On the Stability of Polynomial Equations

Abstract

In this article we prove the Hyers–Ulam type stability for the following two equations with real coefficients:

$${a}_{n}{x}^{n} + {a}_{ n-1}{x}^{n-1} + \cdots + {a}_{ 1}x + {a}_{0} = 0\quad \mbox{ and }\quad {e}^{x} + \alpha x + \beta = 0$$

on a real interval [a, b]. More precisely, we show that if x is an approximate solution of the equation ${a}_{n}{x}^{n} + {a}_{n-1}{x}^{n-1} + \cdots + {a}_{1}x + {a}_{0} = 0$ (resp. ${e}^{x} + \alpha x + \beta = 0)$, then there exists an exact solution of the equation near x.

Mathematics Subject Classification(2000): Primary 39B82; Secondary 34K20, 26D10