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Advances in Difference Equations

, 2012:225 | Cite as

Approximate perfect differential equations of second order

  • Mohammad Reza Abdollahpour
  • Abbas Najati
  • Choonkil Park
  • Themistocles M Rassias
  • Dong Yun ShinEmail author
Open Access
Research
Part of the following topical collections:
  1. Progress in Functional Differential and Difference Equations

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 ) Open image in new window, where f , y C 2 [ a , b ] Open image in new window, Q C [ a , b ] Open image in new window, f 2 ( t ) = f 1 ( t ) f ( t ) Open image in new window and < a < b < + Open image in new window.

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 Open image in new window and let I be an open interval. Assume that a 0 , a 1 , , a n Open image in new window, h : I K Open image in new window 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 Open image in new window
(1.1)
has the Hyers-Ulam stability if, for any function f : I X Open image in new window 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 Open image in new window

for all t I Open image in new window and some ε 0 Open image in new window, there exists a solution g : I X Open image in new window of (1.1) such that f ( t ) g ( t ) K ( ε ) Open image in new window for all t I Open image in new window, where K ( ε ) Open image in new window 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 Open image in new window, 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 Open image in new window in a special class of analytic functions.

Li and Shen [8] proved that if the characteristic equation λ 2 + α λ + β = 0 Open image in new window 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 ) Open image in new window has the Hyers-Ulam stability where y C 2 [ a , b ] Open image in new window, f C [ a , b ] Open image in new window and < a < b < + Open image in new window (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 ) Open image in new window 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 Open image in new window

with p 0 ( x ) p 1 ( x ) + p 2 ( x ) = 0 Open image in new window. Here p 0 Open image in new window, p 1 Open image in new window, p 2 Open image in new window, f : ( a , b ) R Open image in new window are continuous functions. For more results about the Hyers-Ulam stability of differential equations, we can refer to [13, 14, 15, 16, 17, 18, 19, 20, 21].

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 ) , Open image in new window
(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 ) Open image in new window.

It is clear that the differential equation (1.2) is perfect if and only if f 2 ( t ) = f 1 ( t ) f ( t ) Open image in new window. 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 ] Open image in new window, Q C [ a , b ] Open image in new window, f 1 C 1 [ a , b ] Open image in new window, f 2 ( t ) = f 1 ( t ) f ( t ) Open image in new window and < a < b < + Open image in new window. 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 ) Open image in new window

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 < + Open image in new window.

Theorem 2.1 The perfect differential equation
f ( t ) y ( t ) + f 1 ( t ) y ( t ) + f 2 ( t ) y ( t ) = Q ( t ) Open image in new window

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

Proof Let ε > 0 Open image in new window and y C 2 [ a , b ] Open image in new window with
| f ( t ) y ( t ) + f 1 ( t ) y ( t ) + f 2 ( t ) y ( t ) Q ( t ) | ε . Open image in new window
Let g ( t ) = f ( t ) y + ( f 1 ( t ) f ( t ) ) y Open image in new window for all t [ a , b ] Open image in new window. 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 . Open image in new window
We define
z ( x ) = g ( b ) x b Q ( t ) d t , x [ a , b ] . Open image in new window
Then
z ( x ) = Q ( x ) , x [ a , b ] . Open image in new window
(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 ) Open image in new window
for all x [ a , b ] Open image in new window. 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 Open image in new window
for all x [ a , b ] Open image in new window. It is clear that u C 2 [ a , b ] Open image in new window 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 ) . Open image in new window
Therefore,
f ( x ) u ( x ) + [ f 1 ( x ) f ( x ) ] u ( x ) = z ( x ) , x [ a , b ] . Open image in new window
(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 ] . Open image in new window
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 Open image in new window
(2.3)
for all x [ a , b ] Open image in new window. Since f 1 f C [ a , b ] Open image in new window, there exist constants m Open image in new window and M Open image in new window such that m f 1 ( x ) f ( x ) M Open image in new window. 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 Open image in new window
(2.4)
for all x [ a , b ] Open image in new window. Since f C [ a , b ] Open image in new window and | f | > 0 Open image in new window, there exist constants 0 < m M Open image in new window such that m | f ( x ) | M Open image in new window for all x [ a , b ] Open image in new window. Hence, (2.4) implies that
1 M e | m | ( a b ) | F ( x ) | 1 m e | M | ( b a ) Open image in new window
for all x [ a , b ] Open image in new window. 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 ) Open image in new window

for all x [ a , b ] Open image in new window. □

Notes

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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Copyright information

© 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.

Authors and Affiliations

  • Mohammad Reza Abdollahpour
    • 1
  • Abbas Najati
    • 1
  • Choonkil Park
    • 2
  • Themistocles M Rassias
    • 3
  • Dong Yun Shin
    • 4
    Email author
  1. 1.Department of Mathematics, Faculty of SciencesUniversity of Mohaghegh ArdabiliArdabilIran
  2. 2.Department of Mathematics, Research Institute for Natural SciencesHanyang UniversitySeoulKorea
  3. 3.Department of Mathematics, National Technical University of AthensZografou CampusAthensGreece
  4. 4.Department of MathematicsUniversity of SeoulSeoulKorea

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