# Hyers-Ulam Stability of Power Series Equations

Authors: M. Bidkham; H. A. Soleiman Mezerji; M. Eshaghi Gordji
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We prove the Hyers-Ulam stability of power series equation∑n=0∞anxn=0, where an for n=0,1,2,3,… can be real or complex.


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## Body

## 1. Introduction and Preliminaries

A classical question in the theory of functional equations is that “when is it true that a function which approximately satisfies a functional equationℰ must be somehow close to an exact solution of ℰ.” 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 equation(1.1)xn+αx+β=0,
where x∈[-1,1] and proved the following.Theorem 1.1.
If|α|>n,|β|<|α|-1 and y∈[-1,1] satisfies the inequality
(1.2)|yn+αy+β|≤ɛ,
then there exists a solution v∈[-1,1] of (1.1) such that
(1.3)|y-v|≤Kɛ,
where K>0 is constant.They also asked an open problem whether the real polynomial equation(1.4)anxn+an-1xn-1+⋯+a1x+a0=0
has Hyers-Ulam stability for the case that this real polynomial equation has some solution in [a,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 equation(1.5)f(z)=0,
where f is any analytic function. First we give the definition of the generalized Hyers-Ulam stability.Definition 1.2.
Letp 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 everyɛ>0,y∈[-1,1] if
(1.6)|∑n=0∞anyn|≤ɛ(∑n=0∞|an|p2n),
then there exists some x∈[-1,1] satisfying
(1.7)∑n=0∞anxn=0
such that |y-x|≤Kɛ. For the complex coefficients, [-1,1] can be replaced by closed unit disc
(1.8)D={z∈C;|z|≤1}.


## 2. Main Results

The aim of this work is to investigate the generalized Hyers-Ulam stability for (1.7).Theorem 2.1.
If(2.1)∑n=0,n≠1∞|an|<|a1|,(2.2)∑n=2∞n|an|<|a1|,
then there exists an exact solution v∈[-1,1] of (1.7).Proof.
If we set(2.3)g(x)=-1a1(∑n=0,n≠1∞anxn),
for x∈[-1,1], then we have
(2.4)|g(x)|=1|a1||∑n=0,n≠1∞anxn|≤1|a1|(∑n=0,n≠1∞|an|)≤1
by (2.1).
LetX=[-1,1], d(x,y)=|x-y|. 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 x,y∈X, we have
(2.5)d(g(x),g(y))=|1a1(-a0-a2x2-⋯)-1a1(-a0-a1y2-⋯)|≤1|a1||x-y|{∑n=2∞n|an|}.
For x,y∈[-1,1], x≠y, from (2.2), we obtain
(2.6)d(g(x),g(y))≤λd(x,y),
where
(2.7)λ=∑n=2∞n|an||a1|<1.
Thus g is a contraction mapping from X to X. By the Banach contraction mapping theorem, there exists a unique v∈X, such that
(2.8)g(v)=v.
Hence, (1.7) has a solution on [-1,1].Theorem 2.2.
Under the conditions of Theorem2.1, (1.7) has the generalized Hyers-Ulam stability.Proof.
Letɛ>0 and y∈[-1,1] be such that
(2.9)|∑n=0∞anyn|≤ɛ(∑n=0∞|an|p2n).
We will show that there exists a constant K independent of ɛ, v, and y such that
(2.10)|y-v|≤Kɛ
for some v∈[-1,1] satisfying (1.7).
Let us introduce the abbreviationK=2/(|a1|1-p(1-λ)). Then
(2.11)|y-v|=|y-g(y)+g(y)-g(v)|≤|y-g(y)|+|g(y)-g(v)|≤|y-(-1a1∑n=0,n≠1∞anyn)|+λ|y-v|=1|a1||∑n=0∞anyn|+λ|y-v|.
Thus, we have
(2.12)|y-v|≤1|a1|(1-λ)|∑n=0∞anyn|≤1|a1|(1-λ)(∑n=0∞|an|p2n)ϵ≤1|a1|(1-λ)(∑n=0∞|a1|p2n)ϵ≤Kɛ
by (2.9) and so the result follows.Next, for equation of complex power series(2.13)∑n=0∞anzn=0,
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(2.14)∑n=0,n≠1∞|an|<|a1|.
Then there exists an exact solution in open unit disc for (2.13).Proof.
If we set(2.15)g(z)=-1a1(∑n=0,n≠1∞anzn),
for |z|≤1. Such as above we have
(2.16)|g(z)|=1|a1||∑n=0,n≠1∞anzn|≤1|a1|(∑n=0,n≠1∞|an|),for|z|≤1<1
by (2.14).
Since|g(z)|<1 for |z|=1, hence for |g(z)|<|-z|=1 and by Rouche's theorem, we observe that g(z)-z 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 Theorem2.1, (2.13) has the generalized Hyers-Ulam stability.ForR⩾1, we have the following corollary.Corollary 2.5.
If(2.17)∑n=0,n≠1∞|an|Rn<|a1|R,
then there exists an exact solution in {z∈ℂ;|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 [a,b].Remark 2.7.
By replacingan=f(n)(0) in (2.14), we can prove the generalized Hyers-Ulam stability for (1.5).



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*Source: 194948-2011-05-22.xml*