Table of Contents Author Guidelines Submit a Manuscript
Journal of Function Spaces and Applications
Volume 2012, Article ID 382932, 8 pages
http://dx.doi.org/10.1155/2012/382932
Research Article

Simple Harmonic Oscillator Equation and Its Hyers-Ulam Stability

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 𝑋 be a normed space over a scalar field 𝕂 and let 𝐼 be an open interval, where 𝕂 denotes either or . Assume that 𝑎0,𝑎1,,𝑎𝑛𝐼𝕂 are given continuous functions, 𝑔𝐼𝑋 is a given continuous function, and 𝑦𝐼𝑋 is an 𝑛 times continuously differentiable function satisfying the inequality:𝑎𝑛(𝑡)𝑦(𝑛)(𝑡)+𝑎𝑛1(𝑡)𝑦(𝑛1)(𝑡)++𝑎1(𝑡)𝑦(𝑡)+𝑎0(𝑡)𝑦(𝑡)+𝑔(𝑡)𝜀,(1.1)

for all 𝑡𝐼 and for a given 𝜀>0. If there exists an 𝑛 times continuously differentiable function 𝑦0𝐼𝑋 satisfying𝑎𝑛(𝑡)𝑦0(𝑛)(𝑡)+𝑎𝑛1(𝑡)𝑦0(𝑛1)(𝑡)++𝑎1(𝑡)𝑦0(𝑡)+𝑎0(𝑡)𝑦0(𝑡)+𝑔(𝑡)=0,(1.2)

and 𝑦(𝑡)𝑦0(𝑡)𝐾(𝜀) for any 𝑡𝐼, where 𝐾(𝜀) is an expression of 𝜀 with lim𝜀0𝐾(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 [17].

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 𝑓𝐼 satisfies the inequality |𝑦(𝑡)𝑦(𝑡)|𝜀, where 𝐼 is an open subinterval of , then there exists a solution 𝑓0𝐼 of the differential equation 𝑦(𝑡)=𝑦(𝑡) such that |𝑓(𝑡)𝑓0(𝑡)|3𝜀 for any 𝑡𝐼. 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 𝑦(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, 𝑦(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 [1619]. 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:𝑦(𝑥)+𝜔2𝑦(𝑥)=𝑚=0𝑎𝑚𝑥𝑚,(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:𝑦(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 𝑚1,𝑐2𝑚=𝑚1𝑖=0(1)𝑚𝑖1𝑎2𝑖(2𝑖)!𝜔(2𝑚)!2𝑚2𝑖2,𝑐2𝑚+1=𝑚1𝑖=0(1)𝑚𝑖1𝑎2𝑖+1(2𝑖+1)!𝜔(2𝑚+1)!2𝑚2𝑖2,(2.2) where we refer to (1.3) for the 𝑎𝑚. We can easily check that these 𝑐𝑚 satisfy the following𝑎𝑚=(𝑚+2)(𝑚+1)𝑐𝑚+2+𝜔2𝑐𝑚,(2.3) for any 𝑚{0,1,2,}.

Lemma 2.1. (a) If the power series 𝑚=0𝑎𝑚𝑥𝑚 converges for all 𝑥(𝜌,𝜌) with 𝜌>1, then the power series 𝑚=2𝑐𝑚𝑥𝑚 with 𝑐𝑚 given in (2.2) satisfies the inequality |𝑚=2𝑐𝑚𝑥𝑚|𝐶1/(1|𝑥|) for some positive constant 𝐶1 and for any 𝑥(1,1).
(b) If the power series 𝑚=0𝑎𝑚𝑥𝑚 converges for all 𝑥(𝜌,𝜌) with 𝜌1, then for any positive 𝜌0<𝜌, the power series 𝑚=2𝑐𝑚𝑥𝑚 with 𝑐𝑚 given in (2.2) satisfies the inequality |𝑚=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 𝑚=2𝑐𝑚𝑥𝑚 is convergent for all 𝑥(𝜌,𝜌).

