Hyers-Ulam Stability of Power Series Equations

Abstract

We prove the Hyers-Ulam stability of power series equation ∑^∞𝑛=0𝑎_𝑛𝑥^𝑛=0, where 𝑎_𝑛 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 “when is it true that a function which approximately satisfies a functional equation ℰ 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 equation𝑥^𝑛+𝛼𝑥+𝛽=0,(1.1) where 𝑥∈[−1,1] and proved the following.

Theorem 1.1. If |𝛼|>𝑛,|𝛽|<|𝛼|−1 and 𝑦∈[−1,1] satisfies the inequality ||𝑦^𝑛+𝛼𝑦+𝛽||≤𝜀,(1.2) then there exists a solution 𝑣∈[−1,1] of (1.1) such that ||𝑦−𝑣||≤𝐾𝜀,(1.3) where 𝐾>0 is constant.

They also asked an open problem whether the real polynomial equation𝑎_𝑛𝑥^𝑛+𝑎_𝑛−1𝑥^𝑛−1+⋯+𝑎₁𝑥+𝑎₀=0(1.4) 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 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 𝜀>0,𝑦∈[−1,1] if |||||_∞_𝑛=0𝑎_𝑛𝑦^𝑛|||||≤𝜀_∞_𝑛=0||𝑎_𝑛||^𝑝2^𝑛,(1.6) then there exists some 𝑥∈[−1,1] satisfying ^∞_𝑛=0𝑎_𝑛𝑥^𝑛=0(1.7) such that |𝑦−𝑥|≤𝐾𝜀. For the complex coefficients, [−1,1] can be replaced by closed unit disc 𝐷={𝑧∈ℂ;|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 ^∞_𝑛_=0,𝑛≠1||𝑎_𝑛||<||𝑎₁||,(2.1)_∞_𝑛=2𝑛||𝑎_𝑛||<||𝑎₁||,(2.2) then there exists an exact solution 𝑣∈[−1,1] of (1.7).

Proof. If we set 𝑔(𝑥)=−1𝑎₁^∞_{𝑛=0,𝑛≠1}𝑎_𝑛𝑥^𝑛,(2.3) for 𝑥∈[−1,1], then we have ||𝑔(𝑥)||=1||𝑎₁|||||||^∞_{𝑛=0,𝑛≠1}𝑎_𝑛𝑥^𝑛≤1|||||||𝑎₁||^∞_{𝑛=0,𝑛≠1}≤1||𝑎_𝑛||(2.4) by (2.1).
Let 𝑋=[−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 𝑑(𝑔(𝑥),𝑔(𝑦))=||||1𝑎₁−𝑎₀−𝑎₂𝑥²−⋯−1𝑎₁−𝑎₀≤1−𝑎₁𝑦²−⋯||||||𝑎₁||||𝑥−𝑦||^∞_𝑛=2𝑛||𝑎_𝑛||.(2.5) For 𝑥,𝑦∈[−1,1], 𝑥≠𝑦, from (2.2), we obtain 𝑑(𝑔(𝑥),𝑔(𝑦))≤𝜆𝑑(𝑥,𝑦),(2.6) where 𝜆=∑^∞𝑛=2𝑛||𝑎_𝑛||||𝑎₁||<1.(2.7) Thus 𝑔 is a contraction mapping from 𝑋 to 𝑋. By the Banach contraction mapping theorem, there exists a unique 𝑣∈𝑋, such that 𝑔(𝑣)=𝑣.(2.8) Hence, (1.7) has a solution on [−1,1].

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

Proof. Let 𝜀>0 and 𝑦∈[−1,1] be such that |||||_∞_𝑛=0𝑎_𝑛𝑦^𝑛|||||≤𝜀_∞_𝑛=0||𝑎_𝑛||^𝑝2^𝑛.(2.9) We will show that there exists a constant 𝐾 independent of 𝜀, 𝑣, and 𝑦 such that ||𝑦−𝑣||≤𝐾𝜀(2.10) for some 𝑣∈[−1,1] satisfying (1.7).
Let us introduce the abbreviation 𝐾=2/(|𝑎₁|^1−𝑝(1−𝜆)). Then ||𝑦−𝑣||=≤|||||||𝑦𝑦−−𝑔(𝑦−)1+𝑔(𝑦)−𝑔(𝑣)||≤||𝑦−𝑔(𝑦)||+||𝑔(𝑦)−𝑔(𝑣)||𝑎_{1∞𝑛=0,𝑛≠1}𝑎_𝑛𝑦^𝑛=1|||||+𝜆||𝑦−𝑣||||𝑎₁|||||||^∞_𝑛=0𝑎_𝑛𝑦^𝑛|||||+𝜆||𝑦−𝑣||.(2.11) Thus, we have ||𝑦−𝑣||≤1||𝑎₁||(1−𝜆)|||||^∞_𝑛=0𝑎_𝑛𝑦^𝑛|||||≤1||𝑎₁||(1−𝜆)^∞_𝑛=0||𝑎_𝑛||^𝑝≤12^𝑛𝜖||𝑎₁||(1−𝜆)^∞_𝑛=0||𝑎₁||^𝑝≤𝐾𝜀2^𝑛𝜖(2.12) by (2.9) and so the result follows.

Next, for equation of complex power series^∞_𝑛=0𝑎_𝑛𝑧^𝑛=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 ^∞_𝑛_=0,𝑛≠1||𝑎_𝑛||<||𝑎₁||.(2.14) Then there exists an exact solution in open unit disc for (2.13).

Proof. If we set 𝑔(𝑧)=−1𝑎₁^∞_{𝑛=0,𝑛≠1}𝑎_𝑛𝑧^𝑛,(2.15) for |𝑧|≤1. Such as above we have ||𝑔(𝑧)||=1||𝑎₁|||||||^∞_{𝑛=0,𝑛≠1}𝑎_𝑛𝑧^𝑛≤1|||||||𝑎₁||^∞_{𝑛=0,𝑛≠1}<1||𝑎_𝑛||,for|𝑧|≤1(2.16) 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 Theorem 2.1, (2.13) has the generalized Hyers-Ulam stability.

For 𝑅⩾1, we have the following corollary.

Corollary 2.5. If ^∞_𝑛_=0,𝑛≠1||𝑎_𝑛||𝑅^𝑛<||𝑎₁||𝑅,(2.17) 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 replacing 𝑎_𝑛=𝑓^(𝑛)(0) in (2.14), we can prove the generalized Hyers-Ulam stability for (1.5).