Hyers-Ulam Stability of Polynomial Equations

Abstract

We prove the Hyers-Ulam stability of the polynomial equation 𝑎_𝑛𝑥^𝑛+𝑎_𝑛−1𝑥^𝑛−1+⋯+𝑎₁𝑥+𝑎₀=0. We give an affirmative answer to a problem posed by Li and Hua (2009).

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 stability theory for functional equations. The result of Hyers was generalized by Rassias [3]. The topic of the Hyers-Ulam stability of functional equations and its applications has been studied by a number of mathematicians; see [3–40] and references therein.

Recently, Li and Hua [41] discussed and proved the Hyers-Ulam stability of the 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] satisfy 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 the Hyers-Ulam stability for the case that this real polynomial equation has some solutions in [𝑎,𝑏]. The aim of this paper is to give a positive answer to this problem. First of all, we give the definition of the Hyers-Ulam stability.

Definition 1.2. One says that (1.4) has the Hyers-Ulam stability if there exists a constant 𝐾>0 with the following property:
for every 𝜀>0,𝑦∈[−1,1], if ||𝑎_𝑛𝑦^𝑛+𝑎_𝑛−1𝑦^𝑛−1+⋯+𝑎₁𝑦+𝑎₀||≤𝜀(1.5) then there exists some 𝑧∈[−1,1] satisfying 𝑎_𝑛𝑧^𝑛+𝑎_𝑛−1𝑧^𝑛−1+⋯+𝑎₁𝑧+𝑎₀=0(1.6) such that |𝑦−𝑧|≤𝐾𝜀. One calls such 𝐾 a Hyers-Ulam stability constant for (1.4). For the complex polynomial equation, [−1,1] is replaced by closed unit disc 𝐷={𝑧∈ℂ;|𝑧|≤1}.(1.7)

2. Main Results

The aim of this work is to investigate the Hyers-Ulam stability for (1.4).

Theorem 2.1. If ||𝑎||𝑎₀||<||𝑎₁||−||𝑎₂||+||𝑎₃||+⋯+||𝑎_𝑛||,(2.1)₁||>2||𝑎₂||+3||𝑎₃||+⋯+(𝑛−1)||𝑎_𝑛−1||+𝑛||𝑎_𝑛||,(2.2) then there exists an exact solution 𝑣∈[−1,1] of (1.4).

Proof. If we set 𝑔(𝑥)=1𝑎₁−𝑎₀−𝑎₂𝑥²−𝑎₃𝑥³−⋯−𝑎_𝑛−1𝑥^𝑛−1−𝑎_𝑛𝑥^𝑛,(2.3) for 𝑥∈[−1,1], then we have ||𝑔(𝑥)||=1||𝑎₁||||−𝑎⁰≤1−𝑎₂𝑥²−⋯−𝑎_𝑛−1𝑥^𝑛−1−𝑎_𝑛𝑥^𝑛||||𝑎₁||||𝑎⁰≤1||+||𝑎₂||+⋯+||𝑎_𝑛−1||+||𝑎_𝑛||(2.4) by (2.1).
Let 𝑋=[−1,1] and 𝑑(𝑥,𝑦)=|𝑥−𝑦|. Then (𝑋,𝑑) is a complete metric space and 𝑔 maps 𝑋 to 𝑋. Now, we will show that 𝑔 is a contraction from 𝑋 to 𝑋. For any 𝑥,𝑦∈𝑋, we have 𝑑(𝑔(𝑥),𝑔(𝑦))=||||1𝑎₁−𝑎₀−𝑎₂𝑥²−⋯−𝑎_𝑛𝑥^𝑛−1𝑎₁−𝑎₀≤1−⋯−𝑎_𝑛𝑦^𝑛||||||𝑎₁||||𝑥−𝑦||||𝑎²≤1||||𝑥+𝑦||+⋯+||𝑎_𝑛||||𝑥^𝑛−1+⋯+𝑦^𝑛−1||||𝑎₁||||𝑥−𝑦||2||𝑎²||+3||𝑎₃||+⋯+(𝑛−1)||𝑎_𝑛−1||+𝑛||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) Here 𝜆=2||𝑎²||+3||𝑎₃||+⋯+(𝑛−1)||𝑎_𝑛−1||+𝑛||𝑎_𝑛||||𝑎₁||<1.(2.7) Thus 𝑔 is a contraction from 𝑋 to 𝑋. By the Banach contraction mapping theorem, there exists a unique 𝑣∈𝑋 such that 𝑔(𝑣)=𝑣.(2.8) Hence (1.4) has a solution on [−1,1].

As an application of Rouche's theorem, we prove the following theorem for complex polynomial equation𝑎_𝑛𝑧^𝑛+𝑎_𝑛−1𝑧^𝑛−1+⋯+𝑎₁𝑧+𝑎₀=0,(2.9) which is much better than the above result. In fact, we prove the following theorem.

