R E S E A R C H Open Access
Hyers-ulam stability of exact second-order linear differential equations
Mohammad Bagher Ghaemi1, Madjid Eshaghi Gordji2, Badrkhan Alizadeh3and Choonkil Park4*
* Correspondence: baak@hanyang.
ac.kr
4Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, South Korea
Full list of author information is available at the end of the article
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 consid- ering the stability problem with unbounded Cauchy differences║f(x + y) -f(x) - f(y)║
≤ε(║x║p+║y║p) (ε> 0,pÎ[0, 1)).
Let Xbe a normed space over a scalar fieldKand letIbe an open interval. Assume that for any functionf:I® X(y=f(x)) satisfying the differential inequality
an(t)y(n)(t) +an−1(t)y(n−1)(t) +· · ·+a1(t)y(t) +a0(t)y(t) +h(t)≤ε
for alltÎIand someε≥0, there exists a function f0:I®X(y=f0(x)) satisfying an(t)y(n)(t) +an−1(t)y(n−1)(t) +· · ·+a1(t)y(t) +a0(t)y(t) +h(t) = 0.
and ║f(t) -f0(t)║≤ K(ε) for all tÎ I. Here limε®0 K(ε) = 0. Then we say that the above differential equation has theHyers-Ulam stability.
If the above statement is also true when we replace ε and K(ε) by (t) and j(t), where,j:I®[0,∞) are functions not depending onfand f0explicitly, then we say that the corresponding differential equation has theHyers-Ulam stability.
The Hyers-Ulam stability of the differential equationy’=ywas first investigated by Alsina and Ger [4]. This result has been generalized by Takahasi et al. [5] for the Banach space-valued differential equationy’=ly. Jung [6] proved the Hyers-Ulam sta- bility of a linear differential equation of first-order.
Theorem 1.1. ([6])Let y:I→Rbe a continuously differentiable function satisfying the differential inequality
© 2012 Ghaemi et al; licensee Springer. 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.
y(t) +g(t)y(t) +h(t)≤ϕ(t)
for all tÎI, whereg,h:I→Rare continuous functions and:I®[0,∞)is a function.
Assume that
(a)g(t)andexp t
a
g(u)du
h(t)are integrable on(a, c)for each cÎI;
(b)ϕ(t) exp t
a
g(u)du
is integrable on I.
Then there exists a unique real number x such that
y(t)−exp
⎧⎨
⎩− t a
g(u)du
⎫⎬
⎭
⎡
⎣x− t
a
exp
⎧⎨
⎩ υ a
g(u)du
⎫⎬
⎭h(υ)dυ
⎤
⎦
≤ exp
⎧⎨
⎩−
⎛
⎝ t
a
g(u)du
⎞
⎠
⎫⎬
⎭ b
t
ϕ(υ) exp
⎧⎨
⎩
⎛
⎝ υ
a
g(u)du
⎞
⎠
⎫⎬
⎭dυ for all tÎI.
In this article, we prove the Hyers-Ulam stability of exact second-order linear differ- ential equations (see [7]). A general second-order differential equation is of the form
p0(x)y+p1(x)y+p2(x)y+f(x) = 0 (1)
and it isexactif
p0(x)−p1(x) +p2(x) = 0. (2)
2. Main results
In this section, letI= (a, b) be an open interval with -∞ ≤a < b≤ ∞. In the following theorem, p(x) = (p0(x))−1(p1(x)−p0(x))andk=−[p0(a)y(a)−p0(a)y(a) +p1(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 p0,p1,p2,f :I→Rbe continuous functions with p0(x)≠0 for all x Î I, and let:I®[0, ∞)be a function. Assume thaty:I→Ris a twice continuously differentiable function satisfying the differential inequality
p0(x)y+p1(x)y+p2(x)y+f(x)≤ϕ(x) (3) for all xÎI and (2)is true. Then there exists a solution y0:I→Rof(1)such that
y(x)−y0(x)
≤exp
⎧⎨
⎩−
⎛
⎝ x a
p(u)du
⎞
⎠
⎫⎬
⎭ b x
⎛⎝p0(υ)−1υ
a
ϕ(t)dt
⎞
⎠exp
⎧⎨
⎩
⎛
⎝ υ a
p(u)du
⎞
⎠
⎫⎬
⎭dυ for all xÎI.
