Abstract
In this paper we prove the Hyers-Ulam stability of the perfect linear differential equation , where , , and .
MSC:34K20, 26D10, 39B82, 34K06, 39B72.
Similar content being viewed by others
1 Introduction
The question concerning the stability of group homomorphisms was posed by Ulam [1]. Hyers [2] solved the case of approximately additive mappings in Banach spaces and T.M. Rassias generalized the result of Hyers [3].
Definition 1.1 Let X be a normed space over a scalar field and let I be an open interval. Assume that , are continuous functions. We say that the differential equation
has the Hyers-Ulam stability if, for any function satisfying the differential inequality
for all and some , there exists a solution of (1.1) such that for all , where is a function depending only on ε.
Obłoza [4, 5] was the first author who investigated the Hyers-Ulam stability of differential equations (also see [6]).
Jung [7] solved the inhomogeneous differential equation of the form , where n is a positive integer, and he used this result to prove the Hyers-Ulam stability of the differential equation in a special class of analytic functions.
Li and Shen [8] proved that if the characteristic equation has two different positive roots, then the linear differential equation of second order with constant coefficients has the Hyers-Ulam stability where , and (see also [9, 10]). Abdollahpour and Najati [11] proved that the third-order differential equation has the Hyers-Ulam stability. Ghaemi et al. [12] proved the Hyers-Ulam stability of the exact second-order linear differential equation
with . Here , , , are continuous functions. For more results about the Hyers-Ulam stability of differential equations, we can refer to [13–21].
Definition 1.2 We say that the differential equation
is perfect if it can be written as .
It is clear that the differential equation (1.2) is perfect if and only if . The aim of this paper is to investigate the Hyers-Ulam stability of the perfect differential equation (1.2), where , , , and . More precisely, we prove that the equation (1.2) has the Hyers-Ulam stability.
2 Hyers-Ulam stability of the perfect differential equation
In the following theorem, we prove the Hyers-Ulam stability of the differential equation (1.2).
Throughout this section, a and b are real numbers with .
Theorem 2.1 The perfect differential equation
has the Hyers-Ulam stability, where , , and for all .
Proof Let and with
Let for all . It is clear that
We define
Then
Also, we have
for all . Now we define
for all . It is clear that and
Therefore,
Hence, (2.1) implies that
Also, we have
for all . Since , there exist constants and such that . Thus
for all . Since and , there exist constants such that for all . Hence, (2.4) implies that
for all . It follows from (2.3) that
for all . □
References
Ulam SM: Problems in Modern Mathematics. Wiley, New York; 1940.
Hyers DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27: 222–224. 10.1073/pnas.27.4.222
Rassias TM: On the stability of linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1
Obłoza M: Hyers stability of the linear differential equation. Rocznik Nauk.-Dydakt. Prace Mat. 1993, 13: 259–270.
Obłoza M: Connections between Hyers and Lyapunov stability of the ordinary differential equations. Rocznik Nauk.-Dydakt. Prace Mat. 1997, 14: 141–146.
Alsina C, Ger R: On some inequalities and stability results related to the exponential function. J. Inequal. Appl. 1998, 2: 373–380.
Jung S:Hyers-Ulam stability of differential equation . J. Inequal. Appl. 2010., 2010: Article ID 793197
Li Y, Shen Y: Hyers-Ulam stability of linear differential equations of second order. Appl. Math. Lett. 2010, 23: 306–309. 10.1016/j.aml.2009.09.020
Eshaghi Gordji M, Cho Y, Ghaemi MB, Alizadeh B: Stability of the exact second order partial differential equations. J. Inequal. Appl. 2011., 2011: Article ID 306275
Najati, A, Abdollahpour, MR, Cho, Y: Superstability of linear differential equations of second order. Preprint
Abdollahpour MR, Najati A: Stability of linear differential equations of third order. Appl. Math. Lett. 2011, 24: 1827–1830. 10.1016/j.aml.2011.04.043
Ghaemi MB, Eshaghi Gordji M, Alizadeh B, Park C: Hyers-Ulam stability of exact second order linear differential equations. Adv. Differ. Equ. 2012., 2012: Article ID 36
Gavruta P, Jung S, Li Y: Hyers-Ulam stability for second-order linear differential equations with boundary conditions. Electron. J. Differ. Equ. 2011, 2011(80):1–5.
Jung S: Hyers-Ulam stability of linear differential equations of first order. Appl. Math. Lett. 2004, 17: 1135–1140. 10.1016/j.aml.2003.11.004
Jung S: Hyers-Ulam stability of linear differential equations of first order, III. J. Math. Anal. Appl. 2005, 311: 139–146. 10.1016/j.jmaa.2005.02.025
Jung S: Hyers-Ulam stability of linear differential equations of first order, II. Appl. Math. Lett. 2006, 19: 854–858. 10.1016/j.aml.2005.11.004
Miura T: On the Hyers-Ulam stability of a differentiable map. Sci. Math. Jpn. 2002, 55: 17–24.
Miura T, Miyajima S, Takahasi SE: Hyers-Ulam stability of linear differential operators with constant coefficients. Math. Nachr. 2003, 258: 90–96. 10.1002/mana.200310088
Miura T, Jung S, Takahasi SE:Hyers-Ulam-Rassias stability of the Banach space valued differential equations . J. Korean Math. Soc. 2004, 41: 995–1005. 10.4134/JKMS.2004.41.6.995
Popa D, Rasa I: On the Hyers-Ulam stability of the differential equation. J. Math. Anal. Appl. 2011, 381: 530–537. 10.1016/j.jmaa.2011.02.051
Popa D, Rasa I: On the Hyers-Ulam stability of the differential operator with nonconstant coefficients. Appl. Math. Comput. 2012, 219: 1562–1568. 10.1016/j.amc.2012.07.056
Acknowledgements
CP was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299) and DYS was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0021792).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Abdollahpour, M.R., Najati, A., Park, C. et al. Approximate perfect differential equations of second order. Adv Differ Equ 2012, 225 (2012). https://doi.org/10.1186/1687-1847-2012-225
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1847-2012-225