Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2011, Article ID 813137, 10 pages
doi:10.1155/2011/813137
Research Article
Generalized Hyers-Ulam Stability of the
Second-Order Linear Differential Equations
A. Javadian,1 E. Sorouri,2 G. H. Kim,3 and M. Eshaghi Gordji2
1
Department of Physics, Semnan University, P. O. Box 35195-363, Semnan, Iran
Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran
3
Department of Mathematics, Kangnam University, Yongin, Gyeonggi 446-702, Republic of Korea
2
Correspondence should be addressed to G. H. Kim, ghkim@kangnam.ac.kr
Received 26 September 2011; Accepted 23 October 2011
Academic Editor: Kuppalapalle Vajravelu
Copyright q 2011 A. Javadian et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We prove the generalized Hyers-Ulam stability of the 2nd-order linear differential equation of
the form y pxy qxy fx, with condition that there exists a nonzero y1 : I → X
in C2 I such that y1 pxy1 qxy1 0 and I is an open interval. As a consequence of our
main theorem, we prove the generalized Hyers-Ulam stability of several important well-known
differential equations.
1. Introduction
The stability problem of functional equations started with the question concerning stability
of group homomorphisms proposed by Ulam 1 during a talk before a Mathematical
Colloquium at the University of Wisconsin, Madison. In 1941, Hyers 2 gave a partial
solution of Ulam’s problem for the case of approximate additive mappings in the context
of Banach spaces. In 1978, Rassias 3 generalized the theorem of Hyers by considering the
stability problem with unbounded Cauchy differences fx y − fx − fy ≤ xp yp , > 0, p ∈ 0, 1. This phenomenon of stability that was introduced by Rassias 3 is
called the Hyers-Ulam-Rassias stability or the generalized Hyers-Ulam stability.
Let X be a normed space over a scalar field K, and let I be an open interval. Assume
that for any function f : I −→ X satisfying the differential inequality
an tyn t an−1 tyn−1 t · · · a1 ty t a0 tyt ht ≤ 1.1
2
Journal of Applied Mathematics
for all t ∈ I and for some ≥ 0, there exists a function f0 : I −→ X satisfying
an tyn t an−1 tyn−1 t · · · a1 ty t a0 tyt ht 0,
ft − f0 t ≤ K
1.2
for all t ∈ I; here Kt is an expression for with lim−→0 K 0. Then, we say that the above
differential equation has the Hyers-Ulam stability.
If the above statement is also true when we replace and K by ϕt and φt, 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-Rassias stability or the generalized
Hyers-Ulam stability.
The Hyers-Ulam stability of 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. In 6, Miura et al. also proved the Hyers-UlamRassias stability of linear differential of first order, y gtyt 0, where gt is a continuous
function, while the author 7 proved the Hyers-Ulam-Rassias stability of linear differential
of the form cty t yt. Jung 8 proved the Hyers-Ulam-Rassias stability of linear
differential of first order of the form cty t gtyt ht 0.
In this paper, we investigate the generalized Hyers-Ulam stability of differential
equations of the form
y pxy qxy fx.
1.3
We assume that X is a complex Banach space, I a, b is an arbitrary interval, and y1 : I −→
X is a nonzero solution of corresponding homogeneous equation of 1.3, where
y1 pxy1 qxy1 0.
1.4
2. Main Results
Taking some idea from 8, we are going to investigate the stability of the 2nd-order linear
differential equations. For the sake of convenience, all the integrals and derivations will be
viewed as existing and Rω denotes the real part of complex number ω. Moreover, let I a, b be an open interval, where a, b ∈ R {±∞} are arbitrarily given with a < b.
Theorem 2.1. Let X be a complex Banach space. Assume that p, q : I −→ C and f : I −→ X are
continuous functions and y1 : I −→ X is a nonzero twice continuously differentiable function which
satisfies the differential equation 1.4. If a twice continuously differentiable function y : I −→ X
satisfies
y pxy qxy − fx ≤ ϕx
2.1
Journal of Applied Mathematics
3
for all x ∈ I, where k ya/y1 a ∈ X and ϕ : I −→ 0, ∞ is a continuous function, then there
exists a unique x0 ∈ X such that
s
x 2y1 u
exp −
pu du
yx − y1 x ·
y1 u
a
a
v 2y1 u
fv
ds k · x0 exp
pu du dv
y1 u
a y1 v
a
s
x
2y
u
1
≤ y1 x ·
exp −R
pu du
y1 u
a
a
t b
2y1 u
ϕt
dt ds.
