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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 269701, 10 pages
http://dx.doi.org/10.1155/2012/269701
Research Article

Probabilistic (Quasi)metric Versions for a Stability Result of Baker

Department of Mathematics, West University of Timişoara, 4 V. Pârvan Boulevard, 300223 Timişoara, Romania

Received 8 June 2012; Revised 15 October 2012; Accepted 18 October 2012

Academic Editor: Bing Xu

Copyright © 2012 Dorel Miheţ and Claudia Zaharia. 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

By using the fixed point method, we obtain a version of a stability result of Baker in probabilistic metric and quasimetric spaces under triangular norms of Hadžić type. As an application, we prove a theorem regarding the stability of the additive Cauchy functional equation in random normed spaces.

1. Introduction

The use of the fixed point theory in the study of Ulam-Hyers stability was initiated by Baker in the paper [1]. Baker used the classical Banach fixed point theorem to prove the stability of the nonlinear functional equation His result reads as follows.

Theorem 1.1 (see [1, Theorem 2]). Suppose is a nonempty set, is a complete metric space, , , , and Also, suppose that , , and Then there exists a unique mapping such that

Starting with the papers [2, 3], the fixed point method has become a fundamental tool in the study of Ulam-Hyers stability. In the probabilistic and fuzzy setting, this approach was first used in the papers [4, 5] for the case of random and fuzzy normed spaces endowed with the strongest triangular norm . In fact, by identifying a suitable deterministic metric, the stability problem in such spaces was reduced to a fixed point theorem in generalized metric spaces. This idea was adopted by many authors, see for example, [611]. It is worth noting that, in applying this method, the fact that the triangular norm is is essential.

In this paper we study the stability of (1.1) when the unknown takes values in a probabilistic (quasi-) metric space endowed with a triangular norm of Hadžić type. To this end, we employ the fixed point theory in probabilistic metric spaces, rather than that in metric spaces.

2. Hyers-Ulam Stability of the Equation in Probabilistic Metric Spaces

In this section, we study the stability of the equation , where the unknown function is a mapping from a nonempty set to a probabilistic metric space , and and are given mappings.

We assume that the reader is familiar with the basic concepts of the theory of probabilistic metric spaces. As usual, denotes the space of all functions , such that is left-continuous and nondecreasing on , , and denotes the subspace of consisting of functions with . Here we adopt the terminology from [12], hence the probabilistic metric takes values in .

We recall some facts from the fixed point theory in probabilistic metric spaces.

Definition 2.1. A -norm is said to be of -type [13] if the family of its iterates , given by , and for all , is equicontinuous at .

A trivial example of a -norm of -type is the -norm , , but there exist -norms of -type different from Min [14].

The theorem below provides a characterization of continuous -norms of -type.

Proposition 2.2 (see [15]). Suppose that there exists a strictly increasing sequence in such that and . Then is of -type.
Conversely, if is continuous and of -type, then there exists a sequence as in .

Definition 2.3 (see [16]). Let be a probabilistic metric space. A mapping is said to be a Sehgal contraction (or -contraction) if the following relation holds:

Theorem 2.4 (see [17]). Let be a complete probabilistic metric space with of Hadži-type and be a -contraction. Then has a fixed point if and only if there is such that . If , then is the unique fixed point of in the set .

The following lemma completes Theorem 2.4 with an estimation relation, in the case .

Lemma 2.5 (see [18]). Let be a complete probabilistic metric space and be a contraction. Suppose that and let . Then

This lemma can be extended to the case of probabilistic metric spaces under a continuous -norm of -type.

Lemma 2.6. Let be a complete probabilistic metric space, with a continuous t-norm of -type and be a strictly increasing sequence of idempotents of . Suppose is a -contraction with Lipschitz constant . If there exists such that , then is the unique fixed point of in the set Moreover, if is so that , then .

Proof. We have to prove only the last part of the theorem. We show by induction on that implies , for all .
The case is obvious. Now, suppose that . Then
Let be such that , and let . Then for all . Since converges to , goes to 1 as tends to infinity, so By taking we obtain

In order to state our first stability result, we define an appropriate concept of approximate solution for the functional equation (1.1).

Definition 2.7. A probabilistic uniform approximate solution of (1.1) is a function with the property that uniformly on .

Example 2.8. Let be a metric space, and let be defined by Then is a probabilistic metric space (the induced probabilistic metric space). One can easily verify that is a probabilistic uniform approximate solution of (1.1) if and only if it satisfies relation (1.3), thus being an approximate solution in the sense of Theorem 1.1.

Theorem 2.9. Let be a nonempty set, be a complete probabilistic metric space, with a continuous t-norm of -type, and be a strictly increasing sequence of idempotents of . Suppose is a mapping for which there exists with for all , and .
If is a probabilistic uniform approximate solution of (1.1), then there exists a function which is an exact solution of (1.1), with the property that, if is such that then

Proof. Denote by the set of all mappings , and let be Baker’s operator, given by for all . We define the distribution function by for all .
The assumptions on the space ensure that is a complete probabilistic metric space. Also, that is, is a Sehgal contraction on .
Moreover, the relation , uniformly on implies
Now we can apply Lemma 2.6 to obtain a fixed point of , that is a mapping which is a solution of (1.1), with for all .
Next, let be such that for all . Then, from the left continuity of , it follows that , for some . Therefore , so . By Lemma 2.6, , whence we conclude that the estimation (2.12) holds.

