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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 182947, 15 pages
http://dx.doi.org/10.1155/2012/182947
Research Article

Cyclic Contractions on -Metric Spaces

1Department of Mathematics, Atilim University, 06836 incek, Turkey
2Department of Mathematics, Galatasaray University, 34349 Istanbul, Turkey

Received 29 June 2012; Accepted 1 September 2012

Academic Editor: Irena Rachůnková

Copyright © 2012 E. Karapınar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Conditions for existence and uniqueness of fixed points of two types of cyclic contractions defined on -metric spaces are established and some illustrative examples are given. In addition, cyclic maps satisfying integral type contractive conditions are presented as applications.

1. Introduction

The extensive application potential of fixed point theory in various fields resulted in several generalizations of the metric spaces. Amongst them, one can mention quasimetric spaces, partial metric spaces, rectangular metric spaces, -metric spaces, and -metric spaces. Perhaps one of the most interesting generalizations is the -metric space. Introduced by Mustafa and Sims [1] in 2006, the concept of -metric space has drawn the attention of mathematicians and became a very popular subject especially from the point of view of fixed point theory [213].

Another attractive topic in fixed point theory is the concept of cyclic maps and best proximity points introduced by Kirk et al. [14] in 2003. Cyclic maps and in particular the fixed points of cyclic maps have been a subject of growing interest recently (see, e.g., [1527]).

The purpose of this work is to combine these two notions and investigate cyclic maps on -metric spaces. We concentrate on two types of cyclic contractions: cyclic type Banach contractions and cyclic weak -contractions.

Mustafa and Sims [1] introduced the concept of -metric spaces as follows.

Definition 1.1 (see [1]). Let be a nonempty set, be a function satisfying the following properties:(G1) if ,(G2) for all with ,(G3) for all with ,(G4) (symmetry in all three variables),(G5) for all (rectangle inequality). Then the function is called a generalized metric, or, more specifically, a -metric on , and the pair is called a -metric space.

Note that every -metric on induces a metric on defined by

To have better idea about the subject, we give the following examples of -metrics.

Example 1.2. Let be a metric space. The function , defined by for all , is a -metric on .

Example 1.3. Let . The function , defined by for all , is a -metric on .

In their initial paper, Mustafa and Sims [1] defined also the basic topological concepts in -metric spaces as follows.

Definition 1.4 (see [1]). Let be a -metric space, and let be a sequence of points of . We say that is -convergent to if that is, for any , there exists such that , for all . We call the limit of the sequence and write or .

Proposition 1.5 (see [1]). Let be a -metric space. The following are equivalent:(1) is -convergent to ,(2) as ,(3) as ,(4) as .

Definition 1.6 (see [1]). Let be a -metric space. A sequence is called a -Cauchy sequence if, for any , there exists such that for all , that is, as .

Proposition 1.7 (see [1]). Let be a -metric space. Then the following are equivalent:(1)the sequence is -Cauchy,(2)for any , there exists such that , for all .

Definition 1.8 (see [1]). A -metric space is called -complete if every -Cauchy sequence is -convergent in .

Definition 1.9. Let be a -metric space. A mapping is said to be continuous if for any three -convergent sequences , , and converging to , , and , respectively, is -convergent to .

Note that each -metric on generates a topology on whose base is a family of open -balls , where for all and . A nonempty set is -closed in the the -metric space if . Observe that Finally, we have the following proposition.

Proposition 1.10. Let be a -metric space and be a nonempty subset of . A is -closed if for any -convergent sequence in with limit , one has .

2. Banach Contractive Cyclic Maps on -Metric Spaces

Our first result is a fixed point theorem which is the Banach contraction mapping analog for cyclic maps on -metric spaces.

Theorem 2.1. Let be a -complete -metric space and be a family of nonempty -closed subsets of . Let and be a map satisfying If there exists such that holds for all and ,   then, has a unique fixed point in .

Proof. We prove first the existence part. Take an arbitrary and without loss of generality assume that . Define the sequence as Since is cyclic, , , , , and so on. If for some , then obviously, the fixed point of is . Assume that for all .
Put and in (2.2). Then Then, we have, which upon letting implies On the other hand, by symmetry (G4) and the rectangle inequality (G5), we have The inequality (2.7) with and becomes Letting in (2.8), we get
We show next that the sequence is a Cauchy sequence in the metric space where is given in (1.1). For we have and making use of (2.4) and (2.8) we obtain Hence, That is, the sequence is Cauchy in . Since the space is -complete then is complete (see Proposition 10 in [1]) and hence, converges to a number say, . Moreover, is -Cauchy in (see Proposition 9 in [1]) and it is easy to see that . Indeed, if , then the subsequence , the subsequence , and, continuing in this way, the subsequence . All the subsequences are -convergent and hence, they all converge to the same limit . In addition, the sets are -closed, thus, the limit .
We show now that is a fixed point of , that is, . Consider now (1.1) and (2.2) with , and suppose that or , then we have, Passing to limit as , we end up with which contradicts the assumption . Hence, , that is, is a fixed point of .
To prove the uniqueness, we assume that is another fixed point of such that . Both and lie in ; thus, we can substitute and in (2.2). This yields which is true only for but by definition. Thus, the fixed point of is unique.

As a particular case of Theorem 2.1, we give the following corollary.

Corollary 2.2. Let be a -complete -metric space and be a family of nonempty -closed subsets of . Let and be a map satisfying If there exists such that holds for all and , then, has a unique fixed point in .

