Journal of Function Spaces

Journal of Function Spaces / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 382932 | 8 pages | https://doi.org/10.1155/2012/382932

Simple Harmonic Oscillator Equation and Its Hyers-Ulam Stability

Academic Editor: George Isac
Received06 Mar 2008
Accepted21 Apr 2008
Published02 Feb 2012

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 𝑋 be a normed space over a scalar field 𝕂 and let 𝐼 be an open interval, where 𝕂 denotes either ℝ or ℂ. Assume that ğ‘Ž0,ğ‘Ž1,…,ğ‘Žğ‘›âˆ¶ğ¼â†’ğ•‚ are given continuous functions, 𝑔∶𝐼→𝑋 is a given continuous function, and 𝑦∶𝐼→𝑋 is an 𝑛 times continuously differentiable function satisfying the inequality:â€–â€–ğ‘Žğ‘›(𝑡)𝑦(𝑛)(𝑡)+ğ‘Žğ‘›âˆ’1(𝑡)𝑦(𝑛−1)(𝑡)+⋯+ğ‘Ž1(𝑡)ğ‘¦î…ž(𝑡)+ğ‘Ž0‖‖(𝑡)𝑦(𝑡)+𝑔(𝑡)≤𝜀,(1.1)

for all 𝑡∈𝐼 and for a given 𝜀>0. If there exists an 𝑛 times continuously differentiable function 𝑦0∶𝐼→𝑋 satisfyingğ‘Žğ‘›(𝑡)𝑦0(𝑛)(𝑡)+ğ‘Žğ‘›âˆ’1(𝑡)𝑦0(𝑛−1)(𝑡)+⋯+ğ‘Ž1(𝑡)ğ‘¦î…ž0(𝑡)+ğ‘Ž0(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 lim𝜀→0𝐾(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 [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 𝑓∶𝐼→ℝ satisfies the 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 ğ‘¦î…ž(𝑡)=𝑦(𝑡) such that |𝑓(𝑡)−𝑓0(𝑡)|≤3𝜀 for any 𝑡∈𝐼. 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 ğ‘¦î…ž(𝑡)=𝜆𝑦(𝑡) (see also [12, 13]).

Moreover, Miura et al. [14] investigated the Hyers-Ulam stability of 𝑛th order linear differential equation with complex coefficients. They [15] also proved the Hyers-Ulam stability of linear differential equations of first order, ğ‘¦î…ž(𝑡)+𝑔(𝑡)𝑦(𝑡)=0, where 𝑔(𝑡) 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:ğ‘¦î…žî…ž(𝑥)+𝜔2𝑦(𝑥)=âˆžî“ğ‘š=0ğ‘Žğ‘šğ‘¥ğ‘š,(1.3)

where 𝜔 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:ğ‘¦î…žî…ž(𝑥)+𝜔2𝑦(𝑥)=0.(2.1)

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 𝑐0=𝑐1=0 and for 𝑚≥1,𝑐2𝑚=𝑚−1𝑖=0(−1)𝑚−𝑖−1ğ‘Ž2𝑖(2𝑖)!𝜔(2𝑚)!2𝑚−2𝑖−2,𝑐2𝑚+1=𝑚−1𝑖=0(−1)𝑚−𝑖−1ğ‘Ž2𝑖+1(2𝑖+1)!𝜔(2𝑚+1)!2𝑚−2𝑖−2,(2.2) where we refer to (1.3) for the ğ‘Žğ‘š. We can easily check that these 𝑐𝑚 satisfy the followingğ‘Žğ‘š=(𝑚+2)(𝑚+1)𝑐𝑚+2+𝜔2𝑐𝑚,(2.3) for any 𝑚∈{0,1,2,…}.

