Bessel's Differential Equation and Its Hyers-Ulam Stability
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Research Article Bessel’s Differential Equation and Its Hyers-Ulam Stability Byungbae Kim and Soon-Mo Jung Received 23 August 2007; Accepted 25 October 2007 Recommended by Panayiotis D. Siafarikas
We solve the inhomogeneous Bessel differential equation and apply this result to obtain a partial solution to the Hyers-Ulam stability problem for the Bessel differential equation. Copyright © 2007 B. Kim and S.-M. Jung. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction In 1940, Ulam gave a wide ranging talk before the Mathematics Club of the University of Wisconsin, in which he discussed a number of important unsolved problems (see [1]). Among those was the question concerning the stability of homomorphisms: let G1 be a group and let G2 be a metric group with a metric d(·, ·) . Given any δ > 0, does there exist an ε > 0 such that if a function h : G1 →G2 satisfies the inequality d(h(xy), h(x)h(y)) < ε for all x, y ∈ G1 , then there exists a homomorphism H : G1 →G2 with d(h(x),H(x)) < δ for all x ∈ G1 ? In the following year, Hyers [2] partially solved the Ulam problem for the case where G1 and G2 are Banach spaces. Furthermore, the result of Hyers has been generalized by Rassias (see [3]). Since then, the stability problems of various functional equations have been investigated by many authors (see [4–6]). We will now consider the Hyers-Ulam stability problem for the differential equations: assume that X is a normed space over a scalar field K and that I is an open interval, where K denotes either R or C. Let a0 ,a1 ,...,an : I →K be given continuous functions, let g : I →X be a given continuous function, and let y : I →X be an n times continuously differentiable function satisfying the inequality an (t)y (n) (t) + an−1 (t)y (n−1) (t) + · · · + a1 (t)y (t) + a0 (t)y(t) + g(t) ≤ ε
(1.1)
2
Journal of Inequalities and Applications
for all t ∈ I and for a given ε > 0. If there exists an n times continuously differentiable function y0 : I →X satisfying an (t)y0(n) (t) + an−1 (t)y0(n−1) (t) + · · · + a1 (t)y0 (t) + a0 (t)y0 (t) + g(t) = 0
(1.2)
and y(t) − y0 (t) ≤ K(ε) for any t ∈ I, where K(ε) is an expression of ε with lim ε→0 K(ε) = 0, then we say that the above differential equation has the Hyers-Ulam stability. For more detailed definitions of the Hyers-Ulam stability, we refer the reader to [4–8]. Alsina and Ger were the first authors who investigated the Hyers-Ulam stability of differential equations. They proved in [9] that if a differentiable function f : I →R is a solution of the differential inequality | y (t) − y(t)| ≤ ε, where I is an open subinterval of R, then there exists a solution f0 : I →R of the differential equation y (t) = y(t) such that | f (t) − f0 (t)| ≤ 3ε for any t ∈ I. This result of Alsina and Ger has been generalized by Takahasi et al. They proved in [10] that the Hyers-Ulam stability holds true for the Ba
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