Journal of Function Spaces and ApplicationsVolume 2012 (2012), Article ID 382932, 8 pagesdoi:10.1155/2012/382932

Research Article

Simple Harmonic Oscillator Equation and Its Hyers-Ulam Stability

Soon-Mo Jung and Byungbae Kim

Mathematics Section, College of Science and Technology, Hongik University, Jochiwon 339-701, Republic of Korea

Received 6 March 2008; Accepted 21 April 2008

Academic Editor: George Isac

Copyright © 2012 Soon-Mo Jung and Byungbae Kim. 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.

Abstract.

We solve the inhomogeneous simple harmonic oscillator equation and apply this result to obtain a partial solution to the Hyers-Ulam stability problem for the simple harmonic oscillator equation.

1. Introduction

Let be a normed space over a scalar field and let be an open interval, where denotes either or . Assume that are given continuous functions, is a given continuous function, and is an times continuously differentiable function satisfying the inequality:

for all and for a given . If there exists an times continuously differentiable function satisfying

and for any , where is an expression of with , 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 [1–7].

Obloza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [8, 9]). Here, we will introduce a result of Alsina and Ger (see [10]). If a differentiable function satisfies the inequality , where is an open subinterval of , then there exists a solution of the differential equation such that for any . This result has been generalized by Takahasi et al. Indeed, it was proved in [11] that the Hyers-Ulam stability holds true for the Banach space valued differential equation (see also [12, 13]).

Moreover, Miura et al. [14] investigated the Hyers-Ulam stability of th order linear differential equation with complex coefficients. They [15] also proved the Hyers-Ulam stability of linear differential equations of first order, , where is a continuous function.

Jung also proved the Hyers-Ulam stability of various linear differential equations of first order [16–19]. Moreover, he could successfully apply the power series method to the study of the Hyers-Ulam stability of Legendre differential equation (see [20]). Subsequently, the authors [21] investigated the Hyers-Ulam stability problem for Bessel differential equation by applying the same method.

In Section 2 of this paper, by using the ideas from [20, 21], we investigate the general solution of the inhomogeneous simple harmonic oscillator equation of the form:

where is a given positive number. Section 3 will be devoted to a partial solution of the Hyers-Ulam stability problem for the simple harmonic oscillator equation (2.1) in a subclass of analytic functions.

2. Inhomogeneous Simple Harmonic Oscillator Equation

A function is called a simple harmonic oscillator function if it satisfies the simple harmonic oscillator equation:

The simple harmonic oscillator equation plays a great role in physics and engineering. In particular, it describes quantum particles confined in potential wells in quantum mechanics and the Hyers-Ulam stability of solutions of this equation is very important.

In this section, we define and for , where we refer to (1.3) for the . We can easily check that these satisfy the following for any .

Lemma 2.1. (a) If the power series converges for all with , then the power series with given in (2.2) satisfies the inequality for some positive constant and for any . (b) If the power series converges for all with , then for any positive , the power series with given in (2.2) satisfies the inequality for any and for some positive constant which depends on . Since is arbitrarily close to , this means that is convergent for all .

Proof. (a) Since the power series is absolutely convergent on its interval of convergence, with , converges absolutely, that is, by some number .We know that since for , each factor of the form in the summand is either less than 1 if , or is bigger than or equal to 1 if , where denotes the largest integer less than or equal to . Thus, we obtain Similarly, we have and for all .Therefore, we get for every .(b) The power series is absolutely convergent on its interval of convergence, and, therefore, for any given , the series is convergent on and for any . Now, it follows from (2.2), (2.4), (2.6), and (2.8) that for any .

Lemma 2.2. Suppose that the power series converges for all with some positive . Let . Then, the power series with given in (2.2) is convergent for all . Further, for any positive , for any and for some positive constant which depends on .

Proof. The first statement follows from the latter statement. Therefore, let us prove the latter statement. If , then . By Lemma 2.1(b), for any positive , for each and for some positive constant which depends on . If , then by Lemma 2.1(a), for any positive , we get for all and for some positive constant which depends on .

Using these definitions and the lemmas above, we will now show that is a particular solution of the inhomogeneous simple harmonic oscillator equation (1.3).

Theorem 2.3. Assume that is a given positive number and the radius of convergence of the power series is . Let . Then, every solution of the simple harmonic oscillator equation (1.3) can be expressed by where is a simple harmonic oscillator function and are given by (2.2).

Proof. We show that satisfies (1.3). By Lemma 2.2, the power series is convergent for each .Substituting for in (1.3) and collecting like powers together, it follows from (2.2) and (2.3) that (with ) for all .Therefore, every solution of the inhomogeneous simple harmonic oscillator equation (1.3) can be expressed by where is a simple harmonic oscillator function.

3. Partial Solution to Hyers-Ulam Stability Problem

In this section, we will investigate a property of the simple harmonic oscillator equation (2.1) concerning the Hyers-Ulam stability problem. That is, we will try to answer the question whether there exists a simple harmonic oscillator function near any approximate simple harmonic oscillator function.

Theorem 3.1. Let be a given analytic function which can be represented by a power series whose radius of convergence is at least . Suppose there exists a constant such that for all and for some positive number . Let . Define for all and suppose further that for all and for some constant . Then, there exists a simple harmonic oscillator function such that for all , where is any positive number and is some constant which depends on .

Proof. We assumed that can be represented by a power series and also satisfies for all from (3.1).According to Theorem 2.3, can be written as for all , where is some simple harmonic oscillator function and are given by (2.2). Then by Lemmas 2.1 and 2.2 and their proofs (replace and with in Lemma 2.1), for all , where is any positive number and is some constant which depends on .Actually from the proof of Lemma 2.1, with both and replaced by , we find and . Further from the proof of Lemma 2.2, we have we find which completes the proof of our theorem.

4. Example

In this section, we show that there certainly exist functions which satisfy all the conditions given in Theorem 3.1. We introduce an example related to the simple harmonic oscillator equation (1.3) for .

Let be a simple harmonic oscillator function for and let be an analytic function given by where is a positive constant. (We can easily show that the radius of convergence of the power series is . Then, we have

where

for any . So we can here choose .

Furthermore, we getand it follows from (4.1) that

for all , where we set .

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