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Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 194948, 6 pages
http://dx.doi.org/10.1155/2011/194948
Research Article

Hyers-Ulam Stability of Power Series Equations

1Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran
2Center 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 βˆ‘βˆžπ‘›=0π‘Žπ‘›π‘₯𝑛=0, where π‘Žπ‘› for 𝑛=0,1,2,3,… can be real or complex.

1. Introduction and Preliminaries

A classical question in the theory of functional equations is that β€œwhen is it true that a function which approximately satisfies a functional equation β„° 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 Hyers’ theorem by proving the existence of unique linear mappings near approximate additive mappings. On the other hand, J. M. Rassias [4–6] 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 [7–10] and references therein).

Recently, Li and Hua [11] discussed and proved the Hyers-Ulam stability of a polynomial equationπ‘₯𝑛+𝛼π‘₯+𝛽=0,(1.1) where π‘₯∈[βˆ’1,1] and proved the following.

Theorem 1.1. If |𝛼|>𝑛,|𝛽|<|𝛼|βˆ’1 and π‘¦βˆˆ[βˆ’1,1] satisfies the inequality ||𝑦𝑛||+𝛼𝑦+π›½β‰€πœ€,(1.2) then there exists a solution π‘£βˆˆ[βˆ’1,1] of (1.1) such that ||||π‘¦βˆ’π‘£β‰€πΎπœ€,(1.3) where 𝐾>0 is constant.

They also asked an open problem whether the real polynomial equationπ‘Žπ‘›π‘₯𝑛+π‘Žπ‘›βˆ’1π‘₯π‘›βˆ’1+β‹―+π‘Ž1π‘₯+π‘Ž0=0(1.4) has Hyers-Ulam stability for the case that this real polynomial equation has some solution in [π‘Ž,𝑏].

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 equation𝑓(𝑧)=0,(1.5) where 𝑓 is any analytic function. First we give the definition of the generalized Hyers-Ulam stability.

Definition 1.2. Let 𝑝 be a real number. We say that (1.7) has the generalized Hyers-Ulam stability if there exists a constant 𝐾>0 with the following property:
for every πœ€>0,π‘¦βˆˆ[βˆ’1,1] if |||||βˆžξ“π‘›=0π‘Žπ‘›π‘¦π‘›|||||ξƒ©β‰€πœ€βˆžξ“π‘›=0||π‘Žπ‘›||𝑝2𝑛ξƒͺ,(1.6) then there exists some π‘₯∈[βˆ’1,1] satisfying βˆžξ“π‘›=0π‘Žπ‘›π‘₯𝑛=0(1.7) such that |π‘¦βˆ’π‘₯|β‰€πΎπœ€. For the complex coefficients, [βˆ’1,1] can be replaced by closed unit disc 𝐷={π‘§βˆˆβ„‚;|𝑧|≀1}.(1.8)

2. Main Results

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

Theorem 2.1. If βˆžξ“π‘›=0,𝑛≠1||π‘Žπ‘›||<||π‘Ž1||,(2.1)βˆžξ“π‘›=2𝑛||π‘Žπ‘›||<||π‘Ž1||,(2.2) then there exists an exact solution π‘£βˆˆ[βˆ’1,1] of (1.7).

Proof. If we set 𝑔(π‘₯)=βˆ’1π‘Ž1ξƒ©βˆžξ“π‘›=0,𝑛≠1π‘Žπ‘›π‘₯𝑛ξƒͺ,(2.3) for π‘₯∈[βˆ’1,1], then we have ||||=1𝑔(π‘₯)||π‘Ž1|||||||βˆžξ“π‘›=0,𝑛≠1π‘Žπ‘›π‘₯𝑛|||||≀1||π‘Ž1||ξƒ©βˆžξ“π‘›=0,𝑛≠1||π‘Žπ‘›||ξƒͺ≀1(2.4) by (2.1).
Let 𝑋=[βˆ’1,1], 𝑑(π‘₯,𝑦)=|π‘₯βˆ’π‘¦|. Then (𝑋,𝑑) is a complete metric space and 𝑔 map 𝑋 to 𝑋. Now, we will show that 𝑔 is a contraction mapping from 𝑋 to 𝑋. For any π‘₯,π‘¦βˆˆπ‘‹, we have 𝑑||||1(𝑔(π‘₯),𝑔(𝑦))=π‘Ž1ξ€·βˆ’π‘Ž0βˆ’π‘Ž2π‘₯2ξ€Έβˆ’1βˆ’β‹―π‘Ž1ξ€·βˆ’π‘Ž0βˆ’π‘Ž1𝑦2ξ€Έ||||≀1βˆ’β‹―||π‘Ž1||||||ξƒ―π‘₯βˆ’π‘¦βˆžξ“π‘›=2𝑛||π‘Žπ‘›||ξƒ°.(2.5) For π‘₯,π‘¦βˆˆ[βˆ’1,1], π‘₯≠𝑦, from (2.2), we obtain 𝑑(𝑔(π‘₯),𝑔(𝑦))β‰€πœ†π‘‘(π‘₯,𝑦),(2.6) where βˆ‘πœ†=βˆžπ‘›=2𝑛||π‘Žπ‘›||||π‘Ž1||<1.(2.7) Thus 𝑔 is a contraction mapping from 𝑋 to 𝑋. By the Banach contraction mapping theorem, there exists a unique π‘£βˆˆπ‘‹, such that 𝑔(𝑣)=𝑣.(2.8) Hence, (1.7) has a solution on [βˆ’1,1].

