Hyers-ulam stability of exact second-order linear differential equations

Mohammad Bagher Ghaemi,Choonkil Park,Badrkhan Alizadeh,Madjid Eshaghi Gordji

doi:10.1186/1687-1847-2012-36

Abstract

Abstract In this article, we prove the Hyers-Ulam stability of exact second-order linear differential equations. As a consequence, we show the Hyers-Ulam stability of the following equations: second-order linear differential equation with constant coefficients, Euler differential equation, Hermite's differential equation, Cheybyshev's differential equation, and Legendre's differential equation. The result generalizes the main results of Jung and Min, and Li and Shen. Mathematics Subject Classification (2010): 26D10; 34K20; 39B52; 39B82; 46B99.

Highlights

The stability of functional equations was first introduced by Ulam [1]
Rassias [3] generalized the theorem of Hyers by considering the stability problem with unbounded Cauchy differences ║f(x + y) - f(x) - f(y)║ ≤ ε(║x║p+║y║p) (ε > 0, p Î [0, 1))
We say that the above differential equation has the Hyers-Ulam stability

Summary

Introduction

The stability of functional equations was first introduced by Ulam [1]. We say that the above differential equation has the Hyers-Ulam stability. If the above statement is true when we replace ε and K(ε) by (t) and j(t), where , j : I ® [0, ∞) are functions not depending on f and f0 explicitly, we say that the corresponding differential equation has the Hyers-Ulam stability. The Hyers-Ulam stability of the differential equation y’ = y was first investigated by Alsina and Ger [4]. This result has been generalized by Takahasi et al [5] for the Banach space-valued differential equation y’ = ly. A t (b) φ(t) exp g(u)du is integrable on I

Then there exists a unique real number x such that

So we have

Rbe a continuous function and let