· exp R
pudu
·
y1 t
s
y1 u
a
s
2.2
Proof. We assume that
vx yx
y1 x
2.3
for all x ∈ I. It follows from 1.4, 2.1, and 2.3 that
vxy1 x px vxy1 x qx vxy1 x − fx
vx y1 x vxy1 x px vx y1 x vxy1 x
qxvxy1 x − fx
vx y1 x vx 2y1 x pxy1 x
vx y1 x pxy1 x qxy1 x − fx
vx y1 x vx 2y1 x pxy1 x − fx
fx
2y
x
1
px −
y1 xvx vx
y1 x
y1 x 2.4
≤ ϕx,
so, we have
fx 2y1 x
ϕx .
px −
vx vx
≤
y1 x
y1 x y1 x
2.5
4
Journal of Applied Mathematics
For simplicity, we use the following notation:
ys 2y1 u
pu du ·
zs : exp
y1 u
y1 s
a
v s
fv
2y1 u
dv
exp
pu du
−
y1 v
y1 u
a
a
s 2.6
for all s ∈ I. By making use of this notation and by 2.5, we get
s ys 2y1 u
pu du ·
zs − zl exp
y1 u
y1 s
a
v fv
2y1 u
dv
exp
pu du
−
y1 v
y1 u
a
a
l yl 2y1 u
− exp
pu du ·
y1 u
y1 l
a
l
v
fv
2y1 u
exp
pu du dv y
y
v
u
1
1
a
a
t
s
yt 2y1 u
dt exp
pu du ·
l
y1 u
y1 t
a
v t
fv
2y1 u
exp
pu du
dv −
y1 v
y1 u
a
a
t s
2y1 t
2y1 u
pt · exp
pu du
l
y1 t
y1 u
a
t yt 2y1 u
yt ·
exp
pu du ·
y1 t
y1 u
y1 t
a
t 2y1 u
ft
dt
exp
pu du
−
y1 t
y1 u
a
t s
2y1 u
exp
pu du
l
y1 u
a
yt 2y1 t
yt ft
·
pt ·
dt
−
y1 t
y1 t
y1 t
y1 t
t s
2y1 u
ϕt
dt ≤ exp R
pudu
·
y1 t
l
y1 u
a
s
2.7
Journal of Applied Mathematics
5
t
for all l, x ∈ I. Since exp{R a 2y1 u /y1 u pudu · ϕt/y1 t is assumed to be
integrable on I, we may select l0 ∈ I, for any given > 0, such that l, x ≥ l0 implies zx −
zl < . That is, {zl}l∈I is a Cauchy net. By completeness of X, there exists an x0 ∈ X such
that zl converges to x0 as l −→ b. It follows from 2.7 and the previous argument that, for
any x ∈ I,
x s
2y1 u
exp −
pu du
yx − y1 x
y1 u
a
a
v 2y1 u
fv
ds k × x0 exp
pu du dv
y1 u
a y1 v
a
s
x 2y1 u
y1 x ·
exp −
pu du · zs − x0 ds y1 u
a
a
s
x
2y1 u
≤ y1 x ·
exp −R
pu du · zs − zl ds
y1 u
a
a
s
x
2y1 u
y1 x ·
exp −R
pu du · zl − x0 ds
y1 u
a
a
s
x
2y
u
1
≤ y1 x ·
exp −R
pu du
y1 u
a
a
t s
2y1 u
ϕt
dt ds
· exp R
pudu
·
y1 t l
y1 u
a
s
2.8
2y1 u
exp −R
pu du · zl − x0 ds
y1 u
a
a
s
x
2y
u
1
−→ y1 x ·
exp −R
pu du
y1 u
a
a
t b
2y1 u
ϕt
dt ds
· exp R
pudu
·
y1 t s
y1 u
a
y1 x ·
x
s
as l −→ b. Moreover,
y0 x y1 x ·
2y1 u
exp
pu du
y1 u
a
a
v s
2y1 u
fv
ds k
· x0 exp
pu du dv
y1 u
a y1 v
a
x is a solution of 1.3.
