Research Article | Open Access

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

**Academic Editor:**Tadeusz Kaczorek

#### 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. Hyersâ€™ 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 Rassiasâ€™ theorem was obtained by GÄƒvruÅ£a [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, 9â€“28]).

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 . Itâ€™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