Proof. (a) Since the power series 𝑚=0𝑎𝑚𝑥𝑚 is absolutely convergent on its interval of convergence, with 𝑥=1, 𝑚=0𝑎𝑚 converges absolutely, that is, 𝑚=0|𝑎𝑚|<𝑀1 by some number 𝑀1.
We know that ||𝑎2𝑖||(2𝑖)!𝜔(2𝑚)!2𝑚2𝑖2=||𝑎2𝑖||𝜔2𝑚(2𝑚1)𝜔(2𝑚2)||𝑎(2𝑖+1)2𝑖||(||𝑎2𝑚(2𝑚1)for0<𝜔1)2𝑖||𝜔2𝑚(2𝑚1)[𝜔]||𝑎(for𝜔>1)2𝑖||2𝑚(2𝑚1)max{1,𝜔𝜔},(2.4) since for 𝜔>1, each factor of the form 𝜔/ in the summand is either less than 1 if >[𝜔], or is bigger than or equal to 1 if [𝜔], where [𝜔] denotes the largest integer less than or equal to 𝜔. Thus, we obtain ||𝑐2𝑚||𝑚1𝑖=0||𝑎2𝑖||(2𝑖)!𝜔(2𝑚)!2𝑚2𝑖2𝑚1𝑖=0||𝑎2𝑖||2𝑚(2𝑚1)max{1,𝜔𝜔}max{1,𝜔𝜔}𝑖=0||𝑎𝑖||𝑀max1,𝑀1𝜔𝜔𝐶1.(2.5)
Similarly, we have ||𝑎2𝑖+1||(2𝑖+1)!𝜔(2𝑚+1)!2𝑚2𝑖2||𝑎2𝑖+1||2𝑚(2𝑚1)max{1,𝜔𝜔},(2.6) and |𝑐2𝑚+1|𝐶1 for all 𝑚1.
Therefore, we get |||||𝑚=2𝑐𝑚𝑥𝑚|||||𝑚=2||𝑐𝑚||||𝑥𝑚||𝐶1𝑚=2||𝑥𝑚||𝐶11|𝑥|,(2.7) for every 𝑥(1,1).
(b) The power series 𝑚=0𝑎𝑚𝑥𝑚 is absolutely convergent on its interval of convergence, and, therefore, for any given 𝜌0<𝜌, the series 𝑚=0|𝑎𝑚𝑥𝑚| is convergent on [𝜌0,𝜌0] and 𝑚=0||𝑎𝑚|||𝑥|𝑚𝑚=0||𝑎𝑚||𝜌𝑚0𝑀2(2.8) for any 𝑥[𝜌0,𝜌0]. Now, it follows from (2.2), (2.4), (2.6), and (2.8) that |||||𝑚=2𝑐𝑚𝑥𝑚|||||𝑚=1||𝑐2𝑚||𝜌02𝑚+𝑚=1||𝑐2𝑚+1||𝜌02𝑚+1𝑚=1𝜌02𝑚𝑚1𝑖=0||𝑎2𝑖||(2𝑖)!𝜔(2𝑚)!2𝑚2𝑖2+𝑚=1𝜌02𝑚+1𝑚1𝑖=0||𝑎2𝑖+1||(2𝑖+1)!𝜔(2𝑚+1)!2𝑚2𝑖2𝑚=1𝑚1𝑖=0||𝑎2𝑖||𝜌02𝑖max{1,𝜔𝜔}+2𝑚(2𝑚1)𝑚=1𝑚1𝑖=0||𝑎2𝑖+1||𝜌02𝑖+1max{1,𝜔𝜔}2𝑚(2𝑚1)𝑚=1𝑀2max{1,𝜔𝜔}+2𝑚(2𝑚1)𝑚=1𝑀2max{1,𝜔𝜔}𝑀2𝑚(2𝑚1)max2,𝑀2𝜔𝜔𝑚=113𝑚(2𝑚1)2𝑀max2,𝑀2𝜔𝜔𝐶2,(2.9) for any 𝑥[𝜌0,𝜌0].

Lemma 2.2. Suppose that the power series 𝑚=0𝑎𝑚𝑥𝑚 converges for all 𝑥(𝜌,𝜌) with some positive 𝜌. Let 𝜌1=min{1,𝜌}. Then, the power series 𝑚=2𝑐𝑚𝑥𝑚 with 𝑐𝑚 given in (2.2) is convergent for all 𝑥(𝜌1,𝜌1). Further, for any positive 𝜌0<𝜌1, |𝑚=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 𝜌1, then 𝜌1=𝜌. By Lemma 2.1(b), for any positive 𝜌0<𝜌=𝜌1, |𝑚=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 |||||𝑚=2𝑐𝑚𝑥𝑚|||||𝐶1𝐶1|𝑥|11𝜌0𝐶max11𝜌0,𝐶2𝐶,(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 𝑚=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 𝑚=0𝑎𝑚𝑥𝑚 is 𝜌>0. Let 𝜌1=min{1,𝜌}. Then, every solution 𝑦(𝜌1,𝜌1) of the simple harmonic oscillator equation (1.3) can be expressed by 𝑦(𝑥)=𝑦(𝑥)+𝑚=2𝑐𝑚𝑥𝑚,(2.11)where 𝑦(𝑥) is a simple harmonic oscillator function and 𝑐𝑚 are given by (2.2).

