Best proximity point theorems for cyclic p-contractions with some consequences and applications

Abstract. In this paper, we introduce the concept of cyclic p-contraction pair for single-valued mappings. Then we present some best proximity point results for such mappings defined on proximally complete pair of subsets of a metric space. Also, we provide some illustrative examples that compared our results with some earliest. Finally, by taking into account a fixed point consequence of our main result we give an existence and uniqueness result for a common solution of a system of second order boundary value problems.


Introduction
In 1922, Banach [5] proved a very fundamental theorem, which is known as Banach contraction principle on complete metric spaces. According to this theorem, every self mapping T on complete metric space (X, d) satisfying d(T x, T y) kd(x, y) for all x, y ∈ X, where k ∈ (0, 1), has a unique fixed point in X.
This result has been extended and generalized by many authors in different ways. For example, by introducing the concept of cyclic mappings Kirk et al. [16] have obtained a nice and important generalization of it. Although the mapping T has to be continuous in Banach contraction principle, it is not necessary to be continuous in the following. Theorem 1. (See [16].) Let (X, d) be a complete metric space, A, B be two closed subsets of X and T : A ∪ B → A ∪ B be a mapping. Assume that T is a cyclic mapping, that is, T (A) ⊆ B and T (B) ⊆ A. If there exists k ∈ (0, 1) such that d(T x, T y) kd(x, y) for all x ∈ A and y ∈ B, then T has a fixed point in A ∩ B.
Also, the following common fixed point result has been obtained by taking into account a similar approach.
Theorem 2. (See [16].) Let A and B be two nonempty closed subsets of a complete metric space (X, d). Let T : A → B and S : B → A be two functions such that d(T x, Sy) kd(x, y) for all x ∈ A and y ∈ B, where k ∈ (0, 1). Then there exists a unique z ∈ A ∩ B such that T z = Sz = z.
On the other hand, some other extensions of Banach contraction principle have been presented by considering the concept of best proximity point [6]. Now, we recall some fundamental concepts and properties of best proximity points. Let (X, d) be a metric space, A, B be two subsets of X and T : A → B be a mapping. In this context, the mapping T does not have a fixed point in case of A ∩ B = ∅. It is reasonable to research a point x, which is closest to T x, that is, d(x, T x) is minimum. Since d(x, T x) d(A, B) for all x ∈ A, then d(x, T x) is at least d(A, B). Best proximity point theorems are related to find a point satisfying d(x, T x) = d(A, B), which is called a best proximity point. There are many results about this subject in literature [2,11,15,[18][19][20]22].
Note that A ∩ B has to be nonempty in Theorem 4. Hence, by combining the ideas of both cyclic mapping and best proximity point Eldred and Veeremani [8] introduced the concept of cyclic contraction mapping and then obtained a result, which includes the situation A ∩ B = ∅. Definition 1. Let A and B be nonempty subsets of a metric space (X, d). A mapping T : A ∪ B → A ∪ B is called cyclic contraction if it satisfies T (A) ⊆ B and T (B) ⊆ A and the following condition: for all x ∈ A and y ∈ B, where k ∈ (0, 1). Theorem 3. Let A and B be nonempty closed subsets of a metric space (X, d), and let T : After that, Karapınar [12] defined some generalizations of cyclic contraction mappings such as Ciric type and Kannan type cyclic contraction mappings and then presented some nice results for existence of a best proximity point of such mappings in a uniformly convex Banach space. Until now, in almost all results about to guarantee the existence of best proximity points of a cyclic contraction mapping, it has been used some certain properties such as UC-property or boundedly compactness [4,14]. Very recently, the notion of proximally completeness has been defined in [9], and so the UC-property and boundedly compactness have been neglected. Then many authors have studied to show the existence of best proximity points of a cyclic contraction self mapping of A∪B, where (A, B) is a proximally complete pair [1,7,17,23,25].
Lately, Popescu [21] introduced a new type contractive condition named as p-contraction, which expands Banach contraction as follows: Let (X, d) be a metric space and T : X → X be a self mapping. If there exists a k in [0, 1) such that for all x, y ∈ X, then T is said to be a p-contraction mapping. Then he proved that every p-contraction self mapping of a complete metric space has a unique fixed point.
In this paper, by unifying the concepts of p-contraction and cyclic contraction we introduce cyclic p-contraction pair and present some best proximity point results for such mappings. Therefore, our results extend, unify and generalize many fixed point and best proximity point theorems in the literature as properly. Also, we provide some illustrative and comparative examples to show the significance of our results. Finally, we provide the sufficient conditions to guarantee the existence and uniqueness of a common solution of a system of second order boundary value problems.

