Open Access

Approximate perfect differential equations of second order

  • Mohammad Reza Abdollahpour1,
  • Abbas Najati1,
  • Choonkil Park2,
  • Themistocles M Rassias3 and
  • Dong Yun Shin4Email author
Advances in Difference Equations20122012:225

https://doi.org/10.1186/1687-1847-2012-225

Received: 3 September 2012

Accepted: 21 November 2012

Published: 27 December 2012

Abstract

In this paper we prove the Hyers-Ulam stability of the perfect linear differential equation f ( t ) y ( t ) + f 1 ( t ) y ( t ) + f 2 ( t ) y ( t ) = Q ( t ) , where f , y C 2 [ a , b ] , Q C [ a , b ] , f 2 ( t ) = f 1 ( t ) f ( t ) and < a < b < + .

MSC:34K20, 26D10, 39B82, 34K06, 39B72.

Keywords

Hyers-Ulam stabilitydifferential equation

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 a 0 , a 1 , , a n , h : I K are continuous functions. We say that the differential equation
a n ( t ) y ( n ) ( t ) + a n 1 ( t ) y ( n 1 ) ( t ) + + a 1 ( t ) y ( t ) + a 0 ( t ) y ( t ) + h ( t ) = 0
(1.1)
has the Hyers-Ulam stability if, for any function f : I X satisfying the differential inequality
a n ( t ) y ( n ) ( t ) + a n 1 ( t ) y ( n 1 ) ( t ) + + a 1 ( t ) y ( t ) + a 0 y ( t ) + h ( t ) ε

for all t I and some ε 0 , there exists a solution g : I X of (1.1) such that f ( t ) g ( t ) K ( ε ) for all t I , where K ( ε ) 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 y + 2 x y 2 n y = m = 0 a m x m , where n is a positive integer, and he used this result to prove the Hyers-Ulam stability of the differential equation y + 2 x y 2 n y = 0 in a special class of analytic functions.

Li and Shen [8] proved that if the characteristic equation λ 2 + α λ + β = 0 has two different positive roots, then the linear differential equation of second order with constant coefficients y ( x ) + α y ( x ) + β y ( x ) = f ( x ) has the Hyers-Ulam stability where y C 2 [ a , b ] , f C [ a , b ] and < a < b < + (see also [9, 10]). Abdollahpour and Najati [11] proved that the third-order differential equation y ( 3 ) ( t ) + α y ( t ) + β y ( t ) + γ y ( t ) = f ( t ) has the Hyers-Ulam stability. Ghaemi et al. [12] proved the Hyers-Ulam stability of the exact second-order linear differential equation
p 0 ( x ) γ + p 1 ( x ) γ + p 2 ( x ) γ + f ( x ) = 0

with p 0 ( x ) p 1 ( x ) + p 2 ( x ) = 0 . Here p 0 , p 1 , p 2 , f : ( a , b ) R are continuous functions. For more results about the Hyers-Ulam stability of differential equations, we can refer to [1321].

Definition 1.2 We say that the differential equation
f ( t ) y ( t ) + f 1 ( t ) y ( t ) + f 2 ( t ) y ( t ) = Q ( t ) ,
(1.2)

is perfect if it can be written as d d t [ f ( t ) y ( t ) + ( f 1 ( t ) f ( t ) ) y ( t ) ] = Q ( t ) .

It is clear that the differential equation (1.2) is perfect if and only if f 2 ( t ) = f 1 ( t ) f ( t ) . The aim of this paper is to investigate the Hyers-Ulam stability of the perfect differential equation (1.2), where f , y C 2 [ a , b ] , Q C [ a , b ] , f 1 C 1 [ a , b ] , f 2 ( t ) = f 1 ( t ) f ( t ) and < a < b < + . More precisely, we prove that the equation (1.2) has the Hyers-Ulam stability.

