Open Access

Hyers-ulam stability of exact second-order linear differential equations

  • Mohammad Bagher Ghaemi1,
  • Madjid Eshaghi Gordji2,
  • Badrkhan Alizadeh3 and
  • Choonkil Park4Email author
Advances in Difference Equations20122012:36

https://doi.org/10.1186/1687-1847-2012-36

Received: 1 December 2011

Accepted: 23 March 2012

Published: 23 March 2012

Abstract

In this article, we prove the Hyers-Ulam stability of exact second-order linear differential equations. As a consequence, we show the Hyers-Ulam stability of the following equations: second-order linear differential equation with constant coefficients, Euler differential equation, Hermite's differential equation, Cheybyshev's differential equation, and Legendre's differential equation. The result generalizes the main results of Jung and Min, and Li and Shen.

Mathematics Subject Classification (2010): 26D10; 34K20; 39B52; 39B82; 46B99.

Keywords

Hyers-Ulam stability exact second-order linear differential equation

1. Introduction

The stability of functional equations was first introduced by Ulam [1]. Hyers [2] gave a partial solution of Ulam's problem for the case of approximate additive mappings in the context of Banach spaces. Rassias [3] generalized the theorem of Hyers by considering the stability problem with unbounded Cauchy differences ║f(x + y) - f(x) - f(y)║ ≤ ε(║x p +║y p ) (ε > 0, p [0, 1)).

Let X be a normed space over a scalar field and let I be an open interval. Assume that for any function f : IX (y = f(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 ( t ) y ( t ) + h ( t ) ε
for all t I and some ε ≥ 0, there exists a function f0 : I → X (y = f0(x)) satisfying
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 .

and ║f(t) - f0(t)║ ≤ K(ε) for all t I. Here limε 0K(ε) = 0. Then we say that the above differential equation has the Hyers-Ulam stability.

If the above statement is also true when we replace ε and K(ε) by φ(t) and ϕ(t), where φ, ϕ : I → [0, ∞) are functions not depending on f and f0 explicitly, then we say that the corresponding differential equation has the Hyers-Ulam stability.

The Hyers-Ulam stability of the differential equation y' = y was first investigated by Alsina and Ger [4]. This result has been generalized by Takahasi et al. [5] for the Banach space-valued differential equation y' = λy. Jung [6] proved the Hyers-Ulam stability of a linear differential equation of first-order.

Theorem 1.1. ([6]) Let y : I be a continuously differentiable function satisfying the differential inequality
y ( t ) + g ( t ) y ( t ) + h ( t ) φ ( t )

for all t I, where g , h : I are continuous functions and φ : I → [0, ∞) is a function. Assume that

(a) g(t) and exp a t g ( u ) d u h ( t ) are integrable on (a, c) for each c I;

(b) φ ( t ) exp a t g ( u ) d u is integrable on I.

Then there exists a unique real number x such that
y ( t ) - exp - a t g ( u ) d u x - a t exp a υ g ( u ) d u h ( υ ) d υ exp - a t g ( u ) d u t b φ ( υ ) exp a υ g ( u ) d u d υ

for all t I.

In this article, we prove the Hyers-Ulam stability of exact second-order linear differential equations (see [7]). A general second-order differential equation is of the form
p 0 ( x ) y + p 1 ( x ) y + p 2 ( x ) y + f ( x ) = 0
(1)
and it is exact if
p 0 ' ' ( x ) p 1 ' ( x ) + p 2 ( x ) = 0.
(2)

2. Main results

In this section, let I = (a, b) be an open interval with -∞ ≤ a < b ≤ ∞. In the following theorem, p ( x ) = ( p 0 ( x ) ) - 1 ( p 1 ( x ) - p 0 ( x ) ) and k = - [ p 0 ( a ) y ( a ) - p 0 ( a ) y ( a ) + p 1 ( a ) y ( a ) ] . Taking some idea from [6], we investigate the Hyers-Ulam stability of exact second-order linear differential equations. For the sake of convenience, we assume that all the integrals and derivations exist.