Best proximity results
We begin this section by introducing the cyclic p-contraction pair. for all x ∈ A, y ∈ B. Note that if (T, S) is a cyclic p-contraction pair, then (S, T ) is also a cyclic p-contraction pair. It is well known that every pair of nonempty subsets A, B of a metric space (X, d) such that d(A, B) = 0 and also every pair of nonempty subsets A, B of a uniform convex Banach space X such that A is convex satisfy the UC-property.
The following lemma includes an important information about the UC-property.  The following propositions, which show the relation between the UC-property (or boundedly compactness) and the notion of proximally completeness, play an important role to get some corollaries from our main result.
then the mappings T and S have best proximity points.
Proof. Since (T, S) is cyclic p-contraction pair, then for all n ∈ N, we have Now, if there exists n 0 ∈ N such that Then we have Hence, x 2n0 is a best proximity point of T , and x 2n0+1 is a best proximity point of S.
for some n 0 ∈ N, then the cyclic p-contraction pair (T, S) has a best proximity point. Therefore, for the same sequence, we will investigate the condition d(x 2n+1 , x 2n+2 ) d(x 2n , x 2n+1 ) for all n ∈ N in the following two propositions.
Proposition 3. Let A, B be nonempty subsets of a metric space (X, d). Let T : A → B and S : B → A be two mappings such that (T, S) is a cyclic p-contraction pair, and let x 0 ∈ A. Then for the sequence, which is constructed as in Proposition 2, we have Proof. Since (T, S) is cyclic p-contraction pair, by considering Remark 1 we have for all n ∈ N, where λ = 2k/(1 + k) < 1 and µ = (1 − k)/(1 + k). By using last inequality we have for all n ∈ N. Therefore, we conclude that Similarly, we can show that .
The following proposition is crucial for our main result. Recall that a sequence {x n } in a metric space Proposition 4. Let A, B be nonempty subsets of a metric space X. Let T : A → B and S : B → A be two mappings such that (T, S) is a cyclic p-contraction pair, and let x 0 ∈ A. Then the subsequences {x 2n } and {x 2n+1 } of {x n }, which is constructed as in Proposition 2, are bounded.
where k is the contractive constant of cyclic p-contraction pair. Then since {x 2n } is not bounded, there exists a natural number n 0 > 1 such that Because of cyclic p-contraction pair of (T, S), we have This contradicts (2). Thus, {x 2n } is bounded.

Now, since (A, B) is a proximally complete pair, there exist subsequences
Since lim i→∞ d(x 2ni , x 2ni−1 ) = d(A, B), we get lim i→∞ d(x * , x 2ni−1 ) = d (A, B). Also, we have and so A, B). Therefore, we have d(x * , T x * ) d(A, B) and so d(x * , T x * ) = d (A, B). That is, x * is a best proximity point of T . Similarly, it can be shown that y * is a best proximity point of S.
By taking into account Proposition 1 we can immediately obtain the following corollaries.
Corollary 1. Let A, B be nonempty closed subsets of complete metric space (X, d) such that (A, B) satisfies the UC-property. Assume that T : A → B and S : B → A are two mappings such that (T, S) is a cyclic p-contraction pair. Then T and S have best proximity points.
Proof. By Proposition 1(i) we get that (A, B) is a proximally complete pair. Thus, by using Theorem 4 we say that T and S have best proximity points. Proof. From Proposition 1(ii) we know that every boundedly compact subsets pair (A, B) of a metric space is proximally complete. By using Theorem 4 we say that T and S have best proximity points.
The following example is important to show that the best proximity point results including the conditions boundedly compactness or the UC-property in the literature cannot be applied to any cyclic mappings even if they are a kind of cyclic contraction. Example 1. Let X = R 2 be endowed with taxi-cab metric d, that is, for x = (x 1 , x 2 ), y = (y 1 , y 2 ) ∈ X, d(x, y) = |x 1 − y 1 | + |x 2 − y 2 |. Now, we show that the pair (A, B) is a proximally complete pair. Indeed, let {x n } be a cyclically Cauchy sequence in A ∪ B, then {x 2n } ⊂ A, {x 2n+1 } ⊂ B, and for every ε > 0, there exists an N ∈ N such that

