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Research Article | Open Access

Volume 2012 |Article ID 491542 | https://doi.org/10.1155/2012/491542

Erdal KarapΔ±nar, I. Savas Yuce, "Fixed Point Theory for Cyclic Generalized Weak -Contraction on Partial Metric Spaces", Abstract and Applied Analysis, vol. 2012, Article ID 491542, 12 pages, 2012. https://doi.org/10.1155/2012/491542

Fixed Point Theory for Cyclic Generalized Weak π -Contraction on Partial Metric Spaces

Accepted14 May 2012
Published01 Jul 2012

Abstract

A new fixed point theorem is obtained for the class of cyclic weak -contractions on partially metric spaces. It is proved that a self-mapping on a complete partial metric space has a fixed point if it satisfies the cyclic weak -contraction principle.

1. Introduction and Preliminaries

Fixed point theorems analyze the conditions on maps (single or multivalued) under which the existence of a solution for the equation can be guaranteed. The Banach contraction mapping principle [1] is one of the earliest, widely known, and important results in this direction. After this pivotal principle, a number of remarkable results started to appear in the literature published by many authors (see, e.g., [2β36]). In this trend, one of the distinguished contributions was announced by Kirk et al. [15] in 2003. They introduced the notion of cyclic contraction, which is defined as follows.

Let and be two subsets of a metric space . A self-mapping on is called cyclic provided that and . In addition, if for some , the mapping satisfies the inequality then it is called cyclic contraction where . In their outstanding paper [15], the authors also proved that if and are closed subsets of a complete metric space with , then the cyclic contraction has a unique fixed point in .

The notion of -contraction was defined by Boyd and Wong [11]: A self-mapping on a metric space is called -contraction if there exists an upper semi-continuous function such that This concept was generalized by Alber and Guerre-Delabriere [7], by introducing weak -contraction. A self-mapping on a metric space is called weak -contraction if is a strictly increasing map with and

PΔcurar and Rus [19] gave a characterization of -contraction mappings in the context of cyclic operators and proved some fixed point results for such mappings on a complete metric space. As a natural next step, KarapΔ±nar [13] generalized the results in [19] by replacing the notion of cyclic -contraction mappings with cyclic weak -contraction mappings. For more results for cyclic mapping analysis we refer to [3, 7, 12, 14, 16, 17, 21β23] and the references therein.

Recently, in fixed point theory, one of the celebrated subjects is the partial metric spaces. The notion of a partial metric space was defined by Matthews [37] in 1992 as a generalization of usual metric spaces. The motivation behind the theory of partial metric spaces is to transfer mathematical techniques into computer science to develop the branches of computer science such as domain theory and semantics (see, e.g., [35, 38β46]).

Definition 1.1 (see [37, 47]). A partial metric on a nonempty set is a function such that for all (PM1) (symmetry),(PM2) if , then (equality),(PM3) (small self-distances),(PM4) (triangularity). The pair is called a partial metric space (abbreviated by PMS).

Remark 1.2. If , then may not be .

Example 1.3 (see [47]). Let , and define β. Then, is a partial metric space.

Example 1.4 (see [47]). Let , and define by Then, (X, p) is a complete partial metric space.

Example 1.5 (see [48]). Let and be a metric space and a partial metric space, respectively. Mappings () defined by induce partial metrics on , where is an arbitrary function and .
Each partial metric on generates a topology on which has the family of open -balls as a base, where for all and .
If is a partial metric on , then the function given by is a metric on . Furthermore, it is possible to observe that the following also defines a metric on . In fact, and are equivalent (see, e.g., [9]).

Example 1.6 (see, e.g., [5, 27, 30, 47]). Consider with . Then, is a partial metric space. It is clear that is not a (usual) metric. Note that in this case .

For our purposes, we need to recall some basic topological concepts in partial metric spaces (for details see, e.g., [5, 8, 27, 30, 37, 47]).