Proof. We show that 𝑚=2𝑐𝑚𝑥𝑚 satisfies (1.3). By Lemma 2.2, the power series 𝑚=2𝑐𝑚𝑥𝑚 is convergent for each 𝑥(𝜌1,𝜌1).
Substituting 𝑚=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) 𝑦(𝑥)+𝜔2𝑦(𝑥)=𝑚=0(𝑚+2)(𝑚+1)𝑐𝑚+2+𝜔2𝑐𝑚𝑥𝑚=𝑚=0𝑎𝑚𝑥𝑚,(2.12) for all 𝑥(𝜌1,𝜌1).
Therefore, every solution 𝑦(𝜌1,𝜌1) of the inhomogeneous simple harmonic oscillator equation (1.3) can be expressed by 𝑦(𝑥)=𝑦(𝑥)+𝑚=2𝑐𝑚𝑥𝑚,(2.13)where 𝑦(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) be a given analytic function which can be represented by a power series 𝑚=0𝑏𝑚𝑥𝑚 whose radius of convergence is at least 𝜌>0. Suppose there exists a constant 𝜀>0 such that ||𝑦(𝑥)+𝜔2||𝑦(𝑥)𝜀,(3.1) for all 𝑥(𝜌,𝜌) and for some positive number 𝜔. Let 𝜌1=min{1,𝜌}. Define 𝑎𝑚=(𝑚+2)(𝑚+1)𝑏𝑚+2+𝜔2𝑏𝑚 for all 𝑚{0,1,2,} and suppose further that 𝑚=0||𝑎𝑚𝑥𝑚|||||||𝐾𝑚=0𝑎𝑚𝑥𝑚|||||,(3.2) for all 𝑥(𝜌,𝜌) and for some constant 𝐾. Then, there exists a simple harmonic oscillator function 𝑦(𝜌1,𝜌1) such that ||𝑦(𝑥)𝑦||(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 𝑦(𝑥)+𝜔2𝑦(𝑥)=𝑚=0𝑎𝑚𝑥𝑚(3.4) also satisfies 𝑚=0||𝑎𝑚𝑥𝑚|||||||𝐾𝑚=0𝑎𝑚𝑥𝑚|||||𝐾𝜀,(3.5) for all 𝑥(𝜌,𝜌) from (3.1).
According to Theorem 2.3, 𝑦(𝑥) can be written as 𝑦(𝑥)+𝑚=2𝑐𝑚𝑥𝑚 for all 𝑥(𝜌1,𝜌1), where 𝑦 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)𝑦||=|||||(𝑥)𝑚=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=max11𝜌0,𝐶2=max𝐾𝜀1𝜌0𝜔𝜔,𝐾𝜀1𝜌0,32𝐾𝜀𝜔𝜔,32𝐾𝜀,(3.7) we find 𝐾𝐶=max1𝜌0𝜔𝜔,𝐾1𝜌0,32𝐾𝜔𝜔,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 𝑦(𝑥) be a simple harmonic oscillator function for 𝜔=1/4 and let 𝑦(1,1) be an analytic function given by𝑦(𝑥)=𝑦(𝑥)+𝜀𝑚=0𝑥2𝑚4𝑚+1,(4.1) where 𝜀 is a positive constant. (We can easily show that the radius of convergence of the power series 𝑚=0𝑥2𝑚/4𝑚+1 is 2). Then, we have𝑦(1𝑥)+16𝑦(𝑥)=𝑚=0𝑎𝑚𝑥𝑚,(4.2)

where𝑎𝑚=4𝑚2+12𝑚+92𝑚+60((for𝑚{0,2,4,})for𝑚{1,3,5,}),𝑚=0||𝑎𝑚𝑥𝑚||=|||||𝑚=0𝑎𝑚𝑥𝑚|||||(4.3)