Proof. It follows from (2) and (3) that p0(x)y+p1(x)y+p2(x)y+f(x)
=(p0(x)y−p0(x)y)+ (p1(x)y)+ [p0(x)−p1(x) +p2(x)]y+f(x)
=(p0(x)y−p0(x)y)+ (p1(x)y)+f(x)≤ϕ(x) So we have
−ϕ(x)≤(p0(x)y−p0(x)y)+ (p1(x)y)+f(x)≤ϕ(x). (4) Integrating (4) fromatoxfor eachxÎI, we get
p0(x)y−p0(x)y+p1(x)y+k+ x a
f(t)dt
=p0(x)
y+ (p0(x))−1(p1(x)−po(x))y+ (p0(x))−1
⎛
⎝k+ x a
f(t)dt
⎞
⎠
≤ x a
ϕ(t)dt.
(5)
Dividing both sides of the inequality (5) by│p0(x)│, we obtain
y+ (p0(x))−1(p1(x)−po(x))y+ (p0(x))−1
⎛
⎝k+ x a
f(t)dt
⎞
⎠
≤p0(x)−1x
a
ϕ(t)dt.
(6)
If we set
p(x) = (p0(x))−1(p1(x)−p0(x)),h(x) = (p0(x))−1
⎛
⎝k+ x
a
f(t)dt
⎞
⎠
andϕ1(x) =p0(x)−1 x a
ϕ(t)dtin (6), then we have y+p(x)y+h(x)≤ϕ1(x).
Now we are in the situation of Theorem 1.1, that is, there exists a uniquez∈Rsuch that
y(x)−exp
⎧⎨
⎩− x a
p(u)du
⎫⎬
⎭
⎡
⎣z− x a
exp
⎧⎨
⎩ υ
a
p(u)du
⎫⎬
⎭h(υ)dυ
⎤
⎦
≤exp
⎧⎨
⎩−
⎛
⎝ x
a
p(u)du
⎞
⎠
⎫⎬
⎭ b x
ϕ1(υ) exp
⎧⎨
⎩
⎛
⎝ υ a
p(u)du
⎞
⎠
⎫⎬
⎭dυ
= exp
⎧⎨
⎩−
⎛
⎝ x a
p(u)du
⎞
⎠
⎫⎬
⎭ b x
⎛
⎝p0(υ)−1υ
a
ϕ(t)dt
⎞
⎠exp
⎧⎨
⎩
⎛
⎝ υ a
p(u)du
⎞
⎠
⎫⎬
⎭dυ
for all xÎI.
It is easy to show that
y0(x) = exp
⎧⎨
⎩− x a
p(u)du
⎫⎬
⎭
⎡
⎣z− x a
exp
⎧⎨
⎩ υ
a
p(u)du
⎫⎬
⎭h(υ)dυ
⎤
⎦ (11a)
is a solution of (1) with the condition (2).□
If (1) is multiplied by a functionμ(x) such that the resulting equation is exact, that is, (μ(x))[p0(x)y+p1(x)y+p2(x)y+f(x)] = 0 (7) and
(μ(x)p0(x))−(p1(x)μ(x))+p2(x)μ(x) = 0, (8) then we say that μ(x) is anintegrating factorof the Equation (1) (see [7]).