Lemma 2.1. (a) If the power series âˆ‘âˆžğ‘š=0ğ‘Žğ‘šğ‘¥ğ‘š converges for all 𝑥∈(−𝜌,𝜌) with 𝜌>1, then the power series âˆ‘âˆžğ‘š=2𝑐𝑚𝑥𝑚 with 𝑐𝑚 given in (2.2) satisfies the inequality |âˆ‘âˆžğ‘š=2𝑐𝑚𝑥𝑚|≤𝐶1/(1−|𝑥|) for some positive constant 𝐶1 and for any 𝑥∈(−1,1).
(b) If the power series âˆ‘âˆžğ‘š=0ğ‘Žğ‘šğ‘¥ğ‘š converges for all 𝑥∈(−𝜌,𝜌) with 𝜌≤1, then for any positive 𝜌0<𝜌, the power series âˆ‘âˆžğ‘š=2𝑐𝑚𝑥𝑚 with 𝑐𝑚 given in (2.2) satisfies the inequality |âˆ‘âˆžğ‘š=2𝑐𝑚𝑥𝑚|≤𝐶2 for any 𝑥∈[−𝜌0,𝜌0] and for some positive constant 𝐶2 which depends on 𝜌0. Since 𝜌0 is arbitrarily close to 𝜌, this means that âˆ‘âˆžğ‘š=2𝑐𝑚𝑥𝑚 is convergent for all 𝑥∈(−𝜌,𝜌).

Proof. (a) Since the power series âˆ‘âˆžğ‘š=0ğ‘Žğ‘šğ‘¥ğ‘š is absolutely convergent on its interval of convergence, with 𝑥=1, âˆ‘âˆžğ‘š=0ğ‘Žğ‘š converges absolutely, that is, âˆ‘âˆžğ‘š=0|ğ‘Žğ‘š|<𝑀1 by some number 𝑀1.
We know that ||ğ‘Ž2𝑖||(2𝑖)!𝜔(2𝑚)!2𝑚−2𝑖−2=||ğ‘Ž2𝑖||𝜔2𝑚(2𝑚−1)⋯𝜔(2𝑚−2)≤⎧⎪⎨⎪⎩||ğ‘Ž(2𝑖+1)2𝑖||(||ğ‘Ž2𝑚(2𝑚−1)for0<𝜔≤1)2𝑖||𝜔2𝑚(2𝑚−1)[𝜔]≤||ğ‘Ž(for𝜔>1)2𝑖||2𝑚(2𝑚−1)max{1,𝜔𝜔},(2.4) since for 𝜔>1, each factor of the form 𝜔/ℓ in the summand is either less than 1 if ℓ>[𝜔], or is bigger than or equal to 1 if ℓ≤[𝜔], where [𝜔] denotes the largest integer less than or equal to 𝜔. Thus, we obtain ||𝑐2𝑚||≤𝑚−1𝑖=0||ğ‘Ž2𝑖||(2𝑖)!𝜔(2𝑚)!2𝑚−2𝑖−2≤𝑚−1𝑖=0||ğ‘Ž2𝑖||2𝑚(2𝑚−1)max{1,𝜔𝜔}≤max{1,𝜔𝜔}âˆžî“ğ‘–=0||ğ‘Žğ‘–||𝑀≤max1,𝑀1𝜔𝜔≡𝐶1.(2.5)
Similarly, we have ||ğ‘Ž2𝑖+1||(2𝑖+1)!𝜔(2𝑚+1)!2𝑚−2𝑖−2≤||ğ‘Ž2𝑖+1||2𝑚(2𝑚−1)max{1,𝜔𝜔},(2.6) and |𝑐2𝑚+1|≤𝐶1 for all 𝑚≥1.
Therefore, we get |||||âˆžî“ğ‘š=2𝑐𝑚𝑥𝑚|||||â‰¤âˆžî“ğ‘š=2||𝑐𝑚||||𝑥𝑚||≤𝐶1âˆžî“ğ‘š=2||𝑥𝑚||≤𝐶11−|𝑥|,(2.7) for every 𝑥∈(−1,1).
(b) The power series âˆ‘âˆžğ‘š=0ğ‘Žğ‘šğ‘¥ğ‘š is absolutely convergent on its interval of convergence, and, therefore, for any given 𝜌0<𝜌, the series âˆ‘âˆžğ‘š=0|ğ‘Žğ‘šğ‘¥ğ‘š| is convergent on [−𝜌0,𝜌0] and âˆžî“ğ‘š=0||ğ‘Žğ‘š|||𝑥|ğ‘šâ‰¤âˆžî“ğ‘š=0||ğ‘Žğ‘š||𝜌𝑚0≡𝑀2(2.8) for any 𝑥∈[−𝜌0,𝜌0]. Now, it follows from (2.2), (2.4), (2.6), and (2.8) that |||||âˆžî“ğ‘š=2𝑐𝑚𝑥𝑚|||||â‰¤âˆžî“ğ‘š=1||𝑐2𝑚||𝜌02𝑚+âˆžî“ğ‘š=1||𝑐2𝑚+1||𝜌02𝑚+1â‰¤âˆžî“ğ‘š=1𝜌02𝑚𝑚−1𝑖=0||ğ‘Ž2𝑖||(2𝑖)!𝜔(2𝑚)!2𝑚−2𝑖−2+âˆžî“ğ‘š=1𝜌02𝑚+1𝑚−1𝑖=0||ğ‘Ž2𝑖+1||(2𝑖+1)!𝜔(2𝑚+1)!2𝑚−2𝑖−2â‰¤âˆžî“ğ‘š=1𝑚−1𝑖=0||ğ‘Ž2𝑖||𝜌02𝑖max{1,𝜔𝜔}+2𝑚(2𝑚−1)âˆžî“ğ‘š=1𝑚−1𝑖=0||ğ‘Ž2𝑖+1||𝜌02𝑖+1max{1,𝜔𝜔}≤2𝑚(2𝑚−1)âˆžî“ğ‘š=1𝑀2max{1,𝜔𝜔}+2𝑚(2𝑚−1)âˆžî“ğ‘š=1𝑀2max{1,𝜔𝜔}𝑀2𝑚(2𝑚−1)≤max2,𝑀2ğœ”ğœ”î€¾âˆžî“ğ‘š=11≤3𝑚(2𝑚−1)2𝑀max2,𝑀2𝜔𝜔≡𝐶2,(2.9) for any 𝑥∈[−𝜌0,𝜌0].