s 2.9
6
Journal of Applied Mathematics
Now, we prove the uniqueness property of x0 . Assume that x1 , x2 ∈ X satisfy
inequality 2.2 in place of x0 . Then, we have
s
x
2y1 u
exp −
pu du · x2 − x1 ds
y1 x ·
y1 u
a
a
≤ 2y1 x ·
2y1 u
exp −R
pu du
y1 u
a
t b
2y1 u
ϕt
dt ds,
· exp R
pudu
·
y1 t s
y1 u
a
x a
s
2.10
thus,
x2 − x1 2·
≤
t
s
b
exp −R Adu · exp R
Adu · ϕt/ y1 t dt ds 2.11
s
a
a
,
s
x Adu
ds
exp
−R
x
a
a
a
where A denotes 2y1 u /y1 u pu.
It follows from the integrability hypothesis that
t b
2y1 u
ϕt
dt −→ 0
pudu
·
exp R
y1 t s
y1 u
a
2.12
as s −→ b. This implies that x1 x2 and the proof is complete.
Remark 2.2. It follows from Theorem 2.1 that
yx y1 x ·
2y1 u
exp
pu du
y1 u
a
a
v s
2y1 u
fv
ds c2
· c1 exp
pu du dv
y1 u
a y1 v
a
x s 2.13
is the general solution of the differential equation 1.3, where c1 , c2 are arbitrary elements of
X and y1 x is a nonzero solution of the corresponding homogeneous equation 1.3.
Remark 2.3. If we replace C by R in the proof of Theorem 2.1 and we assume that p, q are
real-valued continuous functions, then we can see that Theorem 2.1 is true for a real Banach
space X.
Journal of Applied Mathematics
7
Hence, every 2nd-order linear differential equation has the generalized Hyers-Ulam
stability with the condition that there exists a solution of corresponding homogeneous
equation or there exists a general solution in the ordinary differential equations.
Example 2.4. Consider the second-order linear differential equation with constant coefficients
y by cy fx.
2.14
√
Let b2 − 4c ≥ 0, m −b ± b2 − 4c/2, and let f : I −→ R, ϕ : I −→ 0, ∞ be continuous
functions. Assume that y : I −→ R is a twice continuously differential function satisfying the
differential inequality
y by cy − fx ≤ ϕx
2.15
for all x ∈ I. On the other hand, by ordinary differential equations, we know that y1 x expmx is a solution of corresponding homogeneous equation of 2.14. It follows from
Theorem 2.1, Remark 2.3, and 2.14 that there exists a solution y0 : I −→ R of 2.14 such
that
y0 x expmx ·
x
exp−2m bs − a
a
s
· x0 fv · expvm b − a2m bdv ds k
2.16
a
for all x ∈ I and that
yx − y0 x ≤ expmx ·
x
a
exp−2m bs − a
b
ϕt
dt ds.
· exp2m bt − a · expmx s
2.17
√
Example 2.5. Consider 2.14. Let b2 − 4c < 0, m −b ± b2 − 4c/2 α ± iβ, and let f :
I −→ R, ϕ : I −→ 0, ∞ be continuous functions. Let y : I −→ R be a twice continuously
differential function satisfying the differential inequality of 2.15 for all x ∈ I. It follows
8
Journal of Applied Mathematics
from the ordinary differential equations that y1 x expαx cosβx. Then it follows from
Theorem 2.1, Remark 2.3, and 2.15 that there exists a solution y0 : I −→ R of 2.14 such that
y0 x expαx cos βx ·
x0 cos βa
2
x
a
expαx cos βx ·
x
a
exp2α ba − s
ds
cos2 βs
exp−2α bs
·
cos2 βs
s
fv · exp vα b
a
· cos βv dv · exp vα b ds k
2.18
for all x ∈ I, where k ya/expαa cosβa and x0 ∈ R is unique and
yx−y x ≤ expαx cos βx ·
0
x
a
exp−2αbs b 2 cos
·
βt
·
expαbt·ϕtdt
ds.
s
cos2 βs
2.19
Example 2.6. Consider the equation
y −
2x 2
2
.