Corollary 2.2. Letp0,p1,p2,μ:I→Rbe continuous functions with p0(x)≠ 0 and μ(x)≠0for all xÎ I,and let:I® [0,∞)be a function. Assume that y:I→Ris a twice continuously differentiable function satisfying the differential inequality
(μ(x))[p0(x)y+p1(x)y+p2(x)y+f(x)]≤ϕ(x) (9) for all xÎI and (8)is true. Then there exists a solution y0:I→Rof(7)such that
y(x)−y0(x)
≤exp
⎧⎨
⎩−
⎛
⎝ x
a
p(u)du
⎞
⎠
⎫⎬
⎭ b
x
⎛⎝μ(v)p0(v)−1v
a
ϕ(t)dt
⎞
⎠exp
⎧⎨
⎩
⎛
⎝ v
a
p(u)du
⎞
⎠
⎫⎬
⎭dυ for all xÎI, 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 uniquez∈Rsuch that
y0(x) = exp
⎧⎨
⎩− x a
p(u)du
⎫⎬
⎭
⎡
⎣z− x a
exp
⎧⎨
⎩ υ
a
p(u)du
⎫⎬
⎭h(v)dv
⎤
⎦
is a solution of (7) with the condition (8), where p(x) = (μ(x)p0(x))−1[μ(x)p1(x)−(μ(x)p0(x))] and
h(x) = (μ(x)p0(x))−1
⎛
⎝k+ x a
μ(t)f(t)dt
⎞
⎠
with
k=−[μ(a)p0(a)y(a)−(μ(a)p0(a))y(a) +μ(a)p1(a)y(a)], as desired.□
1. Li and Shen [8] proved the Hyers-Ulam stability of second-order linear differen- tial equations with constant coefficients
y+cy+by+f(x) = 0, (10) where the characteristic equationl2+cl+b= 0 has two positive roots.
Now, it follows from (7) and (8) that μ(x) is an integrating factor for (10) if it satis- fies
μ(x)−cμ(x) +bμ(x) = 0. (11)
It is well-known that μ(x) = exp(mx), wherem= c±√ c2−4b
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±√
c2−4b
2 ,f :I→Rbe a continuous function and let:I® [0,∞)be a func- tion. Assume thaty:I→Ris a twice continuously differentiable function satisfying the differential inequality
y+cy+by+f(x)≤ϕ(x) (12) for all xÎI. Then there exists a solutiony0:I→Rof(10)such that
y(x)−y0(x)≤exp{(m−c)(x−a)} b x
⎛
⎝exp(−mv) v
a
exp(mt)ϕ(t)
⎞
⎠exp{(c−m)(x−a)}dv
for all xÎI.
Proof. μ(x) = exp(mx) is an integrating factor of (10) when c2 - 4b ≥ 0 and m= c±√
c2−4b
2 (the paragraph preceding of this corollary). By (12), we obtain
exp(mx)y+cy+by+f(x)≤exp(mx)ϕ(x) (13)
for all xÎ I. Using Corollary 2.2 with 1(x) = exp(mx)(x) instead of (x) and with (13) instead of (9), we conclude that there exists a unique z∈Rsuch that
y0(x) = exp{(m−c)(x−a)}
⎡
⎣z− x a
exp{(c−m)(v−a)}
⎛
⎝k+ v a
exp(mx)f(x)dx
⎞
⎠dv
⎤
⎦,
wherek= -[exp(ma)y’(a) -mexp(ma)y(a) +cexp(ma)y(a)], for allxÎI, is a solution of (10) and
y(x)−y0(x)≤exp{(m−c)(x−a)} b x
⎛
⎝exp(mv) v a
exp(mt)ϕ(t)
⎞
⎠exp{(c−m)(v−a)}dv,
as desired.□
Corollary 2.4. Consider the Equation (10). Let c2 - 4b <0, m= c±√
c2−4b
2 =α±iβ,f :I→Rbe a continuous function and let: I ® [0,∞) be a function. Assume thaty:I→Ris a twice continuously differentiable function satis- fying the differential inequality
y+cy+by+f(x)≤ϕ(x)
for all xÎI. Then there exists a solutiony0:I→Rof(10)such that y(x)−y0(x)
≤exp
⎧⎨
⎩−
⎛
⎝ x a
p(u)du
⎞
⎠
⎫⎬
⎭ b
x
⎛
⎝μ(v)−1v
a
exp(αt)ϕ(t)dt
⎞
⎠exp
⎧⎨
⎩
⎛
⎝ v
a
p(u)du
⎞
⎠
⎫⎬
⎭dv for all xÎI, where μ(x) =exp(ax) cosbx and p(u) = [c -a+btanbx].