Lemma 2.2. Suppose that the power series âˆ‘âˆžğ‘š=0ğ‘Žğ‘šğ‘¥ğ‘š converges for all 𝑥∈(−𝜌,𝜌) with some positive 𝜌. Let 𝜌1=min{1,𝜌}. Then, the power series âˆ‘âˆžğ‘š=2𝑐𝑚𝑥𝑚 with 𝑐𝑚 given in (2.2) is convergent for all 𝑥∈(−𝜌1,𝜌1). Further, for any positive 𝜌0<𝜌1, |âˆ‘âˆžğ‘š=2𝑐𝑚𝑥𝑚|≤𝐶 for any 𝑥∈[−𝜌0,𝜌0] and for some positive constant 𝐶 which depends on 𝜌0.

Proof. The first statement follows from the latter statement. Therefore, let us prove the latter statement. If 𝜌≤1, then 𝜌1=𝜌. By Lemma 2.1(b), for any positive 𝜌0<𝜌=𝜌1, |âˆ‘âˆžğ‘š=2𝑐𝑚𝑥𝑚|≤𝐶2 for each 𝑥∈[−𝜌0,𝜌0] and for some positive constant 𝐶2 which depends on 𝜌0.
If 𝜌>1, then by Lemma 2.1(a), for any positive 𝜌0<1=𝜌1, we get |||||âˆžî“ğ‘š=2𝑐𝑚𝑥𝑚|||||≤𝐶1≤𝐶1−|𝑥|11−𝜌0𝐶≤max11−𝜌0,𝐶2≡𝐶,(2.10) for all 𝑥∈[−𝜌0,𝜌0] and for some positive constant 𝐶 which depends on 𝜌0.

Using these definitions and the lemmas above, we will now show that âˆ‘âˆžğ‘š=2𝑐𝑚𝑥𝑚 is a particular solution of the inhomogeneous simple harmonic oscillator equation (1.3).