y
y
6
1
x
1 x2
1 x2
2.20
Let I a, b be an open interval, where a, b ∈ 1, ∞ are arbitrarily given with a < b,
f : I −→ R and ϕ : I −→ 0, ∞ are continuous functions. Assume that y : I −→ R is a twice
continuously differential function satisfying the differential inequality
y −
2x 2
2 y y − 6 1 x ≤ ϕx
1 x2
1 x2
2.21
for all x ∈ I. By the trial of y0 x x, we see that it is a solution of corresponding
homogeneous equation of 2.20. Then it follows from Theorem 2.1, Remark 2.3, and 2.21
that there exists a solution y0 : I −→ R of 2.20 such that
y0 x x x0 a
1 − a2
1 a2
k − 6a 2a
3
a2
2
x − 1 x0
x4 3x2
− 3a
1 a2
2
2.22
for all x ∈ I, where k ya/a and x0 ∈ R is unique and
yx − y0 x ≤ x ·
x a
1 s2
s2
b
t
·
·
ϕtdt
ds.
s 1 t2
2.23
Journal of Applied Mathematics
9
Remark 2.7. We know that Eulars differential equation of second order has the general
solution in ordinary differential equations, then we can use Theorem 2.1 and Remark 2.3 for
the Hyers-Ulam-Rassias stability in this case.
Let p be a real constant, and I −1, 1. We know that Legender’s differential equation
1 − x2 y − 2xy p p 1 y 0
2.24
y a0 y1 x a1 y2 x,
2.25
p p1 2
p−2 p p1 p3 4
x x − ··· ,
y1 x 1 −
2
4!
p−3 p−1 p2 p4 5
P − 1 p 2 3
x x − ···
y2 x x −
3!
5!
2.26
has the general solution
where
and a0 , a1 are arbitrary constants. By Theorem 2.1 and Remark 2.3, Legender’s differential
equation has Hyers-Ulam-Rassias stability.
Hermite’s differential equation
y − 2xy 2py 0,
2.27
where p is a real constant, has the general solution
y a0 y1 x a1 y2 x
2.28
∞
−1n 2n p p − 2 · · · p − 2n 2 2n
x ,
y1 x 1 2n!
n1
−1n 2n p − 1 p − 3 · · · p − 2n 1 2n1
y2 x x x
2n 1!
2.29
that
for all x ∈ R, and a0 , a1 are arbitrary constants. Thus Hermites differential equation has
generalized Hyers-Ulam stability.
It is well known from the ordinary differential equations that
y1 x Jp x x 2np
−1n
,
2
n0 n!Γ n p 1
∞
2.30
10
Journal of Applied Mathematics
for all x ∈ R, is a solution of Bessel’s differential equation
x2 y xy x2 − p2 y 0
2.31
that p ≥ 0.
Then Bessel’s differential equation has Hyers-Ulam-Rassias stability.
We know from the ordinary differential equations that Laguerre, Chebyshev, and
Gauss hypergeometric differential equations have the general solution. Then we can show
that those have generalized Hyers-Ulam stability.
Acknowledgment
The third author of this work was partially supported by Basic Science Research Program
through the National Research Foundation of Korea NRF funded by the Ministry of Education, Science and Technology Grant no. 2011–0005197.
References
1 S. M. Ulam, Problems in Modern Mathematics, chapter 6, John Wiley & Sons, New York, NY, USA, 1940.
2 D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of
Sciences of the United States of America, vol. 27, pp. 222–224, 1941.
3 T. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American
Mathematical Society, vol. 72, no. 2, pp. 297–300, 1978.
4 C. Alsina and R. Ger, “On some inequalities and stability results related to the exponential function,”
Journal of Inequalities and Applications, vol. 2, no. 4, pp. 373–380, 1998.
5 S.-E. Takahasi, T. Miura, and S. Miyajima, “On the Hyers-Ulam stability of the Banach space-valued
differential equation y λy,” Bulletin of the Korean Mathematical Society, vol. 39, no. 2, pp. 309–315,
2002.
6 T. Miura, S. Miyajima, and S.-E. Takahasi, “A characterization of Hyers-Ulam stability of first order
linear differential operators,” Journal of Mathematical Analysis and Applications, vol. 286, no. 1, pp. 136–
146, 2003.
7 S.-M. Jung, “Hyers-Ulam stability of linear differential equations of first order,” Applied Mathematics
Letters, vol. 17, no. 10, pp. 1135–1140, 2004.
8 S.-M. Jung, “Hyers-Ulam stability of linear differential equations of first order. II,” Applied Mathematics
Letters, vol. 19, no. 9, pp. 854–858, 2006.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014