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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 825293, 6 pages
http://dx.doi.org/10.1155/2013/825293
Research Article

Ulam-Hyers Stability Results for Fixed Point Problems via - -Contractive Mapping in ( )-Metric Space

1Department of Mathematics, Babeş-Bolyai University, Kogălniceanu Street No. 1, 400084 Cluj-Napoca, Romania
2Department of Mathematics, Atilim University, İncek, 06836 Ankara, Turkey

Received 17 May 2013; Accepted 28 July 2013

Academic Editor: Janusz Brzdek

Copyright © 2013 Monica-Felicia Bota 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 will investigate some existence, uniqueness, and Ulam-Hyers stability results for fixed point problems via - -contractive mapping of type-( ) in the framework of -metric spaces. The presented theorems extend, generalize, and unify several results in the literature, involving the results of Samet et al. (2012).

1. Introduction and Preliminaries

Very recently, Samet et al. [1] introduced the notion of - -contractive type mapping and proved some fixed point results for such mapping. The authors [1] also reported that several fixed point theorems, including the celebrated Banach contraction mapping principle, can be derived from their main results.

The aim of this manuscript is to investigate the existence and/or uniqueness of a fixed point of - -contractive type mapping in the context of a -metric space, a generalization of a usual metric space, which was introduced by Czerwik [2, 3]. In fact, such general settings of metric spaces were considered earlier, for example, by Bourbaki [4], Bakhtin [5], and Heinonen [6]. Following these initial papers, -metric spaces and related fixed point theorems have been investigated by a number of authors; see for example, Boriceanu et al. [7], Boriceanu [8, 9], Bota [10], and Aydi et al. [11, 12]. As a subsidiary purpose, we consider Ulam-Hyers stability of the observed results. The stability problem of functional equations, originated from a question of Ulam [13], in 1940, concerns the stability of group homomorphisms. The first affirmative partial answer to the question of Ulam for Banach spaces was given by Hyers [14] in 1941. Thereafter, this type of stability is called the Ulam-Hyers stability and has attracted attention of several authors. In particular, Ulam-Hyers stability results in fixed point theory have been studied densely; see for example, Bota-Boriceanu and Petruşel [15], Lazăr [16], Rus [17, 18], and F. A. Tişe and I. C. Tişe [19]. Moreover, there are several remarkable results on the stability of certain classes of functional equations via fixed point approach. Most particularly and recently, Brzdęk et al. [20], Brzdęk and Ciepliński [21, 22], and Cadariu [23] reported some interesting results in this direction.

We recollect some essential definitions and fundamental results. We first recall the definition of a -metric space.

Definition 1 (Bakhtin [5], Czerwik [2]). Let be a set, and let be a given real number. A functional is said to be a -metric space if the following conditions are satisfied: (1) if and only if , (2) , (3) , for all , , and . A pair is called a -metric space.

It is clear that -metric turns into usual metric when we take . Hence, we conclude that the class of -metric spaces is larger than the class of usual metric spaces. For more details and examples on -metric spaces, see, for example, [26, 24]. We state the following example for the sake of completeness.

Example 2. Let be a set with the cardinal . Suppose that is a partition of such that . Let be arbitrary. Then, the functional defined by is a -metric on with coefficient .

Let be a -metric space. We consider next the following families of subsets of

In this case is a generalized functional on a -metric space defined by In particular, if , then .

The following basic lemmas will be useful in the proof of the main results.

Lemma 3 (Czerwik [2]). Let be a -metric space. Then, ones has

Lemma 4 (Czerwik [2]). Let be a -metric space, and let . Then

A mapping is called a comparison function if it is increasing and , , for any . We denote by the class of the comparison function . For more details and examples, see, for example, [25, 26]. Among them, we recall the following essential result.

Lemma 5 (Berinde [26], Rus [25]). If is a comparison function, then(1)each iterate of , , is also a comparison function; (2) is continuous at ; (3) , for any .

Later, Berinde [26] introduced the concept of -comparison function in the following way.

Definition 6 (Berinde [26]). A function is said to be a -comparison function if ( ) is increasing, ( ) there exist , , and a convergent series of nonnegative terms such that , for and any .
The notion of a -comparison function was improved as a -comparison function by Berinde [27] in order to extend some fixed point results to the class of -metric space.

Definition 7 (Berinde [27]). Let be a real number. A mapping is called a -comparison function if the following conditions are fulfilled:(1) is monotonically increasing;(2)there exist , , and a convergent series of nonnegative terms such that , for and any .

We denote by for the class of -comparison function . It is evident that the concept of -comparison function reduces to that of -comparison function when .

The following lemma has a crucial role in the proof of our main result.

Lemma 8 (Berinde [24]). If is a -comparison function, then ones has the following:(1)the series converges for any ; (2)the function defined by , is increasing and continuous at .

We note that any -comparison function is a comparison function due to Lemma 8.

Next, we will present the definition of - -contractive and -admissible mappings introduced by Samet et al. [1].

We denote by the family of nondecreasing functions such that for each . It is clear that if (see, e.g., [28]) and, hence, by Lemma 5 (14), for , we have , for any .

Definition 9 (Samet et al. [1]). Let be a metric space and a given mapping. One says that is an - -contractive mapping if there exist two functions and such that

Remark 10. If satisfies the Banach contraction principle, then is an - -contractive mapping, where for all and for all and some .

Definition 11 (Samet et al. [1]). Let and . One says that is -admissible if

Let be the class of fixed points of a self-mapping defined on a nonempty set ; that is, .