Theorem 2.3. Assume that 𝜔 is a given positive number and the radius of convergence of the power series âˆ‘âˆžğ‘š=0ğ‘Žğ‘šğ‘¥ğ‘š is 𝜌>0. Let 𝜌1=min{1,𝜌}. Then, every solution 𝑦∶(−𝜌1,𝜌1)→ℂ of the simple harmonic oscillator equation (1.3) can be expressed by 𝑦(𝑥)=ğ‘¦â„Ž(𝑥)+âˆžî“ğ‘š=2𝑐𝑚𝑥𝑚,(2.11)where ğ‘¦â„Ž(𝑥) is a simple harmonic oscillator function and 𝑐𝑚 are given by (2.2).

Proof. We show that âˆ‘âˆžğ‘š=2𝑐𝑚𝑥𝑚 satisfies (1.3). By Lemma 2.2, the power series âˆ‘âˆžğ‘š=2𝑐𝑚𝑥𝑚 is convergent for each 𝑥∈(−𝜌1,𝜌1).
Substituting âˆ‘âˆžğ‘š=2𝑐𝑚𝑥𝑚 for 𝑦(𝑥) in (1.3) and collecting like powers together, it follows from (2.2) and (2.3) that (with 𝑐0=𝑐1=0) ğ‘¦â€²î…ž(𝑥)+𝜔2𝑦(𝑥)=âˆžî“ğ‘š=0(𝑚+2)(𝑚+1)𝑐𝑚+2+𝜔2𝑐𝑚𝑥𝑚=âˆžî“ğ‘š=0ğ‘Žğ‘šğ‘¥ğ‘š,(2.12) for all 𝑥∈(−𝜌1,𝜌1).
Therefore, every solution 𝑦∶(−𝜌1,𝜌1)→ℂ of the inhomogeneous simple harmonic oscillator equation (1.3) can be expressed by 𝑦(𝑥)=ğ‘¦â„Ž(𝑥)+âˆžî“ğ‘š=2𝑐𝑚𝑥𝑚,(2.13)where ğ‘¦â„Ž(𝑥) 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 𝑦∶(−𝜌,𝜌)→ℂ be a given analytic function which can be represented by a power series âˆ‘âˆžğ‘š=0𝑏𝑚𝑥𝑚 whose radius of convergence is at least 𝜌>0. Suppose there exists a constant 𝜀>0 such that ||ğ‘¦î…žî…ž(𝑥)+𝜔2||𝑦(𝑥)≤𝜀,(3.1) for all 𝑥∈(−𝜌,𝜌) and for some positive number 𝜔. Let 𝜌1=min{1,𝜌}. Define ğ‘Žğ‘š=(𝑚+2)(𝑚+1)𝑏𝑚+2+𝜔2𝑏𝑚 for all 𝑚∈{0,1,2,…} and suppose further that âˆžî“ğ‘š=0||ğ‘Žğ‘šğ‘¥ğ‘š|||||||â‰¤ğ¾âˆžî“ğ‘š=0ğ‘Žğ‘šğ‘¥ğ‘š|||||,(3.2) for all 𝑥∈(−𝜌,𝜌) and for some constant 𝐾. Then, there exists a simple harmonic oscillator function ğ‘¦â„Žâˆ¶(−𝜌1,𝜌1)→ℂ such that ||𝑦(𝑥)âˆ’ğ‘¦â„Ž||(𝑥)≤𝐶𝜀,(3.3) for all 𝑥∈[−𝜌0,𝜌0], where 𝜌0<𝜌1 is any positive number and 𝐶 is some constant which depends on 𝜌0.