2 Hyers-Ulam stability of the perfect differential equation f ( t ) y ( t ) + f 1 ( t ) y ( t ) + f 2 ( t ) y ( t ) = Q ( t )

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 < a < b < + .

Theorem 2.1 The perfect differential equation
f ( t ) y ( t ) + f 1 ( t ) y ( t ) + f 2 ( t ) y ( t ) = Q ( t )

has the Hyers-Ulam stability, where f , y C 2 [ a , b ] , f 1 C 1 [ a , b ] , Q C [ a , b ] and f ( t ) 0 for all t [ a , b ] .

Proof Let ε > 0 and y C 2 [ a , b ] with
| f ( t ) y ( t ) + f 1 ( t ) y ( t ) + f 2 ( t ) y ( t ) Q ( t ) | ε .
Let g ( t ) = f ( t ) y + ( f 1 ( t ) f ( t ) ) y for all t [ a , b ] . It is clear that
| g ( t ) Q ( t ) | = | f ( t ) y ( t ) + f 1 ( t ) y ( t ) + f 2 ( t ) y ( t ) Q ( t ) | ε .
We define
z ( x ) = g ( b ) x b Q ( t ) d t , x [ a , b ] .
Then
z ( x ) = Q ( x ) , x [ a , b ] .
(2.1)
Also, we have
| z ( x ) g ( x ) | = | g ( b ) g ( x ) x b Q ( t ) d t | = | x b g ( t ) d t x b Q ( t ) d t | x b | g ( t ) Q ( t ) | d t ε ( b a )
for all x [ a , b ] . Now we define
F ( x ) = 1 f ( x ) exp { a x f 1 ( t ) f ( t ) d t } , u ( x ) = y ( b ) F ( b ) F ( x ) 1 F ( x ) x b z ( t ) F ( t ) f ( t ) d t
for all x [ a , b ] . It is clear that u C 2 [ a , b ] and
u ( x ) F ( x ) + u ( x ) F ( x ) = z ( x ) F ( x ) f ( x ) , F ( x ) = f 1 ( x ) f ( x ) f ( x ) F ( x ) .
Therefore,
f ( x ) u ( x ) + [ f 1 ( x ) f ( x ) ] u ( x ) = z ( x ) , x [ a , b ] .
(2.2)
Hence, (2.1) implies that
f ( x ) u ( x ) + f 1 ( x ) u ( x ) + f 2 ( x ) u ( x ) = Q ( x ) , x [ a , b ] .
Also, we have
| y ( x ) u ( x ) | = | y ( x ) y ( b ) F ( b ) F ( x ) + 1 F ( x ) x b z ( t ) F ( t ) f ( t ) d t | = 1 | F ( x ) | | y ( x ) F ( x ) y ( b ) F ( b ) + x b z ( t ) F ( t ) f ( t ) d t | = 1 | F ( x ) | | x b z ( t ) F ( t ) f ( t ) d t x b [ y ( t ) F ( t ) ] d t | = 1 | F ( x ) | | x b ( z ( t ) F ( t ) f ( t ) y ( t ) F ( t ) y ( t ) F ( t ) ) d t | = 1 | F ( x ) | | x b F ( t ) ( z ( t ) f ( t ) y ( t ) f 1 ( t ) f ( t ) f ( t ) y ( t ) ) d t | 1 | F ( x ) | x b | F ( t ) f ( t ) | | z ( t ) y ( t ) f ( t ) [ f 1 ( t ) f ( t ) ] y ( t ) | d t = 1 | F ( x ) | x b | F ( t ) f ( t ) | | z ( t ) g ( t ) | d t ε ( b a ) 1 | F ( x ) | x b | F ( t ) f ( t ) | d t
(2.3)
for all x [ a , b ] . Since f 1 f C [ a , b ] , there exist constants m and M such that m f 1 ( x ) f ( x ) M . Thus
{ 1 exp { a x f 1 ( t ) f ( t ) d t } e M ( b a ) if  m 0 ; e m ( b a ) exp { a x f 1 ( t ) f ( t ) d t } e M ( b a ) if  m < 0 M ; e m ( b a ) exp { a x f 1 ( t ) f ( t ) d t } 1 if  M < 0
(2.4)
for all x [ a , b ] . Since f C [ a , b ] and | f | > 0 , there exist constants 0 < m M such that m | f ( x ) | M for all x [ a , b ] . Hence, (2.4) implies that
1 M e | m | ( a b ) | F ( x ) | 1 m e | M | ( b a )
for all x [ a , b ] . It follows from (2.3) that
| y ( x ) u ( x ) | ε ( b a ) 1 | F ( x ) | x b | F ( t ) f ( t ) | d t ε ( b a ) 2 M m 2 e ( | m | + | M | ) ( b a )

