Abstract and Applied Analysis

VolumeΒ 2011, Article IDΒ 370104, 10 pages

http://dx.doi.org/10.1155/2011/370104

## On Pexider Differences in Topological Vector Spaces

^{1}Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil 56199-11367, Iran^{2}Department of Applied Mathematics, Kangnam University, Gyeonggi, Giheung-gu, Yongin 446-702, Republic of Korea

Received 31 May 2011; Accepted 30 August 2011

Academic Editor: AlbertoΒ d'Onofrio

Copyright Β© 2011 Abbas Najati 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

Let be a normed space and a sequentially complete Hausdorff topological vector space over the field of rational numbers. Let and where . We prove that the Pexiderized Jensen functional equation is stable for functions defined on and taking values in . We consider also the Pexiderized Cauchy functional equation.

#### 1. Introduction

The functional equation is stable if any function satisfying the equation * approximately* is near to true solution of . The stability of functional equations was first introduced by Ulam [1] in 1940. More precisely, Ulam proposed the following problem: given a group , a metric group , and a positive number , does there exist a such that if a function satisfies the inequality for all , then there exists a homomorphism such that for all As it is mentioned above, when this problem has a solution, we say that the homomorphisms from to are stable. In 1941, Hyers [2] gave a partial solution of Ulam's problem for the case of approximate additive mappings under the assumption that and are Banach spaces. Aoki [3] and Rassias [4] provided a generalization of Hyers' theorem for additive and linear mappings, respectively, by allowing the Cauchy difference to be unbounded. During the last decades several stability problems of functional equations have been investigated by several mathematicians. A large list of references concerning the stability of functional equations can be found in [5–8].

#### 2. Hyers-Ulam Stability of Jensen's Functional Equation

Jung investigated the Hyers-Ulam stability for Jensen's equation on a restricted domain [9]. In this section, we prove a local Hyers-Ulam stability of the Pexiderized Jensen functional equation in topological vector spaces. In this section is a normed space and is a sequentially complete Hausdorff topological vector space over the field of rational numbers.

Theorem 2.1. *Let be a nonempty bounded convex subset of containing the origin. Suppose that satisfies
**
for all with , where . Then there exists a unique additive mapping such that
**
for all , where and denotes the sequential closure of .*

*Proof. *Suppose . If , let with , otherwise
It is easy to verify that
It follows from (2.1) and (2.6) that
for all with . Hence, by (2.1) and (2.7), we have
for all . Letting in (2.8), we get
for all . It follows from (2.8) and (2.9) that
for all , where . So we get from (2.10) that
for all . Setting in (2.10), we infer that
for all . It is easy to prove that
for all and all integers . Since is a nonempty bounded convex subset of containing the origin, is a nonempty bounded convex subset of containing the origin. It follows from (2.13) that
for all and all integers . Let be an arbitrary neighborhood of the origin in . Since is bounded, there exists a rational number such that . Choose such that . Let and with . Then (2.15) implies that
Thus, the sequence forms a Cauchy sequence in . By the sequential completeness of , the limit exists for each . So (2.2) follows from (2.14).

To show that is additive, replace and by and , respectively, in (2.11) and then divide by to obtain
for all and all integers . Since is bounded, on taking the limit as , we get that is additive. It follows from (2.2) and (2.9) that
for all . So we obtain (2.3). Similarly, we get (2.4).

To prove the uniqueness of , assume on the contrary that there is another additive mapping satisfying (2.2) and there is an such that . So there is a neighborhood of the origin in such that , since is Hausdorff. Since and satisfy (2.2), we get for all . Since is bounded, there exists a positive integer such that . Therefore, which is a contradiction with . This completes the proof.

We apply the result of Theorem 2.1 to study the asymptotic behavior of additive mappings.

Theorem 2.2. *Suppose that has a bounded convex neighborhood of 0. Let be functions satisfying
**
Then are additive and .*

*Proof. *Let be a bounded convex neighborhood of 0 in . It follows from (2.19) that there exists an increasing sequence such that
for all with . Applying (2.20) and Theorem 2.1, we obtain a sequence of unique additive mappings satisfying
for all , where . Since is convex and , we have
for all and all . The uniqueness of implies for all . Hence, letting in (2.21), we obtain that are additive and .

