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Abstract and Applied Analysis
Volume 2012, Article ID 518734, 9 pages
http://dx.doi.org/10.1155/2012/518734
Research Article

A Generalization of Ćirić Quasicontractions

1Department of Mathematics, Atilim University, İncek, 06836 Ankara, Turkey
2Department of Mathematics, Vinh University, 182 Le Duan, Vinh City, Vietnam

Received 26 October 2011; Accepted 4 December 2011

Academic Editor: Simeon Reich

Copyright © 2012 Erdal 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

We proved a fixed point theorem for a class of maps that satisfy Ćirić's contractive condition dependent on another function. We presented an example to show that our result is a real generalization.

1. Introduction and Preliminaries

The fixed point theorems have extensive applications in many disciplines such as in mathematics, economics, and computer science (see, e.g., [15]). The Banach contraction mapping principle [6] is the main motivation of this theory. A self-mapping on a metric space is called a contraction if, for each , there exists a constant such that . Banach [6] proved that every contraction on a complete metric space has a unique fixed point.

Many authors have studied on various generalizations of Banach contraction mapping principle. For important and famous examples of such generalizations, we may cite Kannan [7], Reich [810], Chatterjea [11], Hardy and Rogers [12], and Ćirić [13, 14]. In this manuscript, we focused on Ćirić type contractions and generalized some of his results [13, 14].

In [14], Ćirić introduced a class of self-maps on a metric space which satisfy the following condition: for every and . The maps satisfying Condition (1.1) are said to be quasicontractions.

Let be a self-mapping on a metric space . For and for each , we set (cf. [14, 15]) (1), (2)for , (3).

A metric space is called -orbitally complete if and only if every Cauchy sequence of (for some ) converges to a point in . It is clear that if is complete, then is -orbitally complete for every . We now state the theorem of Ćirić [14].

Theorem 1.1. Let be a quasicontraction on -orbitally complete metric space . Then (1) has a unique fixed point in , (2), for .

Other generalizations of Banach's contraction principle and analogous results for metric spaces are abundantly present in the literature (see [719] and the references given therein). Rhoades [19] considered many types of contractive definitions and analyzed the relationships among them. Recently, some authors have provided a result on the existence of fixed points for a new class of contractive mappings.

Definition 1.2 (see, e.g., [20, 21]). Let be a metric and be two functions. A mapping is called to be a contraction if there exists such that

In the definition above, if we choose for all , then the contraction becomes a contraction. In [20, 22], some fixed point theorems are proved for contractions in metric spaces and in 2-metric spaces. In this paper, by the using the same techniques, we shall extend the Ćirić's theorem (Theorem 1.1). For this purpose, we recall the following elementary definition.

Definition 1.3 (see, e.g., [21, 22]). Let be a metric space. A mapping is called sequentially convergent if the statement is convergent and implies that is a convergent sequence for every sequence .

2. Main Results

The aim of this section is to prove the following result.

Theorem 2.1. Let be a metric space. Let be a one-to-one, continuous, and sequentially convergent mapping and be a self-mapping of . Suppose that there exists , for every and every Cauchy sequence of the form for converges in . Then (i), for all , for all and ,(ii), for all ,(iii) has a unique fixed point , (iv).

Proof. We will mainly follow the arguments in the proof of the Ćirić's theorem. Let be defined as in Theorem 2.1. We start with the proof of (i). Let be arbitrary element. Let be integer and . Then Note that we use as the identity self-mapping. Due to (2.1), we have Thus, is proved. Let us show . Observe that which implies that So we only need to show that for all . Indeed, since is a finite set for each and with , we infer that there exists such that Applying the triangle inequality and by , we get that Hence, . Thus, is proved.

To prove , choose and define the iterative sequence and as follows: We assert that is a Cauchy sequence. Without loss of generality, we may assume that , where . By (i), we obtain According to the assumption , there exists such that Again, by the mentioned assumption, we have So we can get the following system of inequalities: By the same argument, we obtain Since , we have This implies that is a Cauchy sequence. Since every Cauchy sequence of the form converges in , we deduce that Since is sequentially convergent, it follows that . By the hypothesis that is continuous, we get Therefore, . We shall show that . To achieve this, we consider the following inequalities: Hence, Letting and combining with the fact that , we can conclude that

Therefore, . Since is one to one, we obtain that . Now we will show that is the unique fixed point of . Suppose that and are fixed points of . Then , , and applying (2.1), Since , we infer that . Hence, . Since is one-to-one, we may conclude that . This proves of Theorem 2.1. Proof of is directly obtained from (2.17). The theorem is proved.

Remark 2.2. Theorem 1.1 is obtained from Theorem 2.1 if we choose for every .

The following example shows that Theorem 2.1 is indeed an extension of Theorem 1.1.

Example 2.3. Let with metric induced by . Then is a complete metric space. Consider the function for all . It is easy to compute that Therefore, if is large enough, we have It follows that the quasicontractive condition is equivalent to Letting , we get that which contradicts the fact that . Hence is not a quasicontraction. Therefore, we can not apply Theorem 1.1 for . However, it is easy to see that 0 is the unique fixedpoint of .
Now we shall show that satisfies Theorem 2.1 with and . Indeed, is one-to-one, continuous and sequentially convergent on . We have Now, we will show that If , then the inequality above holds. So we can assume that . Then (2.27) is equivalent to It follows from the fact that and are increasing, (2.28) reduces to Consider the function , . It is easy to check that is decreasing on . So we can deduce that for every . Therefore, we can apply Theorem 2.1 for .

As an immediate consequence of Theorem 2.1, we have the following corollary.

Corollary 2.4. Let be a metric space. Let be a one-to-one, continuous, and sequentially convergent mapping, and let be a self-mapping of . Suppose that there exists , for every and every Cauchy sequence of the form for converges in . Then,(a) has a unique fixed point , (b) for .

This corollary is also a consequence of the main result of [21], which is established by a contractive condition of integral type.

We also get the following corollary.

Corollary 2.5. Let be a metric space. Let be a one-to-one, continuous, and sequentially convergent mapping and be a self-mapping of . Suppose that there exists , for every , for some positive integer and every Cauchy sequence of the form for converges in . Then, has a unique fixed point .

Proof. Applying Theorem 2.1 for the map , we can deduce that has a unique fixed point . This yields that and . It follows that is a fixed point of . By the uniqueness of fixed point of , we infer that . Therefore, is a fixed point of . Now, if is a fixed point of , then . Hence, is a fixed point of , so that . This proves the uniqueness of fixed point of .

The following corollary is an extension of the main result of [12].

Corollary 2.6. Let be a metric space. Let be a one-to-one, continuous, and sequentially convergent mapping, and let be a self-mapping of . Suppose that there exists , satisfying that such that for every and every Cauchy sequence of the form for converges in . Then (a) has a unique fixed point . (b).

Proof. It is easy to see that the condition (2.1) is a consequence of the condition (2.33). This is enough to prove the corollary.

Remark 2.7. In the same way in Corollary 2.6, we can get the extensions of Kannan's contractions (see [7]), and Reich's contractions (see [810]).

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