(See [16].) Let .X, d. be a complete metric space, A, B be two closed subsets of X and T : . ∪ . → . ∪ B be a mapping. Assume that T is a cyclic mapping, that is, T (.) ⊆ B and T (.) ⊆ A. If there exists k ∈ (0, 1) such that d.Tx, Ty) ≤ 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 ac- count a similar approach.

(See [16].) Let A and B be two nonempty closed subsets of a complete metric space .X, d.. Let T : . → B and S : . → A be two functions such that

for all x A and y B, where k (0, 1). Then there exists a unique z A B such that Tz = 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 . and . : A B be a mapping. In this context, the mapping . does not have a fixed point in case of A B = . It is reasonable to research a point ., which is closest to Tx, that is, .(x, Tx) is minimum. Since .(x, Tx) ≥ .(A, B) for all x A, then .(x, Tx) is at least .(A, B). Best proximity point theorems are related to find a point satisfying .(x, Tx) = .(A, B), which is called a best prox- imity point. There are many results about this subject in literature [2, 11, 15, 18–20, 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 . ∩ . = ∅.

Let . and . be nonempty subsets of a metric space (X, d). A mapping . : A B A B is called cyclic contraction if it satisfies . (.) . and . (.) . and the following condition:

for all . ∈ . and . ∈ ., where . ∈ (0, 1).

Let A and B be nonempty closed subsets of a metric space .X, d., and let T : . ∪ . → . ∪ B be a cyclic contraction. If either A or B is boundedly compact, then there exists x in A . B with d.x, Tx) = .(A, B).

After that, Karapınar [12] defined some generalizations of cyclic contraction map- pings 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 .-con- traction, which expands Banach contraction as follows: Let (X, d) be a metric space and . : . → . be a self mapping. If there exists a . in [0, 1) such that

for all x, y X, then . is said to be a .-contraction mapping. Then he proved that every .-contraction self mapping of a complete metric space has a unique fixed point.

In this paper, by unifying the concepts of .-contraction and cyclic contraction we introduce cyclic .-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.

Let (X, d) be a metric space and A, B be nonempty subsets of .. Let . : . → . and . : . → . be two mappings. Then (T, S) is called a cyclic .-contraction pair if there exists . ∈ (0, 1) such that

for all x A, y B. Note that if (T, S) is a cyclic .-contraction pair, then (S, T ) is also a cyclic .-contraction pair.

(See [26].) Let (X, d) be a metric space and A, B be nonempty subsets of .. Let x. , z. be two sequences in . and y. be a sequence in .. If .(x., y.) .(A, B) and .(z., y.) .(A, B) imply .(x., z.) 0, then (A, B) is called to satisfy the UC-property.

It is well known that every pair of nonempty subsets A, B of a metric space (X, d) such that .(A, B) = 0 and also every pair of nonempty subsets A, B of a uniform convex Banach space . such that . is convex satisfy the UC-property.

The following lemma includes an important information about the UC-property

(See [26].) Let A and B be subsets of a metric space .X, d.. Assume that .A, B. has the UC-property. Let x. and y. be sequences in A and B, respectively, such that either of the following holds:

then .x.. is a Cauchy sequence. n→∞ .≥.

(See [13].) Let . be a metric space and A, B be nonempty subsets of .. A sequence {x.} in . ∪ . with {._2.} ⊂ . and {._2.+1} ⊂ . is said to be a cyclically Cauchy sequence if for every ε > 0, there exists an . ∈ N such that

when . is even, . is odd and n, m ≥ . .

(See [9].) A pair (A, B) of subsets of a metric space is said to be proximally complete if for every cyclically Cauchy sequence {x.} in . ∪ ., the sequences {._2.} and {._2.+1} have convergent subsequences in . and ..

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.

(See [9].) Let A and B be subsets of a metric space .X, d.. Then the following hold:

(i) If A and B are complete subsets of X and .A, B. satisfies the UC-property, then (A,B) is proximally complete pair

(ii) If A and B are boundedly compact, then .A, B. is proximally complete pair.

Let A, B be nonempty subsets of a metric space .X, d.. Let T : . → B and S : . → A be two mappings such that .T, S. is a cyclic p-contraction pair, and let x. ∈ A. Consider the sequences {._2.} and {._2.+1} constructed as x_2.+1 = Tx_2., x_2.+2 = Sx_2.+1for all n ∈ N. If there exists n. ∈ N such that

then the mappings T and S have best proximity points.