Theorem 2.1. Let p 0 , p 1 , p 2 , f : I be continuous functions with p 0 (x) 0 for all x I, and let φ : I → [0, ∞) be a function. Assume that y : I is a twice continuously differentiable function satisfying the differential inequality
p 0 ( x ) y + p 1 ( x ) y + p 2 ( x ) y + f ( x ) φ ( x )
(3)
for all x I and (2) is true. Then there exists a solution y 0 : I of (1) such that
y ( x ) - y 0 ( x ) e x p - a x p ( u ) d u x b p 0 ( υ ) - 1 a υ φ ( t ) d t e x p a υ p ( u ) d u d υ

for all x I.

Proof. It follows from (2) and (3) that
p 0 ( x ) y + p 1 ( x ) y + p 2 ( x ) y + f ( x ) = ( p 0 ( x ) y - p 0 ( x ) y ) + ( p 1 ( x ) y ) + [ p 0 ( x ) - p 1 ( x ) + p 2 ( x ) ] y + f ( x ) = ( p 0 ( x ) y - p 0 ( x ) y ) + ( p 1 ( x ) y ) + f ( x ) φ ( x )
So we have
- φ ( x ) ( p 0 ( x ) y - p 0 ( x ) y ) + ( p 1 ( x ) y ) + f ( x ) φ ( x ) .
(4)
Integrating (4) from a to x for each x I, we get
p 0 ( x ) y - p 0 ( x ) y + p 1 ( x ) y + k + a x f ( t ) d t = p 0 ( x ) y + ( p 0 ( x ) ) - 1 ( p 1 ( x ) - p o ( x ) ) y + ( p 0 ( x ) ) - 1 k + a x f ( t ) d t a x φ ( t ) d t .
(5)
Dividing both sides of the inequality (5) by │p0(x)│, we obtain
y + ( p 0 ( x ) ) - 1 ( p 1 ( x ) - p o ( x ) ) y + ( p 0 ( x ) ) - 1 k + a x f ( t ) d t p 0 ( x ) - 1 a x φ ( t ) d t .
(6)
If we set
p ( x ) = ( p 0 ( x ) ) - 1 ( p 1 ( x ) - p 0 ( x ) ) , h ( x ) = ( p 0 ( x ) ) - 1 k + a x f ( t ) d t
and φ 1 ( x ) = p 0 ( x ) - 1 a x φ ( t ) d t in (6), then we have
y + p ( x ) y + h ( x ) φ 1 ( x ) .
Now we are in the situation of Theorem 1.1, that is, there exists a unique z such that
y ( x ) - exp - a x p ( u ) d u z - a x exp a υ p ( u ) d u h ( υ ) d υ exp - a x p ( u ) d u x b φ 1 ( υ ) exp a υ p ( u ) d u d υ = exp - a x p ( u ) d u x b p 0 ( υ ) - 1 a υ φ ( t ) d t exp a υ p ( u ) d u d υ

for all x I.

It is easy to show that
y 0 ( x ) = exp - a x p ( u ) d u z - a x exp a υ p ( u ) d u h ( υ ) d υ
(11a)

is a solution of (1) with the condition (2). □

If (1) is multiplied by a function μ(x) such that the resulting equation is exact, that is,
( μ ( x ) ) [ p 0 ( x ) y + p 1 ( x ) y + p 2 ( x ) y + f ( x ) ] = 0
(7)
and
( μ ( x ) p 0 ( x ) ) - ( p 1 ( x ) μ ( x ) ) + p 2 ( x ) μ ( x ) = 0 ,
(8)

then we say that μ(x) is an integrating factor of the Equation (1) (see [7]).

Corollary 2.2. Let p 0 , p 1 , p 2 , μ : I be continuous functions with p0(x) 0 and μ(x) 0 for all x I, and let φ : I → [0, ∞) be a function. Assume that y : I is a twice continuously differentiable function satisfying the differential inequality
( μ ( x ) ) [ p 0 ( x ) y + p 1 ( x ) y + p 2 ( x ) y + f ( x ) ] φ ( x )
(9)
for all x I and (8) is true. Then there exists a solution y 0 : I of (7) such that
y ( x ) - y 0 ( x ) exp - a x p ( u ) d u x b μ ( v ) p 0 ( v ) - 1 a v φ ( t ) d t exp a v p ( u ) d u d υ

for all x I, where p(x) = (μ(x)p0(x))-1 [μ(x)p1(x) - (μ(x)p0(x))' ].