Example 12 (Samet et al. [1]). Let . Define and by , for all , and Then is -admissible. , for all , and Then is -admissible.

Example 13. Let be a partially ordered set and a metric on such that is complete. Let be a nondecreasing mapping with respect to ; that is, . Suppose that there exists such that . Define the mapping by Then, is -admissible. Since there exists such that , we have . On the other hand, for all , from the monotone property of , we have Thus is -admissible.

Theorem 14 (Samet et al. [1]). Let be a complete metric space and an - -contractive mapping satisfying the following conditions: (i) is -admissible; (ii)there exists such that ;(iii) is continuous. Then, has a fixed point; that is, there exists .

Theorem 15 (Samet et al. [1]). Let be a complete metric space and an - -contractive mapping satisfying the following conditions: (i) is -admissible; (ii)there exists such that ;(iii)if is a sequence in such that for all and as , then for all . Then, has a fixed point; that is, there exists .

2. Main Results

First we give the following definition as a generalization of Definition 9.

Definition 16. Let a -metric space and be a given mapping. We say that is an - -contractive mapping of type- if there exist two functions and such that

Our first main result is the following.

Theorem 17. Let be a complete -metric space with constant . Let be an - -contractive mapping of type- satisfying the following conditions: (i) is -admissible; (ii)there exists such that ;(iii) is continuous. Then the fixed point equation (29) has a solution; that is, there exists .

Proof. Let such that (such a point exists from condition (ii)). Define the sequence in by If for some , then is a fixed point for , and the proof finishes. Hence we assume that
Since is -admissible, we have By induction, we get Applying the inequality (12) with and and using (16), we obtain By induction, we get From (18) and using the triangular inequality, for all , we have
Denoting , , we obtain Due to the assumption (14) and Lemma 8, we conclude that the series is convergent. Thus there exists . Regarding and by (20), we obtain that is a Cauchy sequence in the -metric space . Since is complete, there exists such that as . From the continuity of , it follows that as . By the uniqueness of the limit, we get ; that is, is a fixed point of .

In the following theorem, we are able to omit the continuity hypothesis of by adding a new condition.

Theorem 18. Let be a complete -metric space with constant . Let be an - -contractive mapping of type- satisfying the following conditions: (i) is -admissible; (ii)there exists such that ;(iii)if is a sequence in such that for all and as , then for all . Then the fixed point equation (29) has a solution.

Proof. Following the proof of Theorem 17, we know that is a Cauchy sequence in the complete -metric space . Then, there exists such that as . On the other hand, from (16) and the hypothesis (iii), we have Now, using the triangular inequalities, (12) and (21), we get Letting , since is continuous at , we obtain ; that is, .

To assure the uniqueness of the fixed point, we will consider the following hypothesis:

Theorem 19. Adding condition to the hypotheses of Theorem 17 (resp., Theorem 18) one obtains uniqueness of the fixed point of .

Proof. Suppose that and are two fixed points of . From , there exists such that Since is -admissible, from (23), we get Using (24) and (12), we have This implies that Then, letting , we have Similarly, using (24) and (12), we get Using (27) and (28), the uniqueness of the limit gives us . This finishes the proof.

Remark 20. Theorem 14 (resp., Theorem 15) can be derived from Theorem 17 (resp., Theorem 18) by taking . Consequently, all results in [1] can be considered as corollaries of our main results.

3. Ulam-Hyers Stability Results through the Fixed Point Problems

Definition 21. Let be a metric space and an operator. By definition, the fixed point equation is called generalized Ulam-Hyers stability if and only if there exists which is increasing, continuous at and such that for every and for each an -solution of the fixed point equation (29), that is, , satisfies the inequality There exists a solution of (29) such that
If there exists such that , for each , then the fixed point equation (29) is said to be Ulam-Hyers stability.

For Ulam-Hyers stability results in the case of fixed point problems see Bota-Boriceanu, Petruşel [15], Lazăr [16], and Rus [17, 18].

Regarding the Ulam-Hyers stability problem the ideas given in Petru et al. [29] allow us to obtain the following result.

Theorem 22. Let be a complete -metric space with constant . Suppose that all the hypotheses of Theorem 19 hold and additionally that the function , is strictly increasing and onto. Then the following hold.(a)The fixed point equation (29) is generalized Ulam-Hyers stability. (b) and if , are such that , as , then , as ; that is, the fixed point equation (29) is well posed. (c)If is such that there exists with then

Proof. (a) Since is a Picard operator, so . Let and be a solution of (30); that is, Since is - -contractive mapping of type- and since , from , there exists such that ; we obtain Therefore, Consequently, the fixed point equation (29) is generalized Ulam-Hyers stability.
(b) Since is - -contractive mapping of type- and since , from , there exists such that ; we obtain Therefore So, the fixed point equation (29) is well posed.
(c) Since is - -contractive mapping of type- and since , from , there exists such that ; we obtain Therefore So, we have the following estimation: Writing (41) for we get

Acknowledgments

The first author is supported by a grant from the Romanian National Authority for Scientific Research, CNCS UEFISCDI, Project no. PN-II-ID-PCE-2011-3-0094. The last author is supported by the financial support of the Sectoral Operational Programme for Human Resources Development 2007–2013, cofinanced by the European Social Fund, under the Project no. POSDRU/107/1.5/S/76841 with the title Modern Doctoral Studies: Internationalization and Interdisciplinarity.

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