Proof. Since (T, S) is cyclic .-contraction pair, then for all . ∈ N, we have

Now, if there exists .. ∈ N such that

then from (1) we get

And so

Then we have

Note that if the sequence x. , which mentioned in Proposition 2, satisfies .(._2.0 , x_2.0 +1) ≤ .(._2.0 +1, x_2.0 +2) for some .. N, then the cyclic .-contraction pair (T, S) has a best proximity point. Therefore, for the same sequence, we will investi- gate the condition .(._2.+1, x_2.+2) ≤ .(._2., x_2.+1) for all . N in the following two propositions.

Let A, B be nonempty subsets of a metric space .X, d.. Let T : . → B and S : . → A be two mappings such that .T, S. is a cyclic p-contraction pair, and let x. ∈ A. Then for the sequence, which is constructed as in Proposition ., we have d.x., x.₊₁) → .(A, B. as n → ∞.

Proof. Since (T, S) is cyclic .-contraction pair, by considering Remark 1 we have

And so

for all . ∈ N, where . = 2k/(1 + .) < 1 and . = (1 .).(1 + .). By using last inequality we have

for all . ∈ N. Therefore, we conclude that

Similarly, we can show that

Hence, we have

The following proposition is crucial for our main result. Recall that a sequence x. in a metric space (X, d) is bounded if there exists M > 0 such that .(x., x.) ≤ . for all n, m ∈ N.

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. A. Then the subsequences x_2.and x_2.+1of x. , which is constructed as in Proposition ., are bounded.

Proof. Since .(._2., x_2.+1) converges to .(A, B), it is sufficient to show that the se- quence {._2.} is bounded. Now, we assume the contrary. Let . be a real constant with

where . is the contractive constant of cyclic .-contraction pair. Then since ._2. is not bounded, there exists a natural number .. > 1 such that

Because of cyclic p-contraction pair of (T; S), we have

This contradicts (2). Thus, {._2.} is bounded.

Let A, B be nonempty subsets of a metric space .X, d. such that .A, B. is a proximally complete pair. Let T . A Band S . B A be a cyclic p-contraction pair. Then T and S have best proximity points.

Proof. Let .. be an arbitrary point in ., and let {x.} be a sequences constructed as ._2.+1 = Tx_2., ._2.+2 = Sx_2.+1 for all . ∈. If there exists .. ∈ N such that

then by Proposition 2 the mappings . and . have best proximity points. Now, asume

for all . ∈ N. Then from Proposition 3 we know that

And

Now, we want to show that {x.} is a cyclically Cauchy sequence. For this, it is sufficient to show lim_n,m→∞ .(._2., x_2.+1) = .(A, B). Let . ≥ .. Since (T, S) is a cyclic .-contraction pair, we have

for all n, m ∈ N with . ≥ . and (as in the proof of Proposition 3) for all . ∈ N,

where . = 2k/(1 + .) < 1 and . = (1 − .).(1 + .). Therefore, we have

Continuing this process, we have

Since {x.} is a bounded sequence, then there exists a real number M > 0 such that .(.., x_2.−2.+1) ≤ . for all n, m ∈ N with . ≥ ., and so, we get

Hence, we have

Now, since (A, B) is a proximally complete pair, there exist subsequences {._2.. } ⊂ {._2.} and {._2.. +1} ⊂ {._2.+1} such that ._2.. → .. ∈ . and ._2.. +1 → .. ∈ B. Moreover,

Since lim._→∞ .(._2.i , x_2..−1) = .(A, B), we get lim._→∞ .(.., x_2..−1) = .(A, B). Also, we have

And so

From the last inequality we get

Therefore, we have .(.., Tx.) ≤ .(A, B) and so .(.., Tx.) = .(A, B). That is, .. is a best proximity point of . . Similarly, it can be shown that .. is a best proximity point of ..

By taking into account Proposition 1 we can immediately obtain the following corol- laries.

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 . and . have best proximity points.

Let A and B be nonempty closed subsets of a metric space .X, d.. Assume that T . A B and S . B A are mappings such that .T, S. is a cyclic p-contraction pair. If A and B are boundedly compact, then the mappings 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 . and . 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 . = R2 be endowed with taxi-cab metric ., that is, for . = (.., x.), . = (.., y.) ∈ .,

Let

then .(A, B) = 1. It is clear that . is not boundedly compact. Also, the pair (A, B) and (B, A) do not satisfy the UC-property. Indeed, if we choose the sequences x. = (1/n, 1) and z. = (−1/n, −1) in . and y. = (0, 0) in . for all . ∈ N, then we have

but .(x., z.) 2. That is, (A, B) does not satisfy the UC-property. Also, if we choose the sequences x. = (1/n, 2) and z. = (1/n, 0) in . and y. = (1/n, 1) in . for all . ∈ N, then we have

but .(x., z.) → 2. That is, (B, A) does not satisfy the UC-property.

