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Approximate perfect differential equations of second order

  • 1,
  • 1,
  • 2,
  • 3 and
  • 4Email author
Advances in Difference Equations20122012:225

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

  • Received: 3 September 2012
  • Accepted: 21 November 2012
  • Published:

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 stability
  • differential 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, Ardabil, 56199-11367, Iran
(2)
Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, 133-791, Korea
(3)
Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens, 15780, Greece
(4)
Department of Mathematics, University of Seoul, Seoul, 130-743, Korea

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© 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.

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