Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 672531 | 15 pages | https://doi.org/10.1155/2012/672531

Hyers-Ulam-Rassias RNS Approximation of Euler-Lagrange-Type Additive Mappings

Academic Editor: Tadeusz Kaczorek
Received24 Dec 2011
Revised05 Mar 2012
Accepted19 Mar 2012
Published04 Jun 2012

Abstract

Recently the generalized Hyers-Ulam (or Hyers-Ulam-Rassias) stability of the following functional equation where , proved in Banach modules over a unital -algebra. It was shown that if , for some and a mapping satisfies the above mentioned functional equation then the mapping is Cauchy additive. In this paper we prove the Hyers-Ulam-Rassias stability of the above mentioned functional equation in random normed spaces (briefly RNS).

1. Introduction and Preliminaries

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference.

The paper of Rassias has provided a lot of influence in the development of what we call the generalized Hyers-Ulam stability of functional equations. In 1994, a generalization of Rassias’ theorem was obtained by Găvruţa [5] by replacing the bound by a general control function .

The functional equation: is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [6] for mappings , where is a normed space and is a Banach space. Cholewa [7] noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. Czerwik [8] proved the generalized Hyers-Ulam stability of the quadratic functional equation.

The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [2, 4, 5, 928]).

In the sequel, we will adopt the usual terminology, notions, and conventions of the theory of random normed spaces as in [29]. Throughout this paper, the spaces of all probability distribution functions are denoted by . Elements of are functions , such that is left continuous and nondecreasing on , and . It’s clear that the subset , where , is a subset of . The space is partially ordered by the usual point-wise ordering of functions, that is, for all , if and only if . For every , is the element of defined by One can easily show that the maximal element for in this order is the distribution function .

Definition 1.1. A function is a continuous triangular norm (briefly a -norm) if satisfies the following conditions:(i) is commutative and associative;(ii) is continuous;(iii) for all ;(iv) whenever and for all .

Three typical examples of continuous -norms are , and . Recall that, if is a -norm and is a given of numbers in , is defined recursively by and for .

Definition 1.2. A random normed space (briefly ) is a triple , where is a vector space, is a continuous -norm, and is a mapping such that the following conditions hold.(i) for all if and only if .(ii) for all , , and .(iii), for all and .

Definition 1.3. Let be an RNS.
(i)A sequence in is said to be convergent to in if for all , (ii)A sequence in is said to be Cauchy sequence in if for all ,.(iii)The -space is said to be complete if every Cauchy sequence in is convergent.

Theorem 1.4. If is RNS and is a sequence such that , then .

In this paper, we investigate the generalized Hyers-Ulam stability of the following additive functional equation of Euler-Lagrange type: where , , and for some , in random normed spaces.

Every solution of the functional equation (1.3) is said to be a generalized Euler-Lagrange type additive mapping.

2. RNS Approximation of Functional Equation (1.3)

Remark 2.1. Throughout this paper, will be real numbers such that for fixed .

Theorem 2.2. Let be a real linear space, be an RN space, be a function such that for some , and for all and
Let be a complete RN space. If is a mapping such that for all and then there is a unique generalized Euler-Lagrange-type additive mapping such that, for all and all

Proof. For each with , let in (2.3). Then we get the following inequality: for all , where for all and all , and
Letting in (2.5), we get for all . Similarly, letting in (2.5), we get for all . It follows from (2.5), (2.8), and (2.9) that for all
Replacing and by and in (2.10), we get that for all . Putting in (2.11), we get for all . Replacing and by and in (2.11), respectively, we get for all . It follows from (2.12) and (2.13) that for all . So
Replacing by in (2.15) and using (2.1), we get for all and all . Therefore, we have for all . This implies that
Replacing by in (2.18), we obtain
Since the right-hand side of the above inequality tends to 1, when , then the sequence is a Cauchy sequence in complete RN space (), so there exists some point such that for all .
Fix and put in (2.19). Then we obtain and so, for every , we have
Taking the limit as and using (2.22), we get
Since was arbitrary by taking in (2.23), we get
Replacing by for all , in (2.3), we get for all and for all ,
since We conclude that
To prove the uniqueness of mapping , assume that there exists another mapping which satisfies (2.4). Fix , clearly and , for all . Since , so
Since the right-hand side of the above inequality tends to 1, when , therefore, it follows that for all , and so . This completes the proof.

Corollary 2.3. Let be a real linear space, be an RN space, and a complete RN space. Let , and be a mapping with and satisfying for all and . Then the limit exists for all and defines a unique Euler-Lagrange additive mapping such that for all and .

Proof. Let and be defined as .

Corollary 2.4. Let be a real linear space, be an RN space, and a complete RN space. Let and be a mapping with and satisfying for all for all and all . Then, the limit exists for all and defines a unique Euler-Lagrange additive mapping such that for all and .

Proof. Let and be defined as .

Theorem 2.5. Let be a real linear space, be an RN space, be a function such that for some , and for all and , . Let be a complete RN space. If is a mapping satisfying (2.3), then there is a unique generalized Euler-Lagrange-type additive mapping such that, for all for all and all .

Proof. Replacing by in (2.14) and using (2.33), we obtain
So for all . This implies that
Proceeding as in the proof of Theorem 2.2, one can easily show that the sequence is a Cauchy sequence in complete RN space , so there exists some point such that for all .
Taking the limit from both sides of the above inequality, we obtain (2.34).
The rest of the proof is similar to the proof of Theorem 2.2.

Corollary 2.6. Let be a real linear space, be an RN space and a complete RN space. Let , and be a mapping with and satisfying for all for all and all . Then the limit exists for all and defines a unique Euler-Lagrange additive mapping such that for all and .

Proof. Let and be defined as <