Proof. It is easy to show that
exp(αx)(cosβx)(y+cy+by+f(x))≤exp(αx)y+ay+by+f(x)
≤exp(αx)ϕ(x)
for all xÎI. Now, similar to Corollary 2.3, there exists a uniquez∈Rsuch that
y0= exp
⎧⎨
⎩z− x a
p(u)du
⎫⎬
⎭
⎡
⎣z− x a
exp
⎧⎨
⎩ v
a
p(u)du
⎫⎬
⎭
⎛
⎝k+ v a
exp(αx) cosβxf(x)dx
⎞
⎠
⎤
⎦
has the required properties, wherek= [exp(aa)cosbay’(a) - (exp(aa)cosba)’y(a) + cexp(aa) cos bay(a)].□
2. Letaandbbe real constants. The following differential equation x2y+αxy+βy+f(x) = 0
is called the Euler differential equation. It is exact whena -b= 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
x2μ(x))−(αxμ(x)+βμ(x) = 0. (14)
The Equation (14) can be written as
x2μ(x) + (4−α)xμ(x) + (2−α+β)μ(x) = 0.
By the trial of μ(x) =xm, we show that
m2+ (3−α)m+ (2−α+β) = 0. (15)
From (15) we obtain
m= −(3−α)±
(1−α)2−4β
2 .
Now we can use the above corollaries for the Hyers-Ulam stability of Euler differen- tial equation. This result is comparable with [[9], Theorem 2] and the main results of [10].
3. Hermite’s differential equation y−2xy+ 2λy+f(x) = 0 (λ∈R)
is exact whenl= -1 and it has the Hyers-Ulam stability.
4. Chebyshev’s differential equation
(1−x2)y−xy+n2y+f(x) = 0 (n∈Z)
is exact whenn= ±1. By Theorem 2.1, it has the Hyers-Ulam stability.
5. Legendre’s differential equation
(1−x2)y−2xy+n(n+ 1)y+f(x) = 0 (n∈Z)
is exact whenn(n+ 1) = 0 and it has the Hyers-Ulam stability.
Author details
1Department of Mathematics Iran University of Science and Technology, Narmak, Tehran, Iran2Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran3Technical and Vocational University of Iran, Technical and Vocational Faculty of Tabriz, P. O. Box 51745-135, Tabriz, Iran4Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, South Korea
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.
Competing interests
The authors declare that they have no competing interests.
Received: 1 December 2011 Accepted: 23 March 2012 Published: 23 March 2012
References
1. Ulam, SM: In Problems in Modern Mathematics, vol. VI,Science Ed., Wiley, New York (1940) 2. Hyers, DH: On the stability of the linear functional equation. Proc Natl Acad Sci.27, 222–224 (1941) 3. Rassias, ThM: On the stability of the linear mapping in Banach spaces. Proc Am Math Soc.72, 297–300 (1978) 4. Alsina, C, Ger, R: On some inequalities and stability results related to the exponential function. J Inequal Appl.2,
373–380 (1998)
5. Takahasi, S-E, Miura, T, Miyajima, S: On the Hyers-Ulam stability of Banach space-valued differential equationy’=λy. Bull Korean Math Soc.39, 309–315 (2002)
6. Jung, S: Hyers-Ulam stability of linear differential equations of first-orderII. Appl Math Lett.19, 854–858 (2006) 7. Ionascu, EJ: Ordinary Differential Equations-Lecture Notes. http://math.columbusstate.edu/ejionascu/papers/diffeqbook.
pdf (2006)
8. Li, Y, Shen, Y: Hyers-Ulam stability of linear differential equations of second-order. Appl Math Lett.23, 306–309 (2010) 9. Jung, S: Hyers-Ulam stability of linear differential equations of first-orderIII. J Math Anal Appl.311, 139–146 (2005) 10. Jung, S, Min, S: On approximate Eular differential equations. Abstr Appl Anal2009, 8 (2009). Art. ID 537963
doi:10.1186/1687-1847-2012-36
Cite this article as:Ghaemiet al.:Hyers-ulam stability of exact second-order linear differential equations.
Advances in Difference Equations20122012:36.
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