Definition 1.7. (1) A sequence in the PMS converges to a limit point if and only if .
(2) A sequence in the PMS is called a Cauchy sequence if exists and is finite.
(3)β βA PMS is called complete if every Cauchy sequence in converges, with respect to , to a point such that .
(4) A mapping is said to be continuous at if, for every , there exists such that .

Lemma 1.8. (1) A sequence is a Cauchy sequence in the PMS if and only if it is a Cauchy sequence in the metric space .
(2) A PMS is complete if and only if the metric space is complete. Moreover,

In [32], Romaguera introduced the concepts of a 0-Cauchy sequence in a partial metric space and of a 0-complete partial metric space as follows.

Definition 1.9. A sequence in a partial metric space is called 0-Cauchy if . A partial metric space is said to be 0-complete if every 0-Cauchy sequence in converges, with respect to , to a point such that . In this case, is said to be a 0-complete partial metric on .

Remark 1.10 (see [32]). Each 0-Cauchy sequence in is a Cauchy sequence in , and each complete partial metric space is 0-complete.

The following example shows that there exists a 0-complete partial metric space that is not complete.

Example 1.11 1.11 (see [32]). Let be a partial metric space, where denotes the set of rational numbers and the partial metric is given by .

Now, we state a number of simple but useful lemmas that can be derived by manipulating the properties .

Lemma 1.12. Let be a PMS. Then, the following statements hold true: (A) if and , then for every (see [5, 27, 30]), (B) if , , and , then (see [31]), (C) if then (see [27, 30]), (D) if , then (see [27, 30]).

One of the implications of Lemma 1.12 can be stated as follows.

Remark 1.13. If converges to in , then for all .

We would like to point out that the topology induced by a partial metric differs from the topology induced by a metric in certain aspects. The following remark and example highlight one of these aspects.

Remark 1.14. Limit of a sequence in a partial metric space may not be unique.

Example 1.15. Consider with . Then, is a partial metric space. Clearly, is not a metric. Observe that the sequence converges to both and , for example. Therefore, there is no uniqueness of the limit in this partial metric space.

2. Main Results

In this section we aim to state and prove our main results. These results are more general in the sense that they are applicable to chains of cyclic representations and related generalized weak -contractions introduced in the following two definitions.

Definition 2.1 (see, e.g., [13]). Let be a nonempty set, , a positive integer, and an operator. The union is called a cyclic representation of with respect to if (1) each is a nonempty set for , (2), .

Definition 2.2 (cf. [13]). Let be a partial metric space, be a positive integer, closed nonempty subsets of and . An operator is called a cyclic generalized weak -contraction if (1) is a cyclic representation of with respect to , (2) there exists a continuous, nondecreasing function with for every and , such that where for any for with .

Our main theorem in this work is stated below.

Theorem 2.3. Let be a 0-complete partial metric space, a positive integer, and be closed non-empty subsets of . Let be a cyclic representation of with respect to . Suppose that is a cyclic generalized weak -contraction. Then, has a unique fixed point .