Now, we show that the pair (A, B) is a proximally complete pair. Indeed, let {x.} be a cyclically Cauchy sequence in . ∪ ., then {._2.} ⊂ ., {._2.+1} ⊂ ., and for every ε > 0, there exists an . ∈ N such that

when . is even, . is odd and n, m ≥ . . We have to show that ._2. and ._2.+1 have convergent subsequences in . and ., respectively. First, note that since . is a compact subset of ., then {._2.+1} has a such subsequence. Now, if {._2.} has a subsequence in .., then {._2.} has a convergent subsequence because .. is a compact subset of .. Assume {._2.} has not a subsequence in .., then there exists .. ∈ N such that ._2. ∈ .. for all . ≥ ... In this case, from equation (3) we can say that the sequence ._2. → (0, −1) ∈ .. as . → ∞

Now, we define two mappings . : . → . and . : . → . by

And

for all x A, y B. It can be shown that (T, S) is a cyclic .-contraction pair with . = 1.2. Therefore, since all the hypotheses of Theorem 4 are satisfied, . and . have best proximity points in . and ., respectively. Besides, if we define the mapping . : . ∪ . → . ∪ . as

for all x A B, then . is not cyclic contraction mapping. Actually, let . = (1.2, 1) and . = (1.2, 2). So that, we have

for each . ∈ (0, 1).

Taking into account similar contractive inequality, the following best proximity point result can be obtained for cyclic self mapping of . ∪ ..

Let A, B be nonempty subsets of a metric space .X, d. such that .A, B. is a proximally complete pair, and let F : . ∪ . → . ∪ 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 . and . by Tx = Fx for all x A and Sy . Fy for all y B. Since . is a cyclic mapping satisfying (4), then (T, S) is a cyclic .-contraction pair. Therefore, from Theorem 4 there exist z A and w B, which are best proximity points of . .

Now, we present an example showing that Corollary 3 can be applied, but some existing similar results cannot.

Example 2. Let . = R be endowed with usual metric d, A = [0, 1.3] and . = [2.3,1]. Then (A, B) is a proximally complete pair, and .(A, B) = 1.3. Now, we define a mapping . : A B A B as Fx = 1 . for all x A B. Then it is clear that . (.) . and . (.) ., hence, . is a cyclic mapping. Besides, we can see that inequality (4) holds for . = 1.2. Indeed, we have

for all x A, y B. Therefore, since all the hypotheses of Corollary 3 are satisfied, then . has at least two best proximity points in A B. Actually, . = 1.3 and . = 2.3 are best proximity points of . .

Now, we show that some earlier best proximity point results cannot be applied to this example. For . = 0 and . = 1, we have

Therefore, we cannot find . ∈ (0, 1) satisfying

Or

Hence, Theorems 10 and 12, which are main results of [12], cannot be applied to this example

Let .X, d. be a complete metric space, A, B be two closed subsets of X and T : . → B and S : . → A be two mappings. If there exists k ∈ (0, 1) such that

for all x . A and y . B, then T and S have a unique common fixed point in A . B.

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.

Let .X, d. be a complete metric space and T : . → 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.

Let .X, d. be a complete metric space, A, B be two closed subsets of X and F : . ∪ . → . ∪ B be a mapping. Assume that F is a cyclic mapping, that is, F (.) ⊆ B and F (.) ⊆ A. If there exists k ∈ (0, 1) such that

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 . many existence results provided for problem (5) in the literature (see [3,10,24,27]). Here, we will consider a common type condition on . and ., 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.[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) can be considered as the existence of common fixed point of the operators T and S

defined on X = C[0; 1] by

where . = .[0, 1] is the space of all continuous real-valued functions defined on [0, 1].

It is clear that . .(t, s) d. = .(1 − .).2 and thus sup

Consider the uniform metric .. on ., that is

then it is well known that the space (X, d.) is complete

Suppose the following conditions hold:

(i). there exists a continuous function p : [0, 1] → [0, ∞) satisfying

(ii). there exists k < . such that . .(t, s).(.) d. ≤ k.

Then the system of second order two point boundary value problem given by (5) and (6) has a unique common solution.

Note that if max .∈[0,1] .(.) ≤ 8. for k < 1, we have ∫ 1 .(t, s).(.) d. ≤ ..

Proof of Theorem 5. Consider the complete metric space (X, d.) and the operators . and . on . given by (9). Then for any u, v ∈ . and . ∈ [0, 1], we have

Hence, we have

Or equivalently

Therefore, the contractive condition of Corollary 5 is satisfied. Hence, the operators . and . 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.