International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 576852 | 7 pages | https://doi.org/10.1155/2009/576852

Hyers-Ulam Stability of Nonhomogeneous Linear Differential Equations of Second Order

Academic Editor: Seppo Hassi
Received01 Jul 2009
Accepted25 Sep 2009
Published19 Oct 2009

Abstract

The aim of this paper is to prove the stability in the sense of Hyers-Ulam of differential equation of second order . That is, if is an approximate solution of the equation , then there exists an exact solution of the equation near to .

1. Introduction and Preliminaries

In 1940, Ulam [1] posed the following problem concerning the stability of functional equations: give conditions in order for a linear mapping near an approximately linear mapping to exist. The problem for the case of approximately additive mappings was solved by Hyers [2] when and are Banach spaces, and the result of Hyers was generalized by Rassias (see [3]). Since then, the stability problems of functional equations have been extensively investigated by several mathematicians (cf. [35]).

In connection with the stability of exponential functions, Alsina and Ger [6] remarked that the differential equation has the Hyers-Ulam stability. More explicitly, they proved that if a differentiable function satisfies for all , then there exists a differentiable function satisfying for any such that for every.

The above result of Alsina and Ger has been generalized by Miura et al. [7], by Miura [8], and also by Takahasi et al. [9]. Indeed, they dealt with the Hyers-Ulam stability of the differential equation , while Alsina and Ger investigated the differential equation .

Furthermore, the result of Hyers-Ulam stability for first-order linear differential equations has been generalized by Miura et al. [10], by Takahasi et al. [11], and also by Jung [12]. They dealt with the nonhomogeneous linear differential equation of first order

Jung [13] proved the generalized Hyers-Ulam stability of differential equations of the form

and also applied this result to the investigation of the Hyers-Ulam stability of the differential equation

Recently, Wang et al. [14] discussed the Hyers-Ulam stability of the first-order nonhomogeneous linear differential equation

They proved the following theorem.

Theorem 1.1 (see [14]). Let and be continuous real functions on the interval such that and for all and some independent of . Then (1.4) has the Hyers-Ulam stability.

The aim of this paper is to investigate the Hyers-Ulam stability of the following linear differential equations of second order under some special conditions:

where .

For the sake of convenience, all the integrals in the rest of the work will be viewed as existing. We say that (1.5) has the Hyers-Ulam stability if there exists a constant with the following property: for every , if

then there exists some satisfying

such that . We call such a Hyers-Ulam stability constant for (1.5).

2. Main Results

Now, the main results of this work are given in the following theorems.

Theorem 2.1. If a twice continuously differentiable function satisfies the differential inequality
for all and for some , and the Riccati equation has a particular solution, then there exists a solution of (1.5) such that where is a constant.

Proof. Let and be a continuously differentiable function such that
We will show that there exists a constant independent of and such that for some satisfying .
Assume that is a particular solution of Riccati equation ; if we set
then thus
Using the similar technique in [14], we can prove
satisfying and there exists an such that
By , we get

Using the same technique as above, we know that
satisfying and there exists a such that The desired conclusion is proved.

Theorem 2.2. Let and be continuous real functions on the interval such that and is a nonzero bounded particular solution . If is a twice continuously differentiable function, which satisfies the differential inequality for all and for some , then there exists a solution such that where is a constant, and satisfies .

Proof. Setting , we obtain
and also
By a simple calculation, we see that

Without loss of generality, we may assume that . Using the similar technique in [14], we know that
satisfies and also for some .
From the inequalities and the similar technique in [14], we further get that
satisfies and also for some .
Consequently, we have
for some positive constant .
Define . It then follows from the above inequality that holds for every . We can easily verify that satisfies . This completes the proof of our theorem.

We can prove the following corollaries by using an analogous argument. Hence, we omit the proofs.

Corollary 2.3. Let and be continuous real functions on the interval such that and . If is a twice continuously differentiable function, which satisfies the differential inequality
for all and for some , then there exists a solution such that where is a constant, and satisfies .

Corollary 2.4. Let and be continuous real functions on the interval such that and is a nonzero bounded particular solution . If is a twice continuously differentiable function, which satisfies the differential inequality
for all and for some , then there exists a solution such that where is a constant, and satisfies .

Acknowledgments

The work of this paper was supported by the National Natural Science Foundation of China (10871213). The authors would like to thank the referee for a number of valuable suggestions regarding a previous version of this paper.

References

  1. S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience, New York, NY, USA, 1960. View at: MathSciNet
  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: Google Scholar | Zentralblatt MATH | MathSciNet
  3. Th. 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet
  4. Y.-H. Lee and K.-W. Jun, “A generalization of the Hyers-Ulam-Rassias stability of Jensen's equation,” Journal of Mathematical Analysis and Applications, vol. 238, no. 1, pp. 305–315, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  5. C.-G. Park, “On the stability of the linear mapping in Banach modules,” Journal of Mathematical Analysis and Applications, vol. 275, no. 2, pp. 711–720, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  6. 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. T. Miura, S.-E. Takahasi, and H. Choda, “On the Hyers-Ulam stability of real continuous function valued differentiable map,” Tokyo Journal of Mathematics, vol. 24, no. 2, pp. 467–476, 2001. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  8. 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 | MathSciNet
  9. 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: Google Scholar | Zentralblatt MATH | MathSciNet
  10. 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet
  11. S.-E. Takahasi, H. Takagi, T. Miura, and S. Miyajima, “The Hyers-Ulam stability constants of first order linear differential operators,” Journal of Mathematical Analysis and Applications, vol. 296, no. 2, pp. 403–409, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  12. 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: Google Scholar | Zentralblatt MATH | MathSciNet
  13. 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet
  14. G. Wang, M. Zhou, and L. Sun, “Hyers-Ulam stability of linear differential equations of first order,” Applied Mathematics Letters, vol. 21, no. 10, pp. 1024–1028, 2008. View at: Google Scholar | Zentralblatt MATH | MathSciNet

Copyright © 2009 Yongjin Li and Yan Shen. 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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