Proof. First we show that if has a fixed point, then it is unique. Suppose on the contrary that there exist such that and with . Due to (2.1), we have which is equivalent to by (PM3) in Definition 1.1. This is a contradiction because by Lemma 1.12 and for all . Hence, has a unique fixed point.
Next we prove the existence of a fixed point of . Let and be the Picard iteration. If there exists such that , then the theorem follows. Indeed, we get and, therefore, is a fixed point of . Thus, we may assume that for all where we have by Lemma 1.12.
Since is cyclic, for any , there is such that and . Then, by (2.1), we have where If we assume , then (2.6) turns into which is a contradiction. Because when we let , which is positive by (2.5), we get , we need to take . Define . Then, the inequality in (2.6) turns into Therefore, is a nonnegative nonincreasing sequence. Hence, converges to . We aim to show that . Suppose on the contrary that . Letting in (2.9), we get that Thus, we obtain that Since , then for . Taking in the previous inequality, we derive which contradicts with (2.11). Thus, we have .
We claim that is a 0-Cauchy sequence. In order to prove this assertion, we first prove the following statement.(A) For every , there exists such that if with , then . Assume the contrary that there exists such that for any we can find with satisfying the inequality . Then, we have by the triangular inequality. Since and lie in different adjacently labeled sets and for certain , from (2.13) we get by using the contractive condition in (2.1), where
If is equal to either or , then letting in (2.14) yields that is a contradiction. Hence, and inequality (2.14) give As and is nondecreasing, we have Inequalities (2.17) and (2.18) give us Letting with and since , we have which is a contradiction. Therefore, (A) is proved.
Now, in order to prove that is a Cauchy sequence, we fix an . Using (A), we find such that if with , On the other hand, since , we also find such that for any . We take with . Then, there exists such that . Therefore, for , and so From (2.21) and (2.22) and the last inequality, Hence, This prove that is a 0-Cauchy sequence.
Since is a 0-complete partial metric space, there exists such that . Now, we will prove that is a fixed point of . In fact, since converges to and is a cyclic representation of with respect to , the sequence has infinite number of terms in each for . Regarding that is closed for , we have . Now fix such that and . We take a subsequence of with (the existence of this subsequence is guaranteed by the mentioned comment above). Using the triangular inequality and the contractive conditions, we can obtain where . If is equal to either or , then by letting , inequality (2.26) implies Therefore, , that is, is a fixed point of by Lemma 1.12. If then (2.26) turns into Letting , we get that . Hence, by the properties of we have .

Theorem 2.4. Let be a self-mapping as in Theorem 2.3. Then, the fixed point problem for is well posed, that is, if there exists a sequence in with , as , then as .

Proof. Due to Theorem 2.3, we know that for any initial value , is the unique fixed point of . Thus, is well defined. Consider where . There are three cases to consider: if , then (2.29) is equivalent to Since we are given as , we derive that . Regarding the definition of , we obtain that . The theorem follows.
If it is the case that , then the right-hand side of (2.29) tends to as tends to infinity. Hence, the theorem follows again.
As the last case, we have . Similarly we conclude that the right-hand side of (2.29) tends to as tends to infinity because we know that by Theorem 2.3 and , which completes the proof.

Theorem 2.5. Let be a self-mapping as in Theorem 2.3. Then, has the limit shadowing property, that is, if there exists a convergent sequence in with and , as , then there exists such that as .

Proof. As in the proof of Theorem 2.4, we observe that for any initial value , is the unique fixed point of . Thus, are well defined. Set as a limit of a convergent sequence in . Consider where . There are three cases to consider: If , then (2.31) is equivalent to as . This is possible only if , which implies that and thus . Thus, we have , as .
If it is the case that , then the right-hand side of (2.31) tends to as tends to infinity. Hence, the theorem follows again.
As the last case, we have . Similarly we conclude that the right-hand side of (2.31) tends to as tends to infinity because we know that by Theorem 2.3 and , which completes the proof.

Theorem 2.6. Let be a non-empty set, and partial metric spaces, a positive integer, and closed non-empty subsets of and . Suppose that (1) is a cyclic representation of with respect to . (2) for all , (3) is a complete partial metric space, (4) is continuous, (5) is a cyclic weak -contraction where with is a lower semicontinuous function for and . Then, converges to in for any and is the unique fixed point of .

Proof. Let . As in Theorem 2.3, assumption (5) implies that is a Cauchy sequence in . Taking (2) into account, is a Cauchy sequence in , and due to (3) it converges to in for any . Condition (4) implies the uniqueness of .

Definition 2.7 (cf. [2]). Let be a partial metric space, a positive integer, and closed non-empty subsets of and . An operator is called a cyclic generalized -contraction if (1) is a cyclic representation of with respect to , and (2) there exists a continuous, non-decreasing function with for every and , such that βwhere βfor any for with .

The following is an analog of the main theorem in [2].

Corollary 2.8. Let be a 0-complete partial metric space, a positive integer, and closed non-empty subsets of . Let be a cyclic representation of with respect to . Suppose that is a cyclic generalized -contraction. Then, has a unique fixed point .

Proof. It follows from Theorem 2.3 by taking .

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