Proof. It follows from Theorem 2.1 that there exists a unique z such that
y 0 ( x ) = exp - a x p ( u ) d u z - a x exp a υ p ( u ) d u h ( v ) d v
is a solution of (7) with the condition (8), where
p ( x ) = ( μ ( x ) p 0 ( x ) ) - 1 [ μ ( x ) p 1 ( x ) - ( μ ( x ) p 0 ( x ) ) ]
and
h ( x ) = ( μ ( x ) p 0 ( x ) ) - 1 k + a x μ ( t ) f ( t ) d t
with
k = - [ μ ( a ) p 0 ( a ) y ( a ) - ( μ ( a ) p 0 ( a ) ) y ( a ) + μ ( a ) p 1 ( a ) y ( a ) ] ,
as desired. □
  1. 1.
    Li and Shen [8] proved the Hyers-Ulam stability of second-order linear differential equations with constant coefficients
    y + c y + b y + f ( x ) = 0 ,
    (10)
     

where the characteristic equation λ2 + + b = 0 has two positive roots.

Now, it follows from (7) and (8) that μ(x) is an integrating factor for (10) if it satisfies
μ ( x ) - c μ ( x ) + b μ ( x ) = 0 .
(11)

It is well-known that μ(x) = exp(mx), where m = c ± c 2 - 4 b 2 , is a solution of (11) and consequently, it is an integrating factor of (10). Now the following corollaries are the generalization of [[8], Theorems 2.1 and 2.2].

Corollary 2.3. Consider the Equation (10). Let c2 - 4b ≥ 0, m = c ± c 2 - 4 b 2 , f : I be a continuous function and let φ : I → [0, ∞) be a function. Assume that y : I is a twice continuously differentiable function satisfying the differential inequality
y + c y + b y + f ( x ) φ ( x )
(12)
for all x I. Then there exists a solution y 0 : I of (10) such that
y ( x ) - y 0 ( x ) exp { ( m - c ) ( x - a ) } x b exp ( - m v ) a v exp ( m t ) φ ( t ) exp { ( c - m ) ( x - a ) } d v

for all x I.

Proof. μ(x) = exp(mx) is an integrating factor of (10) when c2 - 4b ≥ 0 and m = c ± c 2 - 4 b 2 (the paragraph preceding of this corollary). By (12), we obtain
exp ( m x ) y + c y + b y + f ( x ) exp ( m x ) φ ( x )
(13)
for all x I. Using Corollary 2.2 with φ1(x) = exp(mx)φ(x) instead of φ(x) and with (13) instead of (9), we conclude that there exists a unique z such that
y 0 ( x ) = exp { ( m - c ) ( x - a ) } z - a x exp { ( c - m ) ( v - a ) } k + a v exp ( m x ) f ( x ) d x d v ,
where k = -[exp(ma)y'(a) -m exp(ma)y(a) + c exp(ma)y(a)], for all x I, is a solution of (10) and
y ( x ) - y 0 ( x ) exp { ( m - c ) ( x - a ) } x b exp ( m v ) a v exp ( m t ) φ ( t ) exp { ( c - m ) ( v - a ) } d v ,

as desired. □

Corollary 2.4. Consider the Equation (10). Let c2 - 4b < 0, m = c ± c 2 - 4 b 2 = α ± i β , f : I be a continuous function and let φ : I → [0, ) be a function. Assume that y : I is a twice continuously differentiable function satisfying the differential inequality
y + c y + b y + f ( x ) φ ( x )
for all x I. Then there exists a solution y 0 : I of (10) such that
| y ( x ) y 0 ( x ) | exp { ( a x p ( u ) d u ) } x b ( | μ ( v ) | 1 a v exp ( α t ) φ ( t ) d t ) exp { ( a v p ( u ) d u ) } d v

for all x I, where μ(x) = exp(αx) cos βx and p(u) = [c - α + β tan βx].