Theorem 2.2. If ||𝑎₀||<||𝑎₁||−||𝑎₂||+||𝑎₃||+⋯+||𝑎_𝑛||,(2.10) then there exists an exact solution in open unit disc for (2.9).

Proof. If we set 𝑔(𝑧)=1𝑎₁−𝑎₀−𝑎₂𝑧²−𝑎₃𝑧³−⋯−𝑎_𝑛−1𝑧^𝑛−1−𝑎_𝑛𝑧^𝑛,(2.11) then we have ||𝑔(𝑧)||=1||𝑎₁||||−𝑎⁰≤1−𝑎₂𝑧²−⋯−𝑎_𝑛−1𝑧^𝑛−1−𝑎_𝑛𝑧^𝑛||||𝑎₁||||𝑎⁰<1||+||𝑎₂||+⋯+||𝑎_𝑛−1||+||𝑎_𝑛||,for|𝑧|≤1(2.12) by (2.10).
Since |𝑔(𝑧)|<1 for |𝑧|=1, then |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.

Theorem 2.3. If the conditions of Theorem 2.1 hold and 𝑦∈[−1,1] satisfies the inequality ||𝑎_𝑛𝑦^𝑛+𝑎_𝑛−1𝑦^𝑛−1+⋯+𝑎₁𝑦+𝑎₀||≤𝜀,(2.13) then (1.4) has the Hyers-Ulam stability.

Proof. Let 𝜀>0 and 𝑦∈[−1,1] such that ||𝑎_𝑛𝑦^𝑛+𝑎_𝑛−1𝑦^𝑛−1+⋯+𝑎₁𝑦+𝑎₀||≤𝜀.(2.14) We will show that there exists a constant 𝐾 independent of 𝜀 and 𝑣 such that ||𝑦−𝑣||≤𝐾𝜀(2.15) for some 𝑣∈[−1,1] satisfying (1.4).
Let us introduce the abbreviation 𝐾=1/|𝑎₁|(1−𝜆). Then ||𝑦−𝑣||=≤||||||𝑦𝑦−−𝑔1(𝑦)+𝑔(𝑦)−𝑔(𝑣)||≤||𝑦−𝑔(𝑦)||+||𝑔(𝑦)−𝑔(𝑣)||𝑎₁−𝑎₀=1−𝑎₂𝑦²−⋯−𝑎_𝑛𝑦^𝑛||||+𝜆||𝑦−𝑣||||𝑎₁||||𝑎^𝑛𝑦^𝑛+𝑎_𝑛−1𝑦^𝑛−1+⋯+𝑎₁𝑦+𝑎₀||+𝜆||𝑦−𝑣||.(2.16) Thus, we have ||𝑦−𝑣||≤1||𝑎₁||(1−𝜆)||𝑎^𝑛≤𝐾𝜀𝑦^𝑛+𝑎_𝑛−1𝑦^𝑛−1+⋯+𝑎₁𝑦+𝑎₀||(2.17) by (2.13) and so the result follows.

Corollary 2.4. In Theorem 2.2, if there exists 𝑦∈𝐷 satisfying the inequality (2.13), then (2.9) has the Hyers-Ulam stability.

Remark 2.5. For 𝑎_𝑛=1, 𝑎_𝑖=0, for 2≤𝑖≤𝑛−1, combining Theorems 2.1 and 2.3 gives Theorem 1.1.

Remark 2.6. By the similar way, one can easily prove the Hyers-Ulam stability of (1.4) on any finite interval [𝑎,𝑏].

Remark 2.7. Let 𝑓 be any complex function such that 𝑓 is analytic in Δ={𝑧∈ℂ∶|𝑧|⟨𝑅,𝑅⟩0}.(2.18) It is an interesting open problem whether 𝑓 has the Hyers-Ulam stability for the case that 𝑓 has some zeros in Δ.

We note that there is an error in the proof of Theorem 2.2 of [41], when Li and Hua stated that if (𝑋,𝑑) is a complete metric linear space then metric 𝑑 is invariant, more precisely 𝑑(𝑥,𝑦)=𝑑(𝑥−𝑦,0)(2.19) for all 𝑥,𝑦∈𝑋. We give a counterexample for this case. Suppose that 𝑋=ℝ, and we define metric 𝑑 on 𝑋 as follows: 𝑑(𝑥,𝑦)=||𝑥+[𝑥]−(𝑦+[𝑦])||,(2.20) for all 𝑥,𝑦∈𝑋(𝑋,𝑑) is a complete metric linear space, and 𝑑 is not an invariant metric on 𝑋, that is, there are 𝑥,𝑦∈𝑋 such that 𝑑(𝑥,𝑦)≠𝑑(𝑥−𝑦,0).(2.21)