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Discrete Dynamics in Nature and Society
Volume 2011, Article ID 929824, 14 pages
http://dx.doi.org/10.1155/2011/929824
Research Article

Functional Inequalities Associated with Cauchy Additive Functional Equation in Non-Archimedean Spaces

1Department of Mathematics, Urmia University, Urmia, Iran
2Department of Mathematics, National Technical University of Athens, 15780 Zografou, Greece
3Department of Mathematics, Semnan University, P. O. Box 35195-363, Iran
4Research Group of Nonlinear Analysis and Applications (RGNAA), Semnan University, Iran
5Center of Excellence in Nonlinear Analysis and Applications (CENAA), Semnan University, Iran

Received 30 October 2010; Accepted 19 January 2011

Academic Editor: Rigoberto Medina

Copyright © 2011 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

We investigate the generalized Hyers-Ulam stability of the functional inequalities and in non-Archimedean normed spaces in the spirit of the Th. M. Rassias stability approach.

1. Introduction

Ulam [1] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms.

Let be a group and let be a metric group with the metric . Given , does there exist a , such that if a mapping satisfies the inequality for all , then there exists a homomorphism with for all

In other words, under what condition does there exist a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. In 1941, Hyers [2] gave the first affirmative answer to the question of Ulam for Banach spaces. Let be a mapping between Banach spaces such that for all , and for some . Then there exists a unique additive mapping such that for all . Moreover, if is continuous in for each fixed , then is linear. In 1978, Rassias [3] proved the following theorem.

Theorem 1.1. Let be a mapping from a normed vector space into a Banach space subject to the inequality for all , where and p are constants with and . Then there exists a unique additive mapping such that for all . If then inequality (1.3) holds for all , and (1.4) for . Also, if the function from into is continuous in real for each fixed , then is linear.

In 1991, Gajda [4] answered the question for the case , which was raised by Rassias. This new concept is known as Hyers-Ulam-Rassias stability of functional equations. The reader is referred to [513] for a number of results in this domain of research.

In 1994, a generalization of the Rassias theorem was obtained by Găvruţa as follows [14].

Suppose is an abelian group, is a Banach space, and that the so-called admissible control function satisfies for all . If is a mapping with for all , then there exists a unique mapping such that and for all .

During the last decades, several stability problems of functional equations have been investigated by a number of mathematicians, see [1517] and references therein for more detailed information.

By a non-Archimedean field we mean a field equipped with a function (valuation) from into such that if and only if , , and for all . Clearly and for all .

Let be a vector space over a scalar field with a non-Archimedean nontrivial valuation . A function is a non-Archimedean norm (valuation) if it satisfies the following conditions:(i) if and only if ;(ii); (iii)the strong triangle inequality (ultrametric), namely,

Then is called a non-Archimedean space. Due to the fact that a sequence is Cauchy if and only if converges to zero in a non-Archimedean space. By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent (see [1822]).

Gilányi [23] and Rätz [24] showed that if satisfies the functional inequality then satisfies the Jordan-von Neumann functional equation Gilányi [23] and Fechner [25] proved the generalized Hyers-Ulam stability of the functional inequality (1.3).

Cho and Kim [26] proved the generalized Hyers-Ulam stability of the following functional inequalities: which are associated with Jordan-von Neumann-type Cauchy-Jensen additive functional equations.

Now, we consider the following functional inequality: which is associated with Cauchy additive functional equation.

The purpose of this paper is to prove that if satisfies the inequalities (1.12) and (1.13), which satisfies certain conditions, then is Cauchy additive, and thus we prove the generalized Hyers-Ulam stability of the functional inequalities (1.12) and (1.13) in non-Archimedean normed spaces.

2. Stability of Functional Inequality (1.12)

In this section, we prove the generalized Hyers-Ulam stability of the functional inequality (1.12). Throughout this section, we assume that is an additive semigroup and is a complete non-Archimedean space.

We need the following lemma in the main results.

Lemma 2.1. Let be a mapping such that for all . If , then the mapping is Cauchy additive.

Proof. Letting in (2.1), we get . So, . Letting and replacing by in (2.1), we get for all . So, for all . Setting in (2.1), we obtain So, for all . Replacing by in (2.3), we get for all . Using (2.4), we obtain and for all . Letting , and , in (2.1) we get for all . Hence, is additive.