Theorem 2.3. *Let be a nonempty bounded convex subset of containing the origin. Suppose that with satisfies
**
for all with , where and . Then there exists a unique additive mapping such that
**
for all with . Moreover, for all .*

*Proof. *Letting in (2.23), we get . Letting in (2.23), we get
for all with . If we put in (2.23), we have
Hence it follows from (2.27) and (2.28) that
for all with . We can replace by in (2.29) for all nonnegative integers . So we have
for all with . Therefore,
for all with and all integers . Let with , and let be an arbitrary neighborhood of the origin in . Since is bounded, there exists a rational number such that . Choose such that . Let with . Then (2.31) implies that
Thus, the sequence forms a Cauchy sequence in for all with . By the sequential completeness of , the limit exists for each . It follows from (2.28) that
for all with . Letting and in (2.31), we obtain that satisfies
for all with . It follows from the definition of that , and we conclude from (2.23) that for all with . Using an extension method of Skof [10], we extend the additivity of to the whole space (see also [11]). Let be the extension of in which for all with . It follows from (2.34) that satisfies (2.24). To prove (2.25), we have from (2.24) and (2.28) that
for all with . Similarly, we obtain (2.26).

#### 3. Hyers-Ulam Stability of Cauchy's Functional Equation

The following theorems are alternative results for the Pexiderized Cauchy functional equation.

Theorem 3.1. *Let be a nonempty bounded convex subset of containing the origin. Suppose that satisfies
**
for all with , where . Then there exists a unique additive mapping such that
**
for all , where .*

*Proof. *Using the method in the proof of Theorem 2.1, we get
for all . Letting in (3.3), we get
for all . It follows from (3.3) and (3.4) that
for all . The rest of the proof is similar to the proof of Theorem 2.1, and we omit the details.

Corollary 3.2. *Suppose that has a bounded convex neighborhood of 0. Let be functions satisfying
**
Then are additive and .*

Theorem 3.3. *Let be a nonempty bounded convex subset of containing the origin. Suppose that with satisfies
**
for all with , where and . Then there exists a unique additive mapping such that
**
for all with . Moreover, for all .*

*Proof. *Using the same method in the proof of Theorem 2.3, we get from (3.7) that for all with . Therefore,
for all with . The rest of the proof is similar to the proof of Theorem 2.3, and we omit the details.

#### References

- S. M. Ulam,
*Problems in Modern Mathematics*, John Wiley and Sons, New York, NY, USA, Chapter 6, 1964. - D. H. Hyers, βOn the stability of the linear functional equation,β
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 27, pp. 222β224, 1941. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - T. Aoki, βOn the stability of the linear transformation in Banach spaces,β
*Journal of the Mathematical Society of Japan*, vol. 2, pp. 64β66, 1950. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - T. M. Rassias, βOn the stability of the linear mapping in Banach spaces,β
*Proceedings of the American Mathematical Society*, vol. 72, no. 2, pp. 297β300, 1978. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - S. Czerwik,
*Functional Equations and Inequalities in Several Variables*, World Scientific Publishing, River Edge, NJ, USA, 2002. View at Publisher Β· View at Google Scholar - D. H. Hyers, G. Isac, and T. M. Rassias,
*Stability of Functional Equations in Several Variables*, Birkhäuser Boston, Boston, Mass, USA, 1998. - S.-M. Jung,
*Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis*, Hadronic Press, Palm Harbor, Fla, USA, 2001. - T. M. Rassias, βOn the stability of functional equations and a problem of Ulam,β
*Acta Applicandae Mathematicae. An International Survey Journal on Applying Mathematics and Mathematical Applications*, vol. 62, no. 1, pp. 23β130, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - S.-M. Jung, βHyers-Ulam-Rassias stability of Jensen's equation and its application,β
*Proceedings of the American Mathematical Society*, vol. 126, no. 11, pp. 3137β3143, 1998. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - F. Skof, βOn the stability of functional equations on a restricted domain and a related topic,β in
*Stability of Mappings of Hyers-Ulam Type*, T. M. Rassias and J. Tabor, Eds., pp. 141β151, Hadronic Press, Palm Harbor, Fla, USA, 1994. View at Google Scholar Β· View at Zentralblatt MATH - A. Najati and T. M. Rassias, βStability of the Pexiderized Cauchy and Jensen's equations on restricted domains,β
*Communications in Mathematical Analysis*, vol. 8, no. 2, pp. 125β135, 2010. View at Google Scholar Β· View at Zentralblatt MATH