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

Proof. Let πœ€>0 and π‘¦βˆˆ[βˆ’1,1] be such that |||||βˆžξ“π‘›=0π‘Žπ‘›π‘¦π‘›|||||ξƒ©β‰€πœ€βˆžξ“π‘›=0||π‘Žπ‘›||𝑝2𝑛ξƒͺ.(2.9) We will show that there exists a constant 𝐾 independent of πœ€, 𝑣, and 𝑦 such that ||||π‘¦βˆ’π‘£β‰€πΎπœ€(2.10) for some π‘£βˆˆ[βˆ’1,1] satisfying (1.7).
Let us introduce the abbreviation 𝐾=2/(|π‘Ž1|1βˆ’π‘(1βˆ’πœ†)). Then ||||=||||≀||||+||||≀|||||ξƒ©π‘¦βˆ’π‘£π‘¦βˆ’π‘”(𝑦)+𝑔(𝑦)βˆ’π‘”(𝑣)π‘¦βˆ’π‘”(𝑦)𝑔(𝑦)βˆ’π‘”(𝑣)π‘¦βˆ’βˆ’1π‘Ž1βˆžξ“π‘›=0,𝑛≠1π‘Žπ‘›π‘¦π‘›ξƒͺ|||||||||=1+πœ†π‘¦βˆ’π‘£||π‘Ž1|||||||βˆžξ“π‘›=0π‘Žπ‘›π‘¦π‘›|||||||||.+πœ†π‘¦βˆ’π‘£(2.11) Thus, we have ||||≀1π‘¦βˆ’π‘£||π‘Ž1|||||||(1βˆ’πœ†)βˆžξ“π‘›=0π‘Žπ‘›π‘¦π‘›|||||≀1||π‘Ž1||(1βˆ’πœ†)βˆžξ“π‘›=0||π‘Žπ‘›||𝑝2𝑛ξƒͺπœ–β‰€1||π‘Ž1||(1βˆ’πœ†)βˆžξ“π‘›=0||π‘Ž1||𝑝2𝑛ξƒͺπœ–β‰€πΎπœ€(2.12) by (2.9) and so the result follows.

Next, for equation of complex power seriesβˆžξ“π‘›=0π‘Žπ‘›π‘§π‘›=0,(2.13) 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 βˆžξ“π‘›=0,𝑛≠1||π‘Žπ‘›||<||π‘Ž1||.(2.14) Then there exists an exact solution in open unit disc for (2.13).

Proof. If we set 𝑔(𝑧)=βˆ’1π‘Ž1ξƒ©βˆžξ“π‘›=0,𝑛≠1π‘Žπ‘›π‘§π‘›ξƒͺ,(2.15) for |𝑧|≀1. Such as above we have ||||=1𝑔(𝑧)||π‘Ž1|||||||βˆžξ“π‘›=0,𝑛≠1π‘Žπ‘›π‘§π‘›|||||≀1||π‘Ž1||ξƒ©βˆžξ“π‘›=0,𝑛≠1||π‘Žπ‘›||ξƒͺ,for|𝑧|≀1<1(2.16) by (2.14).
Since |𝑔(𝑧)|<1 for |𝑧|=1, hence for |𝑔(𝑧)|<|βˆ’π‘§|=1 and by Rouche's theorem, we observe that 𝑔(𝑧)βˆ’π‘§ has exactly one zero in |𝑧|<1 which implies that 𝑔 has a unique fixed point in |𝑧|<1.

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

For 𝑅⩾1, we have the following corollary.

Corollary 2.5. If βˆžξ“π‘›=0,𝑛≠1||π‘Žπ‘›||𝑅𝑛<||π‘Ž1||𝑅,(2.17) then there exists an exact solution in {π‘§βˆˆβ„‚;|𝑧|<𝑅} 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 [π‘Ž,𝑏].

Remark 2.7. By replacing π‘Žπ‘›=𝑓(𝑛)(0) in (2.14), we can prove the generalized Hyers-Ulam stability for (1.5).

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