3. Generalized Cyclic Weak -Contractions on -Metric Spaces

The main goal of a number of studies regarding fixed points is to weaken the contractive conditions on the map under consideration. Inspired by this idea, in 1969, Boyd and Wong [28] defined the concept of -contraction. Later, in 1997, Alber and Guerre-Delabriere [29] defined the weak -contractions on Hilbert spaces and proved fixed point theorem regarding such contractions. A map on a metric space is called a weak -contraction if there exists a strictly increasing function with such that for all . These types of contractions have also been a subject of extensive research (see, e.g., [3032]). In what follows, we discuss cyclic weak -contractions on -metric spaces.

Consider the set of continuous functions with and for . We have the following fixed point theorem.

Theorem 3.1. Let be a -complete -metric space and be a family of nonempty -closed subsets of with . Let be a map satisfying Suppose that there exists a function such that the map satisfies for all and , where Then has a unique fixed point in .

Proof. To prove the existence part, we construct a sequence of Picard iterations as usual. Take an arbitrary and define the sequence as Since is cyclic, , , , and so on. If for some , then, obviously, the fixed point of is . Assume that for all .
Let and in (3.3) where If , then (3.6) yields which implies and hence . This contradicts the assumption for all . Then we must have , in (3.6), so that Thus, the sequence is a nonnegative nonincreasing sequence which converges to . Letting in (3.9) we get It follows that ; therefore, , that is, The equation (2.7) in the proof of Theorem 2.1 with and yields and hence,
We claim that is a -Cauchy sequence in . Assume the contrary, that is, is not -Cauchy. Then, according to Proposition 1.7 there exist and corresponding subsequences and of satisfying for which where is chosen as the smallest integer satisfying (3.14), that is, It is easy to see from (3.14) and (3.15) and the rectangle inequality (G5) that Taking limit as in (3.16) and using (3.11) we obtain Observe that for every there exists satisfying such that Therefore, for large enough values of we have and and lie in the consecutive sets and , respectively, for some . We next substitute and in (3.3) to obtain where Employing rectangle inequality (G5) repeatedly we see that or equivalently Note that the sum on the right-hand side of (3.22) consists of finite number of terms, and due to (3.11) each term of this sum tends to 0 as . Therefore, Using rectangle inequality (G5) again, we have from which we deduce upon letting and using (3.23). Now, passing to limit as in (3.19) and using (3.11), (3.23), and (3.25) we get and hence . We conclude that which contradicts the assumption that is not -Cauchy. Thus, the sequence is -Cauchy and since is -complete; it is -convergent to a limit, say . It can be easily seen that . Since , then the subsequence , the subsequence , and, continuing in this way, the subsequence . All the subsequences are -convergent in the -closed sets and hence, they all converge to the same limit .
To show that the limit of the Picard sequence is the fixed point of , that is, we employ (3.3) with . This leads to where Passing to limit as , we get Thus, and hence, , that is, .
Finally, we prove that the fixed point is unique. Assume that is another fixed point of such that . Then, since both and belong to , we set and in (3.3) which yields where, Then (3.30) becomes and clearly, , so we conclude that , that is, the fixed point of is unique.

For particular choices of the function we obtain the following corollaries.

Corollary 3.2. Let be a -complete -metric space and be a family of nonempty -closed subsets of with . Let be a map satisfying Suppose that there exists such that the map satisfies for all and , where Then has a unique fixed point in .

Proof. The proof is obvious by choosing the function in Theorem 3.1 as .

Corollary 3.3. Let be a -complete -metric space and be a family of nonempty -closed subsets of with . Let be a map satisfying Suppose that there exist constants ,  ,  ,  and with such that the map satisfies for all and , . Then, has a unique fixed point in .

Proof. Clearly we have, where By Corollary 3.2, the map has a unique fixed point.

4. An Example and Applications

To illustrate the cyclic weak -contractions on -metric spaces we give the following example.

Example 4.1. Let and let be given as . Let and . Define the function as Clearly, the function is a -metric on . Define also as . Obviously, the map has a unique fixed point .
It can be easily shown that the map satisfies the condition (3.3). Indeed, note that Then, and clearly, Hence, has a unique fixed point by Theorem 3.1.

Cyclic maps satisfying integral type contractive conditions are amongst common applications of fixed point theorems. In this context, we consider the following applications.

Corollary 4.2. Let be a -complete -metric space and be a family of nonempty -closed subsets of with . Let be a map satisfying Suppose also that where and for all and , . Then has a unique fixed point in .

Corollary 4.3. Let be a -complete -metric space and be a family of nonempty -closed subsets of with . Let be a map satisfying Suppose also that where and for all and , . Then has a unique fixed point in .

Very recently, Jachymski proved the equivalence of auxiliary functions (see Lemma, in [33]). Inspired by the results from the remarkable paper of Jachymski, we state the following theorem.

Theorem 4.4 (see [33]). Let be a self-map on a -complete -metric space and be a family of nonempty -closed subsets of with . Let be a map satisfying Assume that Then the following statements are equivalent: (i)there exist functions such that ,(ii)there exists a function such that for any bounded sequence of positive reals, implies and ,(iii)there exists a continuous function such that and ,(iv)there exists function and a nondecreasing, right continuous function with and for all with ,(v)there exists a continuous and nondecreasing function such that and for all with .

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