Proof. It is easy to show that
exp ( α x ) ( cos β x ) ( y + c y + b y + f ( x ) ) exp ( α x ) y + a y + b y + f ( x ) exp ( α x ) φ ( x )
for all x I. Now, similar to Corollary 2.3, there exists a unique z such that
y 0 = exp z - a x p ( u ) d u z - a x exp a v p ( u ) d u k + a v exp ( α x ) cos β x f ( x ) d x
has the required properties, where k = [exp(αa) cos βay'(a) - (exp(αa) cos βa)'y(a) + c exp(αa) cos βay(a)]. □
  1. 2.
    Let α and β be real constants. The following differential equation
    x 2 y + α x y + β y + f ( x ) = 0
     

is called the Euler differential equation. It is exact when α - β = 2. By Theorem 2.1, it has the Hyers-Ulam stability.

In general, μ(x) is an integrating factor of Euler differential equation if it satisfies
x 2 μ ( x ) ) " ( α x μ ( x ) ' + β μ ( x ) = 0.
(14)
The Equation (14) can be written as
x 2 μ ( x ) + ( 4 - α ) x μ ( x ) + ( 2 - α + β ) μ ( x ) = 0 .
By the trial of μ(x) = x m , we show that
m 2 + ( 3 - α ) m + ( 2 - α + β ) = 0 .
(15)
From (15) we obtain
m = - ( 3 - α ) ± ( 1 - α ) 2 - 4 β 2 .
Now we can use the above corollaries for the Hyers-Ulam stability of Euler differential equation. This result is comparable with [[9], Theorem 2] and the main results of [10].
  1. 3.
    Hermite's differential equation
    y - 2 x y + 2 λ y + f ( x ) = 0 ( λ )
     
is exact when λ = -1 and it has the Hyers-Ulam stability.
  1. 4.
    Chebyshev's differential equation
    ( 1 - x 2 ) y - x y + n 2 y + f ( x ) = 0 ( n )
     
is exact when n = ±1. By Theorem 2.1, it has the Hyers-Ulam stability.
  1. 5.
    Legendre's differential equation
    ( 1 - x 2 ) y - 2 x y + n ( n + 1 ) y + f ( x ) = 0 ( n )
     

is exact when n(n + 1) = 0 and it has the Hyers-Ulam stability.

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Iran University of Science and Technology
(2)
Department of Mathematics, Semnan University
(3)
Technical and Vocational Faculty of Tabriz, Technical and Vocational University of Iran
(4)
Department of Mathematics, Research Institute for Natural Sciences, Hanyang University

References

  1. Ulam SM: Problems in Modern Mathematics. Volume VI. Science Ed., Wiley, New York; 1940.Google Scholar
  2. Hyers DH: On the stability of the linear functional equation. Proc Natl Acad Sci 1941, 27: 222–224.MathSciNetView ArticleGoogle Scholar
  3. Rassias ThM: On the stability of the linear mapping in Banach spaces. Proc Am Math Soc 1978, 72: 297–300.View ArticleGoogle Scholar
  4. Alsina C, Ger R: On some inequalities and stability results related to the exponential function. J Inequal Appl 1998, 2: 373–380.MathSciNetGoogle Scholar
  5. Takahasi S-E, Miura T, Miyajima S: On the Hyers-Ulam stability of Banach space-valued differential equation y' = λy . Bull Korean Math Soc 2002, 39: 309–315.MathSciNetView ArticleGoogle Scholar
  6. Jung S: Hyers-Ulam stability of linear differential equations of first-order II . Appl Math Lett 2006, 19: 854–858.MathSciNetView ArticleGoogle Scholar
  7. Ionascu EJ:Ordinary Differential Equations-Lecture Notes. 2006. [http://math.columbusstate.edu/ejionascu/papers/diffeqbook.pdf]Google Scholar
  8. Li Y, Shen Y: Hyers-Ulam stability of linear differential equations of second-order. Appl Math Lett 2010, 23: 306–309.MathSciNetView ArticleGoogle Scholar
  9. Jung S: Hyers-Ulam stability of linear differential equations of first-order III . J Math Anal Appl 2005, 311: 139–146.MathSciNetView ArticleGoogle Scholar
  10. Jung S, Min S: On approximate Eular differential equations. Abstr Appl Anal 2009, 2009: 8. Art. ID 537963MathSciNetGoogle Scholar

Copyright

© Ghaemi 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.