Abstract and Applied Analysis

VolumeΒ 2011, Article IDΒ 194948, 6 pages

http://dx.doi.org/10.1155/2011/194948

## Hyers-Ulam Stability of Power Series Equations

^{1}Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran^{2}Center of Excellence in Nonlinear Analysis and Applications (CENAA), Semnan University, Semnan, Iran

Received 6 December 2010; Revised 15 February 2011; Accepted 17 March 2011

Academic Editor: JohnΒ Rassias

Copyright Β© 2011 M. Bidkham 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 prove the Hyers-Ulam stability of power series equation , where for can be real or complex.

#### 1. Introduction and Preliminaries

A classical question in the theory of functional equations is that βwhen is it true that a function which approximately satisfies a functional equation must be somehow close to an exact solution of .β Such a problem was formulated by Ulam [1] in 1940 and solved in the next year for the Cauchy functional equation by Hyers [2]. It gave rise to the * Hyers-Ulam stability * for functional equations.

In 1978, Th. M. Rassias [3] provided a generalization of Hyersβ theorem by proving the existence of unique linear mappings near approximate additive mappings. On the other hand, J. M. Rassias [4β6] considered the Cauchy difference controlled by a product of different powers of norm. This new concept is known as generalized Hyers-Ulam stability of functional equations (see also [7β10] and references therein).

Recently, Li and Hua [11] discussed and proved the Hyers-Ulam stability of a polynomial equation where and proved the following.

Theorem 1.1. *If and satisfies the inequality
**
then there exists a solution of (1.1) such that
**
where is constant.*

They also asked an open problem whether the real polynomial equation has Hyers-Ulam stability for the case that this real polynomial equation has some solution in .

In this paper we establish the Hyers-Ulam-Rassias stability of power series with real or complex coefficients. So we prove the generalized Hyers-Ulam stability of equation where is any analytic function. First we give the definition of the generalized Hyers-Ulam stability.

*Definition 1.2. *Let be a real number. We say that (1.7) has the generalized Hyers-Ulam stability if there exists a constant with the following property:

for every if
then there exists some satisfying
such that . For the complex coefficients, can be replaced by closed unit disc

#### 2. Main Results

The aim of this work is to investigate the generalized Hyers-Ulam stability for (1.7).

Theorem 2.1. *If
**
then there exists an exact solution of (1.7).*

*Proof. *If we set
for , then we have
by (2.1).

Let , . Then is a complete metric space and map to . Now, we will show that is a contraction mapping from to . For any , we have
For , , from (2.2), we obtain
where
Thus is a contraction mapping from to . By the Banach contraction mapping theorem, there exists a unique , such that
Hence, (1.7) has a solution on .

Theorem 2.2. *Under the conditions of Theorem 2.1, (1.7) has the generalized Hyers-Ulam stability.*

*Proof. *Let and be such that
We will show that there exists a constant independent of , , and such that
for some satisfying (1.7).

Let us introduce the abbreviation . Then
Thus, we have
by (2.9) and so the result follows.

Next, for equation of complex power series as an application of Rouche's theorem, we prove the following theorem which is much better than above result. In fact, we prove the following.

Theorem 2.3. *If
**
Then there exists an exact solution in open unit disc for (2.13).*

*Proof. *If we set
for . Such as above we have
by (2.14).

Since for , hence for and by Rouche's theorem, we observe that has exactly one zero in which implies that has a unique fixed point in .

Corollary 2.4. *Under the conditions of Theorem 2.1, (2.13) has the generalized Hyers-Ulam stability.*

For , we have the following corollary.

Corollary 2.5. *If
**
then there exists an exact solution in for (2.13).*

The proof is similar to previous and details are omitted.

*Remark 2.6. *By the similar way, one can easily prove the generalized Hyers-Ulam stability of (1.7) on any finite interval .

*Remark 2.7. *By replacing in (2.14), we can prove the generalized Hyers-Ulam stability for (1.5).

#### References

- S. M. Ulam,
*A Collection of Mathematical Problems*, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience, New York, NY, USA, 1960. - 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 - Th. 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 - J. M. Rassias, βOn approximation of approximately linear mappings by linear mappings,β
*Journal of Functional Analysis*, vol. 46, no. 1, pp. 126β130, 1982. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - J. M. Rassias, βOn a new approximation of approximately linear mappings by linear mappings,β
*Discussiones Mathematicae*, vol. 7, pp. 193β196, 1985. View at Google Scholar Β· View at Zentralblatt MATH - J. M. Rassias, βSolution of a problem of Ulam,β
*Journal of Approximation Theory*, vol. 57, no. 3, pp. 268β273, 1989. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - M. Bavand Savadkouhi, M. E. Gordji, J. M. Rassias, and N. Ghobadipour, βApproximate ternary Jordan derivations on Banach ternary algebras,β
*Journal of Mathematical Physics*, vol. 50, no. 4, Article ID 042303, 9 pages, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - M. E. Gordji, S. Kaboli Gharetapeh, J. M. Rassias, and S. Zolfaghari, βSolution and stability of a mixed type additive, quadratic, and cubic functional equation,β
*Advances in Difference Equations*, vol. 2009, Article ID 826130, 17 pages, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - M. E. Gordji, J. M. Rassias, and N. Ghobadipour, βGeneralized Hyers-Ulam stability of generalized $(N,K)$-derivations,β
*Abstract and Applied Analysis*, vol. 2009, Article ID 437931, 8 pages, 2009. View at Google Scholar - M. E. Gordji, S. Zolfaghari, J. M. Rassias, and M. B. Savadkouhi, βSolution and stability of a mixed type cubic and quartic functional equation in quasi-Banach spaces,β
*Abstract and Applied Analysis*, vol. 2009, Article ID 417473, 14 pages, 2009. View at Google Scholar Β· View at Zentralblatt MATH - Y. Li and L. Hua, βHyers-Ulam stability of a polynomial equation,β
*Banach Journal of Mathematical Analysis*, vol. 3, no. 2, pp. 86β90, 2009. View at Google Scholar Β· View at Zentralblatt MATH