for all x [ a , b ] . □

Declarations

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).

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili
(2)
Department of Mathematics, Research Institute for Natural Sciences, Hanyang University
(3)
Department of Mathematics, National Technical University of Athens, Zografou Campus
(4)
Department of Mathematics, University of Seoul

References

  1. Ulam SM: Problems in Modern Mathematics. Wiley, New York; 1940.Google Scholar
  2. Hyers DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27: 222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleGoogle Scholar
  3. Rassias TM: On the stability of linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1View ArticleGoogle Scholar
  4. Obłoza M: Hyers stability of the linear differential equation. Rocznik Nauk.-Dydakt. Prace Mat. 1993, 13: 259–270.Google Scholar
  5. Obłoza M: Connections between Hyers and Lyapunov stability of the ordinary differential equations. Rocznik Nauk.-Dydakt. Prace Mat. 1997, 14: 141–146.Google Scholar
  6. Alsina C, Ger R: On some inequalities and stability results related to the exponential function. J. Inequal. Appl. 1998, 2: 373–380.MathSciNetGoogle Scholar
  7. Jung S:Hyers-Ulam stability of differential equation y + 2 x y 2 n y = 0 . J. Inequal. Appl. 2010., 2010: Article ID 793197Google Scholar
  8. 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.2009.09.020MathSciNetView ArticleGoogle Scholar
  9. 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 306275Google Scholar
  10. Najati, A, Abdollahpour, MR, Cho, Y: Superstability of linear differential equations of second order. PreprintGoogle Scholar
  11. Abdollahpour MR, Najati A: Stability of linear differential equations of third order. Appl. Math. Lett. 2011, 24: 1827–1830. 10.1016/j.aml.2011.04.043MathSciNetView ArticleGoogle Scholar
  12. 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 36Google Scholar
  13. 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.MathSciNetGoogle Scholar
  14. Jung S: Hyers-Ulam stability of linear differential equations of first order. Appl. Math. Lett. 2004, 17: 1135–1140. 10.1016/j.aml.2003.11.004MathSciNetView ArticleGoogle Scholar
  15. 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.2005.02.025MathSciNetView ArticleGoogle Scholar
  16. Jung S: Hyers-Ulam stability of linear differential equations of first order, II. Appl. Math. Lett. 2006, 19: 854–858. 10.1016/j.aml.2005.11.004MathSciNetView ArticleGoogle Scholar
  17. Miura T: On the Hyers-Ulam stability of a differentiable map. Sci. Math. Jpn. 2002, 55: 17–24.MathSciNetGoogle Scholar
  18. 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.200310088MathSciNetView ArticleGoogle Scholar
  19. Miura T, Jung S, Takahasi SE:Hyers-Ulam-Rassias stability of the Banach space valued differential equations y = λ y . J. Korean Math. Soc. 2004, 41: 995–1005. 10.4134/JKMS.2004.41.6.995MathSciNetView ArticleGoogle Scholar
  20. 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.2011.02.051MathSciNetView ArticleGoogle Scholar
  21. 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.2012.07.056MathSciNetView ArticleGoogle Scholar

Copyright

© Abdollahpour et al.; licensee Springer 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.