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

Furthermore, we get|||𝑦(1𝑥)+|||16𝑦(𝑥)𝑚=08𝜀(𝑚+1)(2𝑚+1)4𝑚+3|𝑥|2𝑚+𝑚=0𝜀4𝑚+3|𝑥|2𝑚𝑚=015𝜀32|𝑥|2𝑚2𝑚+𝑚=0𝜀43|𝑥|2𝑚4𝑚𝑚=015𝜀1322𝑚+𝑚=0𝜀4314𝑚<𝜀,(4.4)and it follows from (4.1) that||𝑦(𝑥)𝑦||=|||||𝜀(𝑥)𝑚=0𝑥2𝑚4𝑚+1|||||𝜀4𝑚=014𝑚=𝐶𝜀,(4.5)

for all 𝑥(1,1), where we set 𝐶=1/3.

References

  1. S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, USA, 2002. View at Publisher · View at Google Scholar
  2. D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, pp. 222–224, 1941. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser Boston Inc., Boston, Mass, USA, 1998.
  4. D. H. Hyers and T. M. Rassias, “Approximate homomorphisms,” Aequationes Mathematicae, vol. 44, no. 2-3, pp. 125–153, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press Inc., Palm Harbor, Fla, USA, 2001.
  6. T. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297–300, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. S. M. Ulam, Problems in Modern Mathematics, John Wiley & Sons, 1964.
  8. M. Obłoza, “Hyers stability of the linear differential equation,” Rocznik Naukowo-Dydaktyczny, no. 13, pp. 259–270, 1993. View at Google Scholar · View at Zentralblatt MATH
  9. M. Obłoza, “Connections between Hyers and Lyapunov stability of the ordinary differential equations,” Rocznik Naukowo-Dydaktyczny, no. 14, pp. 141–146, 1997. View at Google Scholar · View at Zentralblatt MATH
  10. C. Alsina and R. Ger, “On some inequalities and stability results related to the exponential function,” Journal of Inequalities and Applications, vol. 2, no. 4, pp. 373–380, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. S.-E. Takahasi, T. Miura, and S. Miyajima, “On the Hyers-Ulam stability of the Banach space-valued differential equation y=λy,” Bulletin of the Korean Mathematical Society, vol. 39, no. 2, pp. 309–315, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. T. Miura, “On the Hyers-Ulam stability of a differentiable map,” Scientiae Mathematicae Japonicae, vol. 55, no. 1, pp. 17–24, 2002. View at Google Scholar
  13. T. Miura, S.-M. Jung, and S.-E. Takahasi, “Hyers-Ulam-Rassias stability of the Banach space valued linear differential equations y=λy,” Journal of the Korean Mathematical Society, vol. 41, no. 6, pp. 995–1005, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. T. Miura, S. Miyajima, and S.-E. Takahasi, “Hyers-Ulam stability of linear differential operator with constant coefficients,” Mathematische Nachrichten, vol. 258, pp. 90–96, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. T. Miura, S. Miyajima, and S.-E. Takahasi, “A characterization of Hyers-Ulam stability of first order linear differential operators,” Journal of Mathematical Analysis and Applications, vol. 286, no. 1, pp. 136–146, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. S.-M. Jung, “Hyers-Ulam stability of linear differential equations of first order,” Applied Mathematics Letters, vol. 17, no. 10, pp. 1135–1140, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. S.-M. Jung, “Hyers-Ulam stability of linear differential equations of first order. II,” Applied Mathematics Letters, vol. 19, no. 9, pp. 854–858, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S.-M. Jung, “Hyers-Ulam stability of linear differential equations of first order. III,” Journal of Mathematical Analysis and Applications, vol. 311, no. 1, pp. 139–146, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. S.-M. Jung, “Hyers-Ulam stability of a system of first order linear differential equations with constant coefficients,” Journal of Mathematical Analysis and Applications, vol. 320, no. 2, pp. 549–561, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. S.-M. Jung, “Legendre's differential equation and its Hyers-Ulam stability,” Abstract and Applied Analysis, vol. 2007, Article ID 56419, 14 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. B. Kim and S.-M. Jung, “Bessel's differential equation and its Hyers-Ulam stability,” Journal of Inequalities and Applications, vol. 2007, Article ID 21640, 8 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet