Discrete Dynamics in Nature and SocietyVolume 2012 (2012), Article ID 346561, 13 pagesdoi:10.1155/2012/346561

Research Article

On the Stability of an -Variables Functional Equation in Random Normed Spaces via Fixed Point Method

A. Ebadian,¹ M. Eshaghi Gordji,² H. Khodaei,² R. Saadati,³ and Gh. Sadeghi⁴

¹Department of Mathematics, Payame Noor University, Tehran, Iran²Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran³Department of Mathematics, Iran University of Science and Technology, Behshahr, Iran⁴Department of Mathematics and Computer Sciences, Tarbiat Moallem University Sabzevar, Sabzevar, P.O. Box 397, Iran

Received 17 September 2011; Revised 5 January 2012; Accepted 29 January 2012

Academic Editor: Seenith Sivasundaram

Copyright © 2012 A. Ebadian 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.

At first we find the solution of the functional equation where is an integer number. Then, we obtain the generalized Hyers-Ulam-Rassias stability in random normed spaces via the fixed point method for the above functional equation.

1. Introduction and Preliminaries

A basic question in the theory of functional equations is as follows: “when is it true that a function that approximately satisfies a functional equation must be close to an exact solution of the equation?”

If the problem accepts a solution, we say the equation is stable. The first stability problem concerning group homomorphisms was raised by Ulam [1] in 1940 and affirmatively solved by Hyers [2]. The result of Hyers was generalized by Aoki [3] for approximate additive function and by Rassias [4] for approximate linear functions by allowing the difference Cauchy equation to be controlled by . Taking into consideration a lot of influence of Ulam, Hyers, and Rassias on the development of stability problems of functional equations, the stability phenomenon that was proved by Rassias is called the Hyers-Ulam-Rassias stability. In 1994, a generalization of Rassias theorem was obtained by Găvruţa [5], who replaced by a general control function (see also [6–24]).

In the sequel we adopt the usual terminology, notations, and conventions of the theory of random normed spaces, as in [25–29]. Throughout this paper, let be the space of distribution functions, that is, and the subset is the set where denotes the left limit of function at the point . The space is partially ordered by the usual pointwise ordering of functions, that is, if and only if for all . The maximal element for in this order is the distribution function given by

Definition 1.1 (see [28]). A mapping is a continuous triangular norm (briefly, a -norm) if satisfies the following conditions:(a) is commutative and associative,(b) is continuous,(c) for all ;(d) whenever and for all .

Typical examples of continuous -norms are , , and (the Łukasiewicz -norm).

Recall (see [30, 31]) that if is a -norm and is a given sequence of numbers in , is defined recurrently by is defined as .

It is known [31] that for the Łukasiewicz -norm the following implication holds:

Definition 1.2 (see [29]). A random normed space (briefly, RN space) is a triple , where is a vector space, is a continuous -norm, and is a mapping from into such that the following conditions hold:(RN1) for all if and only if ,(RN2) for all , ,(RN3) for all and .

Definition 1.3. Let be an RN space.(1)A sequence in is said to be convergent to in if, for every and , there exists positive integer such that whenever .(2)A sequence in is called Cauchy if, for every and , there exists positive integer such that whenever .(3)An RN space is said to be complete if and only if every Cauchy sequence in is convergent to a point in . A complete RN space is said to be a random Banach space.

Theorem 1.4 (see [28]). If is an RN space and is a sequence such that , then almost everywhere.

Theorem 1.5 (see [32, 33]). Let be a complete generalized metric space, and let be a strictly contractive mapping with Lipschitz constant . Then, for each given element , either for all nonnegative integers or there exists a positive integer such that (1), for all ,(2)the sequence converges to a fixed point of ,(3) is the unique fixed point of in the set ,(4) for all .

The theory of random normed spaces (RN spaces) is important as a generalization of deterministic result of linear normed spaces and also in the study of random operator equations. The notion of an RN space corresponds to the situations when we do not know exactly the norm of point and we know only probabilities of possible values of this norm. The RN spaces may also provide us the appropriate tools to study the geometry of nuclear physics and have important application in quantum particle physics. The generalized Hyers-Ulam stability of different functional equations in random normed spaces (RN spaces) and fuzzy normed spaces has been recently studied in, Alsina [34], Mirmostafaee et al. [35–38], Miheţ and Radu [26, 27, 39, 40], Miheţ et al. [41, 42], Baktash et al. [43] and Saadati et al. [44].

In this paper, we consider the -dimensional additive functional equation where is an integer number. It is easy to see that the function is a solution of the functional equation (1.7).

As a special case, if in (1.7), then the functional equation (1.7) reduces to Also by putting in (1.7), we obtain that is,

The main purpose of this paper is to prove the stability of (1.7) in random normed spaces via the fixed point method.

2. Results in RN spaces via Fixed Point Method

Lemma 2.1. Let and be real vector spaces. A function with satisfies (1.7) if and only if is additive.

Proof. Let satisfy the functional equation (1.7). Hence, according to (1.7), we get for all . Setting in (2.1), we have that is, for all . On the other hand, we have the relation for all . Hence, we obtain from (2.3) and (2.4) that for all . Setting in (2.5) we get for all . Replacing and by and in (2.5), respectively, and then using , we obtain that for all , which implies that is additive.Conversely, suppose that is additive, and thus satisfies the equation . Hence we have and for all . Replacing and by and in the additive equation and then using lead to for all .Now, we are going to prove our assumption by induction on . It holds for ; see (2.7). Assume that (1.7) holds for the case, where ; that is, we have for all . Replacing by in (2.8), we obtain for all . Replacing by in (2.9), we obtain for all . Adding (2.9) to (2.10), one gets for all . Therefore, it follows from (2.7) and (2.11) that (1.7) holds for . This completes the proof of the theorem.

From now on, let be a linear space and a complete RN space. For convenience, we use the following abbreviation for a given function : for all , where is an integer number.

Theorem 2.2. Let be a function ( is denoted by ) such that, for some , for all and all . Suppose that a function with satisfies the inequality for all and all . Then, there exists a unique additive function such that for all and all .

Proof. Letting in (2.14), we get for all and all . Setting in (2.16), we obtain from (2.4) and that for all and all , or for all and all . Let be the set of all functions with and introduce a generalized metric on as follows: where, as usual, . It is easy to show that is a generalized complete metric space [26, 45].Now we consider the function defined by for all and .Now let such that . Then, that is, if , we have . This means that for all , that is, is a strictly contractive self-function on with the Lipschitz constant .It follows from (2.18) that for all and all , which implies that .Due to Theorem 1.5, there exists a function such that is a fixed point of , that is, for all .Also, as , implies the equality for all . If we replace with in (2.14), respectively, and divide by , then it follows from (2.13) that for all and all . By letting in (2.25), we find that for all , which implies , and thus satisfies (1.7). Hence by Lemma 2.1, the function is additive.According to the fixed point alterative, since is the unique fixed point of in the set , is the unique function such that for all and all . Again using the fixed point alterative gives which implies the inequality for all and all . So, for all and all . This completes the proof.

Now, we present a corollary that is an application of the last theorem in the classical case.

Corollary 2.3. Let and , normed linear spaces, define for and . Define for all and all in which . Now, for , (2.13) holds for all and all . Suppose that an odd function satisfies (2.14) for all and all . Then, by the last theorem there exists a unique additive function such that for all and all . Hence, for all .

Theorem 2.4. Let be a function such that, for some , for all and all . Suppose that an odd function satisfies (2.14) for all and all . Then, there exists a unique additive function such that for all and all .

Proof. Letting in (2.14), we get for all and all . Setting in the last inequality, we obtain by using oddness of and (2.4) that for all and all , or for all and all . Let be the set of all odd functions , and introduce a generalized metric on as follows: It is easy to show that is a generalized complete metric space [26, 45]. Let be the function defined by for all and . One can show that for all , that is, is a strictly contractive self-function on with the Lipschitz constant .It follows from (2.38) that for all and all , which implies that .Due to Theorem 1.5, the sequence converges to a fixed point of , that is, and for all .Also, is the unique fixed point of in the set , and implies the inequality for all and all . This implies that inequality (2.35) holds. Furthermore, we can obtain that the function satisfies (1.7). Hence by Lemma 2.1, we get that the function is additive.

Now, we present a corollary that is an application of the last theorem in the classical case.

Corollary 2.5. Let and , normed linear spaces, define for and . Define for all and all in which . Now, for , (2.34) holds for all and all . Suppose that an odd function satisfies (2.14) for all and all . Then, by the last theorem there exists a unique additive function such that for all and all . Hence, for all .

Acknowledgments

The authors would like to thank the referees and the Editor Professor Seenith Sivasundaram for giving useful suggestions for the improvement of this paper.

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