Let
when n is even, m is odd and n, m N . We have to show that {x 2n } and {x 2n+1 } have convergent subsequences in A and B, respectively. First, note that since B is a compact subset of X, then {x 2n+1 } has a such subsequence. Now, if {x 2n } has a subsequence in A 1 , then {x 2n } has a convergent subsequence because A 1 is a compact subset of X.
Assume {x 2n } has not a subsequence in A 1 , then there exists n 0 ∈ N such that x 2n ∈ A 2 for all n n 0 . In this case, from equation (3) we can say that the sequence x 2n → (0, −1) ∈ A 2 as n → ∞. Now, we define two mappings T : A → B and S : B → A by and Sy = (0, −1), y = (b, 2), for all x ∈ A, y ∈ B. It can be shown that (T, S) is a cyclic p-contraction pair with k = 1/2. Therefore, since all the hypotheses of Theorem 4 are satisfied, T and S have best proximity points in A and B, respectively. Besides, if we define the mapping F : for all x ∈ A ∪ B, then F is not cyclic contraction mapping. Actually, let x = (1/2, 1) and y = (1/2, 2). So that, we have for each k ∈ (0, 1).
Taking into account similar contractive inequality, the following best proximity point result can be obtained for cyclic self mapping of A ∪ B.
Corollary 3. Let A, B be nonempty subsets of a metric space (X, d) such that (A, B) is a proximally complete pair, and let F : A ∪ B → A ∪ B be a cyclic mapping. Assume that there exists k ∈ (0, 1) such that for all x ∈ A and y ∈ B. Then there exist z ∈ A and w ∈ B, which are best proximity points of F .
Proof. Define two mappings T and S by T x = F x for all x ∈ A and Sy = F y for all y ∈ B. Since F is a cyclic mapping satisfying (4), then (T, S) is a cyclic p-contraction pair. Therefore, from Theorem 4 there exist z ∈ A and w ∈ B, which are best proximity points of F . Now, we present an example showing that Corollary 3 can be applied, but some existing similar results cannot.
Then it is clear that F (A) ⊆ B and F (B) ⊆ A, hence, F is a cyclic mapping. Besides, we can see that inequality (4) holds for k = 1/2. Indeed, we have for all x ∈ A, y ∈ B. Therefore, since all the hypotheses of Corollary 3 are satisfied, then F has at least two best proximity points in A ∪ B. Actually, x = 1/3 and y = 2/3 are best proximity points of F . Now, we show that some earlier best proximity point results cannot be applied to this example. For x = 0 and y = 1, we have Therefore, we cannot find k ∈ (0, 1) satisfying Hence, Theorems 10 and 12, which are main results of [12], cannot be applied to this example.

Applications
Let (X, d) be a metric space and A, B be two subsets of X. If A∩B = ∅, then d(A, B) = 0.
In this case, a best proximity result turns to a fixed point result. In this context, we can obtain the following: for all x ∈ A and y ∈ B, then T and S have a unique common fixed point in A ∩ B.
Corollary 5. Let (X, d) be a complete metric space and T, S : X → X be two mappings.
If there exists k ∈ (0, 1) such that for all x, y ∈ X, then T and S have a unique common fixed point in X.
Corollary 6. Let (X, d) be a complete metric space and T : X → X be a self mapping. If there exists k ∈ (0, 1) such that for all x, y ∈ X. Then the mapping T has a unique fixed point in X.
Corollary 7. Let (X, d) be a complete metric space, A, B be two closed subsets of X and F : A ∪ B → A ∪ B be a mapping. Assume that F is a cyclic mapping, that is, for all x ∈ A and y ∈ B, then F has a unique fixed point in A ∩ B.
Now, by considering the Corollary 5 we present an existence and uniqueness theorem for common solution of a system of second order two point boundary value problem as follows: where f, g : [0, 1] × R → R are continuous functions. By considering some certain conditions on the function f many existence results provided for problem (5) in the literature (see [3,10,24,27]). Here, we will consider a common type condition on f and g, we provide a new existence and uniqueness result for common solution of the system. By considering the Green's function defined as we can see that problems (5) and (6) are equivalent to the integral equations and respectively. Therefore, u ∈ C 2 [0, 1] is a common solution of (5) and (6) if and only if it is a common solution of (7) and (8). Hence, the existence of common solution of (5) and (6) where X = C[0, 1] is the space of all continuous real-valued functions defined on [0, 1]. It is clear that (ii) there exists k < 1 such that 1 0 G(t, s)p(s) ds k. Then the system of second order two point boundary value problem given by (5) and (6) has a unique common solution.
Remark 2. Note that if max s∈[0,1] p(s) 8k for k < 1, we have 1 0 G(t, s)p(s) ds k. Proof of Theorem 5. Consider the complete metric space (X, d ∞ ) and the operators T and S on X given by (9). Then for any u, v ∈ X and t ∈ [0, 1], we have T u(t) − Sv(t) Hence, we have Therefore, the contractive condition of Corollary 5 is satisfied. Hence, the operators T and S have a unique common fixed point, and thus the system of second order two point boundary value problem given by (5) and (6) has a unique common solution.

Conclusion
Unifying the concepts of p-contraction and cyclic contraction, we introduce cyclic p-contraction pair and present some best proximity point results for such mappings. We also provide the sufficient conditions to guarantee the existence and uniqueness of a common solution of a system of second order boundary value problems by taking into account our theoretical results.