Theorem 2.2. Let be a function such that for all and let the limit exists for all . Suppose that with is a mapping satisfying for all . Then there exists an additive mapping such that for all . Moreover, if then is the unique additive mapping satisfying (2.9).

Proof. Putting and in (2.8), we get for all . Replacing by in (2.11), we obtain for all . Replacing by in (2.12), we get for all . Let . It follows from (2.12) and (2.13) that for all . Replacing by in (2.14), we get for all . It follows from (2.6) and (2.15) that the sequence is Cauchy. Since is complete, we conclude that is convergent. Set for all . Using induction one can show that for all and . By taking to approach infinity in (2.16) and using (2.7) one obtains (2.9).
It follows from (2.8) that for all . So, for all . By Lemma 2.1, the mapping is additive.
Now, let be another additive mapping satisfying (2.9). Then we have for all . Therefore . This completes the proof of the uniqueness of .

Corollary 2.3. Let and be positive real numbers, and let be a mapping satisfying for all . If then there exists a unique additive mapping such that for all .

Proof. Defining by we have for all .
Applying Theorem 2.2, we conclude the required result.

Corollary 2.4. Let be a function satisfying Let , let be a normed space and let fulfill the inequality for all . Then there exists a unique additive mapping such that for all .

Proof. Defining by we have for all .
Applying Theorem 2.2, we conclude the required result.

Theorem 2.5. Let be a function such that for all and let the limit exist for all . Suppose that with is a mapping satisfying for all . Then there exists an additive mapping such that for all . Moreover, if then is the unique additive mapping satisfying (2.30).

Proof. It follows from (2.14) that for all . Hence, for all . It follows from (2.27) and (2.33) that the sequence is a Cauchy sequence for all . Since is complete, the sequence converges. So, one can define the mapping by for all .
The rest of the proof is similar to the proof of Theorem 2.2.

Remark 2.6. We can obtain similar results to Corollary 2.3 for and Corollary 2.4.

3. Stability of Functional Inequality (1.13)

We prove the generalized Hyers-Ulam stability of the functional inequality (1.13). Throughout this section, we assume that is an additive semigroup and is a complete non-Archimedean space.

We need the following lemma in the main results.

Lemma 3.1. Let be a mapping such that for all . If , then the mapping is Cauchy additive.

Proof. Letting in (3.1), we get for all . Hence, for all . Letting and in (3.1), we get for all . Hence, for all . Letting and in (3.1), we get for all . Hence, for all . Letting in (3.1), we get for all . So, for all . Let and in (3.8). Then for all . So, is additive.

Theorem 3.2. Let be a function such that for all and let the limit exist for all . Suppose that with is a mapping satisfying for all . Then there exists an additive mapping such that for all . Moreover, if then is the unique additive mapping satisfying (3.13).

Proof. Letting and in (3.12), we get for all . Putting and in (3.12), we get for all . It follows from (3.15) and (3.16) that for all . Replacing by in (3.17), we get for all . It follows from (3.10) and (3.18) that the sequence is Cauchy. Since is complete, we conclude that is convergent. Set for all . Using induction one can show that for all and . By taking to approach infinity in (3.19) and using (3.11) one obtains (3.13).
Replacing ,  , and by ,, and , respectively, in (3.12) we get for all . Taking the limit as and using (3.10) we get for all . By Lemma 3.1, the mapping is additive.
If is another additive mapping satisfying (3.13), then for all . Therefore . This proves the uniqueness of . Hence, the mapping is a unique additive mapping satisfying (3.13).

Corollary 3.3. Let and be positive real numbers, and let be a mapping satisfying for all . If then there exists a unique additive mapping such that for all .

Proof. Letting in Theorem 3.2, we obtain the result.

Corollary 3.4. Let be a function satisfying Let , let be a normed space, and let fulfill the inequality for all . Then there exists a unique additive mapping such that for all .

Proof. If we define in Theorem 3.2, then we get the result.

Remark 3.5. We can formulate a similar statement to Theorem 3.2 in which we can define the sequence under suitable conditions on the function and and then obtain similar results to Corollary 3.3 for and Corollary 3.4.

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