Proof. We assumed that 𝑦(𝑥) can be represented by a power series and ğ‘¦î…žî…ž(𝑥)+𝜔2𝑦(𝑥)=âˆžî“ğ‘š=0ğ‘Žğ‘šğ‘¥ğ‘š(3.4) also satisfies âˆžî“ğ‘š=0||ğ‘Žğ‘šğ‘¥ğ‘š|||||||â‰¤ğ¾âˆžî“ğ‘š=0ğ‘Žğ‘šğ‘¥ğ‘š|||||≤𝐾𝜀,(3.5) for all 𝑥∈(−𝜌,𝜌) from (3.1).
According to Theorem 2.3, 𝑦(𝑥) can be written as ğ‘¦â„Žâˆ‘(𝑥)+âˆžğ‘š=2𝑐𝑚𝑥𝑚 for all 𝑥∈(−𝜌1,𝜌1), where ğ‘¦â„Ž is some simple harmonic oscillator function and 𝑐𝑚 are given by (2.2). Then by Lemmas 2.1 and 2.2 and their proofs (replace 𝑀1 and 𝑀2 with 𝐾𝜀 in Lemma 2.1), ||𝑦(𝑥)âˆ’ğ‘¦â„Ž||=|||||(𝑥)âˆžî“ğ‘š=2𝑐𝑚𝑥𝑚|||||≤𝐶𝜀(3.6) for all 𝑥∈[−𝜌0,𝜌0], where 𝜌0<𝜌1 is any positive number and 𝐶 is some constant which depends on 𝜌0.
Actually from the proof of Lemma 2.1, with both 𝑀1 and 𝑀2 replaced by 𝐾𝜀, we find 𝐶1=max{𝐾𝜀𝜔𝜔,𝐾𝜀} and 𝐶2=3/2𝐶1. Further from the proof of Lemma 2.2, we have 𝐶𝐶𝜀=max11−𝜌0,𝐶2=max𝐾𝜀1−𝜌0𝜔𝜔,𝐾𝜀1−𝜌0,32𝐾𝜀𝜔𝜔,32𝐾𝜀,(3.7) we find 𝐾𝐶=max1−𝜌0𝜔𝜔,𝐾1−𝜌0,32𝐾𝜔𝜔,32𝐾,(3.8)which completes the proof of our theorem.

4. Example

In this section, we show that there certainly exist functions 𝑦(𝑥) which satisfy all the conditions given in Theorem 3.1. We introduce an example related to the simple harmonic oscillator equation (1.3) for 𝜔=1/4.

Let ğ‘¦â„Ž(𝑥) be a simple harmonic oscillator function for 𝜔=1/4 and let 𝑦∶(−1,1)→ℝ be an analytic function given by𝑦(𝑥)=ğ‘¦â„Ž(𝑥)+ğœ€âˆžî“ğ‘š=0𝑥2𝑚4𝑚+1,(4.1) where 𝜀 is a positive constant. (We can easily show that the radius of convergence of the power series âˆ‘âˆžğ‘š=0𝑥2𝑚/4𝑚+1 is 2). Then, we haveğ‘¦î…žî…ž(1𝑥)+16𝑦(𝑥)=âˆžî“ğ‘š=0ğ‘Žğ‘šğ‘¥ğ‘š,(4.2)

whereğ‘Žğ‘š=⎧⎪⎨⎪⎩4𝑚2+12𝑚+92𝑚+60((for𝑚∈{0,2,4,…})for𝑚∈{1,3,5,…}),âˆžî“ğ‘š=0||ğ‘Žğ‘šğ‘¥ğ‘š||=|||||âˆžî“ğ‘š=0ğ‘Žğ‘šğ‘¥ğ‘š|||||(4.3)

for any 𝑥∈(−1,1). So we can here choose 𝐾=1.

Furthermore, we get|||ğ‘¦î…žî…ž(1𝑥)+|||≤16𝑦(𝑥)âˆžî“ğ‘š=08𝜀(𝑚+1)(2𝑚+1)4𝑚+3|𝑥|2𝑚+âˆžî“ğ‘š=0𝜀4𝑚+3|𝑥|2ğ‘šâ‰¤âˆžî“ğ‘š=015𝜀32|𝑥|2𝑚2𝑚+âˆžî“ğ‘š=0𝜀43|𝑥|2𝑚4ğ‘šâ‰¤âˆžî“ğ‘š=015𝜀1322𝑚+âˆžî“ğ‘š=0𝜀4314𝑚<𝜀,(4.4)and it follows from (4.1) that||𝑦(𝑥)âˆ’ğ‘¦â„Ž||=|||||𝜀(𝑥)âˆžî“ğ‘š=0𝑥2𝑚4𝑚+1|||||≤𝜀4âˆžî“ğ‘š=014𝑚=𝐶𝜀,(4.5)

for all 𝑥∈(−1,1), where we set 𝐶=1/3.

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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.


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