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Approximate perfect differential equations of second order

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]. □

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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.

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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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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

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Keywords

  • Hyers-Ulam stability
  • differential equation