Hyers-Ulam Stability of Polynomial Equations

and Applied Analysis 3 for x ∈ −1, 1 , then we have ∣ ∣g x ∣ ∣ 1 |a1| ∣ ∣ ∣−a0 − a2x − · · · − an−1xn−1 − anx ∣ ∣ ∣ ≤ 1 |a1| |a0| |a2| · · · |an−1| |an| ≤ 1 2.4 by 2.1 . LetX −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 )) ∣ ∣ ∣ ∣ 1 a1 ( −a0 − a2x − · · · − anx ) − 1 a1 (−a0 − · · · − any ) ∣ ∣ ∣ ∣ ≤ 1 |a1| ∣ ∣x − y∣∣ { |a2| ∣ ∣x y ∣ ∣ · · · |an| ∣ ∣ ∣xn−1 · · · yn−1 ∣ ∣ ∣ } ≤ 1 |a1| ∣ ∣x − y∣∣{2|a2| 3|a3| · · · n − 1 |an−1| n|an|}. 2.5 For x, y ∈ −1, 1 , x / y, from 2.2 , we obtain d ( g x , g ( y )) ≤ λd(x, y). 2.6


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 equation where x ∈ −1, 1 and proved the following.where K > 0 is constant.
They also asked an open problem whether the real polynomial equation 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: then there exists some z ∈ −1, 1 satisfying 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

Main Results
The aim of this work is to investigate the Hyers-Ulam stability for 1.4 .
then there exists an exact solution v ∈ −1, 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

2.6
Here Thus g is a contraction from X to X.By the Banach contraction mapping theorem, there exists a unique v ∈ X such that 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 equation which is much better than the above result.fact, we prove the following theorem.

Theorem 2.2. If
then there exists an exact solution in open unit disc for 2.9 .
Proof.If we set then we have then 1.4 has the Hyers-Ulam stability.
Proof.Let ε > 0 and y ∈ −1, 1 such that We will show that there exists a constant K independent of ε and v such that for some v ∈ −1, 1 satisfying 1.4 .

2.16
Thus, we have   Remark 2.7.Let f be any complex function such that f is analytic in 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

by 2. 10 . 1 . 2 . 3 .
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| < Theorem If the conditions of Theorem 2.1 hold and y ∈ −1, 1 satisfies the inequality

Remark 2 . 6 .
By the similar way, one can easily prove the Hyers-Ulam stability of 1.4 on any finite interval a, b .
• • • a 1 y a 0 In Theorem 2.2, if there exists y ∈ D satisfying the inequality 2.13 , then 2.9 has the Hyers-Ulam stability.