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Approximate perfect differential equations of second order

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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], QC[a,b], f 2 (t)= f 1 (t) f (t) and <a<b<+.

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

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:IK 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:IX 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 tI and some ε0, there exists a solution g:IX of (1.1) such that f(t)g(t)K(ε) for all tI, 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 +2x y 2ny= 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 +2x y 2ny=0 in a special class of analytic functions.

Li and Shen [8] proved that if the characteristic equation λ 2 +αλ+β=0 has two different positive roots, then the linear differential equation of second order with constant coefficients y (x)+α y (x)+βy(x)=f(x) has the Hyers-Ulam stability where y C 2 [a,b], fC[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)+γy(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], QC[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], QC[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)dt,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 ) dt

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 fC[a,b] and |f|>0, there exist constants 0<mM 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]. □

References

  1. 1.

    Ulam SM: Problems in Modern Mathematics. Wiley, New York; 1940.

  2. 2.

    Hyers DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27: 222–224. 10.1073/pnas.27.4.222

  3. 3.

    Rassias TM: On the stability of linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1

  4. 4.

    Obłoza M: Hyers stability of the linear differential equation. Rocznik Nauk.-Dydakt. Prace Mat. 1993, 13: 259–270.

  5. 5.

    Obłoza M: Connections between Hyers and Lyapunov stability of the ordinary differential equations. Rocznik Nauk.-Dydakt. Prace Mat. 1997, 14: 141–146.

  6. 6.

    Alsina C, Ger R: On some inequalities and stability results related to the exponential function. J. Inequal. Appl. 1998, 2: 373–380.

  7. 7.

    Jung S:Hyers-Ulam stability of differential equation y +2x y 2ny=0. J. Inequal. Appl. 2010., 2010: Article ID 793197

  8. 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.2009.09.020

  9. 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 306275

  10. 10.

    Najati, A, Abdollahpour, MR, Cho, Y: Superstability of linear differential equations of second order. Preprint

  11. 11.

    Abdollahpour MR, Najati A: Stability of linear differential equations of third order. Appl. Math. Lett. 2011, 24: 1827–1830. 10.1016/j.aml.2011.04.043

  12. 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 36

  13. 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.

  14. 14.

    Jung S: Hyers-Ulam stability of linear differential equations of first order. Appl. Math. Lett. 2004, 17: 1135–1140. 10.1016/j.aml.2003.11.004

  15. 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.2005.02.025

  16. 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.2005.11.004

  17. 17.

    Miura T: On the Hyers-Ulam stability of a differentiable map. Sci. Math. Jpn. 2002, 55: 17–24.

  18. 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.200310088

  19. 19.

    Miura T, Jung S, Takahasi SE:Hyers-Ulam-Rassias stability of the Banach space valued differential equations y =λy. J. Korean Math. Soc. 2004, 41: 995–1005. 10.4134/JKMS.2004.41.6.995

  20. 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.2011.02.051

  21. 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.2012.07.056

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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).

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Correspondence to Dong Yun Shin.

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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.

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Keywords

  • Hyers-Ulam stability
  • differential equation