Remark 2.10. The result of Baker [1] can be obtained as a particular case of Theorem 2.9, by considering in this theorem the induced probabilistic metric space (see Example 2.8).

From Theorem 2.9 one can derive a stability result for the Cauchy additive functional equation in random normed spaces.

Recall (see [12]) that a random normed space (-space) is a triple , where is a real linear space, is a mapping from to , and is a -norm, satisfying the following conditions ( will be denoted by ):(i) for all iff , the null vector of ;(ii), for all , , and all ;(iii), for all and all .

Definition 2.11. A probabilistic uniform approximate solution of (2.16) is a function with the property that uniformly on .

Theorem 2.12. Let be a real linear space, be a complete -space with —a continuous t-norm of -type, and be a strictly increasing sequence of idempotents of .
If is a probabilistic uniform approximate solution of (2.16), then there exists a mapping which is an exact solution of (2.16), with the property that, if is such that then

Proof. We apply Theorem 2.9 for , , and , in the probabilistic metric space with defined by for all , . Note that satisfies (2.10) for , since for all , and .
It is easy to see that is a probabilistic uniform approximate solution of (1.1), so there exists an exact solution of (1.1), that is, a mapping satisfying for all . The estimation (2.19) can be immediately derived from the corresponding one in Theorem 2.9.
It remains to show that is additive. This follows from the fact that , for all , and is a probabilistic uniform approximate solution of (2.16). Namely, for all , implying for all .

3. Hyers-Ulam Stability of the Equation in Probabilistic Quasimetric Spaces

The defining feature of quasimetric structures is the absence of symmetry. This allows one to consider different notions of convergence and completeness. We state the terminology and notations, following [19] (also see [20]).

Definition 3.1. A probabilistic quasimetric space is a triple , where is a nonempty set, is a -norm, and is a mapping satisfying(i) if and only if ;(ii), for all , for all .

We note that if verifies the symmetry assumption , for all , then is a probabilistic metric space.

If is a probabilistic quasimetric space, then the mapping defined by for all is called the conjugate probabilistic quasimetric of .

Definition 3.2. Let be a probabilistic quasimetric space. A sequence in is said to be:(i)right -Cauchy (left -Cauchy) if, for each and , there exists so that, for all , ( resp.);(ii)-convergent (-convergent) to if, for each and , there exists so that (), for all .

Definition 3.3. Let and . The space is complete if every -Cauchy sequence is convergent.

Definition 3.4. The probabilistic quasimetric space has the - (-) property if every -(-) convergent sequence has a unique limit.

The following lemma is a quasimetric analogue of Lemma 2.6.

Lemma 3.5. Let be a -complete probabilistic quasimetric space with the property, where is a continuous t-norm of -type. Let be a strictly increasing sequence of idempotents of .
Suppose is a Sehgal contraction with Lipschitz constant , and is an element of such that . Then is a fixed point of and if is so that , then .

Proof. We proceed in the classical manner to show that the sequence of iterates is right -Cauchy, therefore it is -convergent to . The fact that is a fixed point of is a consequence of the property of the space . Next, as in the proof of Lemma 2.6 we show by induction on that implies , for all .
Let be such that , and let . Then for all . Since is -convergent to , goes to 1 as tends to infinity, so By taking we obtain

The probabilistic quasimetric version of Baker’s theorem can be stated as follows.

Theorem 3.6. Let be a nonempty set, be a -complete probabilistic quasimetric space with the property, with a continuous t-norm of -type, and be a strictly increasing sequence of idempotents of . Suppose is a mapping for which there exists with for all , and .
If is a probabilistic uniform approximate solution of (1.1), then there exists a function which is an exact solution of (1.1), with the property that, if is such that then

Proof. We only sketch the proof, as it is very similar to that of Theorem 2.9.
As in the mentioned proof, denote by the set of all mappings , and define the distribution function by for all and Baker’s operator , for all .
The assumptions on the space ensure that is a -complete probabilistic quasimetric space with the property and that is a Sehgal contraction on , and the relation , uniformly on implies
We can now apply Lemma 3.5 to obtain a mapping which is a solution of (1.1), with for all .
The estimation (3.6) follows by using the left continuity of , as in the proof of Theorem 2.9.

Acknowledgments

The work of D. Miheţ was supported by a Grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, no. PN-II-ID-PCE-2011-3-0087. The work of C. Zaharia was supported by the strategic Grant POSDRU/CPP107/DMI1.5/S/78421, Project ID 78421 (2010), cofinanced by the European Social Fund—Investing in People, within the Sectoral Operational Programme Human Resources Development 2007–2013.

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