) is a coincidence best proximity point of the triple (g, J, T).
If we take L 1 = 0, L 2 = 0 and L = 0 in Theorem 3.9, we obtain the following coincidence best proximity point theorem which is more general than that of Basha [5]. If g is the identity mapping, we immediately obtain the following corollary as follows.  there exist α ∈ [0, 1) and L 1 ≥ 0 such that d(gx n , gx n+1 ) ≤ αd(gx n-1 , gx n ) + L 1 min d(gx n , gx n ), d(gx n-1 , gx n+1 ) Taking limit as n → ∞, we have d(gu, gx) = 0. Thus, gu = gx. From (3.10), we obtain that i.e., x is a coincidence best proximity point of the pair (g, J). The uniqueness and remaining path of the proof follows by that of Theorem 3.9. Obviously, g is an isometry with Y 0 ⊆ g(Y 0 ), J preserves the isometric distance with respect to g and it is a proximal Berinde g-contraction of the first and the second kind with J(Y 0 ) ⊆ Z 0 . Therefore, all hypothesis of Theorem 3.13 are valid. Moreover, -1 ∈ Y is a coincidence best proximity point of the pair (g, J).
If we suppose that Y 0 is closed in Theorem 3.13, then we do not need to assume that J is a proximal Berinde g-contractions of the second kind. This gives the following theorem. Proof By the proof of Lemma 3.8, we get that the sequence {x n } in Y 0 defined by d(gx n+1 , Jx n ) = d(Y , Z), for all n ∈ N * (3.11) converges to an element x ∈ Y . Since Y 0 is closed, we obtain that x ∈ Y 0 . Since J(Y 0 ) ⊆ Z 0 , there exists z ∈ Y 0 such that d(gz, Jx) = d(Y , Z).
Taking n → ∞ in the above inequality, by the continuity of g, we get d(gx, gz) = 0, and so, gx = gz. It implies that i.e., x is a coincidence best proximity point of the pair (g, J).
If we take L 1 = 0 and L 2 = 0 in Theorem 3.13, we obtain the following coincidence best proximity point result, which is more general than a coincidence best proximity point result in [5]. Corollary 3. 16 Let Y and Z be nonempty closed subsets of a complete metric space (X, d) such that Y 0 is nonempty. Let J : Y → Z and g : Y → Y satisfy the following conditions: (i) J is a proximal contraction of the first and second kind; (ii) J preserves the isometric distance with respect to g and J(Y 0 ) ⊆ Z 0 ; (iii) g is an isometry with Y 0 ⊆ g(Y 0 ). Then, there exists a unique element x ∈ Y such that d(gx, Jx) = d(Y , Z).
Moreover, for any fixed x 0 ∈ Y 0 , the sequence {x n } defined by d(gx n+1 , Jx n ) = d(Y , Z), for all n ∈ N * converges to the element x.
If g is the identity mapping, we obtain the following result by applying Theorem 3.13 as follows.