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Applied Mathematics Letters 18 (2005) 1304–
www.elsevier.com/locate/aml
Conditional stability for a class of second-order differential
equations
✩
Cezar Avramescu
a
, Octavian G. Mustafa
a
, Svitlana P. Rogovchenko
b
Yuri V. Rogovchenko
b,∗
aDepartment of Mathematics, University of Craiova, Al. I. Cuza 13, Craiova, Romania
b Department of Mathematics, Eastern Mediterranean University, Famagusta, TRNC, Mersin 10, Turkey
Received 2 November 2004; accepted 8 February 2005
Abstract
Using two classes of test functions introduced recently by the authors, a criterion for conditional stability of the
trivial solution of a second-order semi-linear differential equation is established.
© 2005 Elsevier Ltd. All rights reserved.
Keywords: Semi-linear differential equations; Second order; Asymptotic behaviour; Test functions; Conditional stability
1. Preliminaries
Asymptotic integration of ordinary differential equations is an important tool in modern applied
mathematics. A variety of practical problems arising, for instance, in astrophysics, materials science and
mathematical ecology provoke considerable interest in this topic, reflected in recent publications [1–4].
In particular, Agarwal and O’Regan [2, Section 1.12], [5, Section 13.2] applied a fixed point argument,
known as the Furi–Pera theorem [6], to investigate the existence of vanishing-at-infinity solutions of the
✩ This research was supported in part by the programme “Research in Pairs” of the Mathematisches Forschungsinstitut
Oberwolfach, Germany (O.M. and Yu.R.) and Research Fellowships in Mathematics of the Abdus Salam International Centre
for Theoretical Physics (S.R. and Yu.R.). ∗ (^) Corresponding author. Tel.: +90 392 630 1010; fax: +90 392 365 1604.
E-mail address: yuri.rogovchenko@emu.edu.tr (Yu.V. Rogovchenko).
0893-9659/$ - see front matter © 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.aml.2005.02.
differential equation (m > 0 )
u
′′ = m
2 u + F(t, u), t ≥ 0 ,
assuming that for each r > 0 there exists a continuous non-negative function τr (t) such that |u| ≤ r
implies |F(t, u)| ≤ τr (t) for all t ≥ 0, and
lim t→+∞
e
−mt
∫ (^) t
0
e
ms τr (s)ds = lim t→+∞
e
mt
t
e
−ms τr (s)ds = 0.
An extension of these results to a larger class of second-order semi-linear differential equations
u
′′ = a(t)u + F(t, u), t ≥ t 0 ≥ 0 , (1)
has been obtained recently by Agarwal et al. [7] under the assumption that the associated linear equation
u
′′ = a(t)u, t ≥ t 0 ≥ 0 , (2)
possesses a solution x(t) having the Poincaré–Perron property (see [8])
lim t→+∞
x
′ (t)
x(t)
= −m.
It has been established that a(t) can be written in the form
a(t) = m
2
where b : [t 0 , +∞) → R is a continuous function satisfying for all q > 0
lim t→+∞
e
−qt
∫ (^) t
t 0
e
qs b(s)ds = 0. (4)
Remarkably, (4) is equivalent to the celebrated Hartman condition [9, (IV), p. 570],
lim t→+∞
[
sup v> 0
1 + v
∫ (^) t+v
t
b(s)ds
]
but the proof in [7] relies on completely different tools.
The present work continues the study of the properties of solutions of Eq. (1) undertaken by Agarwal
et al. [7] and Avramescu et al. [8] and establishes an interesting link between the results in [2], [7], and
a classical theorem due to Wintner [10, p. 67] stating that Eq. (2) has a solution x(t) such that
x(t) = e
−mt
−mt ) and x
′ (t) = −me
−mt
−mt ) (5)
as t → +∞ provided that
∫ (^) +∞
t 0
e
− 2 mt
∫ (^) t
t 0
e
2 ms |b(s)|dsdt < +∞,
e
− 2 mt
t
t 0
e
2 ms |b(s)|ds ≤ M < +∞,
for all t ≥ t 0. For further details the reader is referred to [7] and [8].
In this work, we are concerned with asymptotic behaviour of bounded solutions of Eq. (1). If b(t) in
(3) is continuous and absolutely integrable on [t 0 , +∞), Eq. (2) has a solution y(t) such that y(t) ∼ e
mt
as t → +∞ [11, p. 126]. Therefore, one cannot expect stability of the trivial solution of Eq. (1), but
Lemma 4. Let g : [t 0 , +∞) → ( 0 , +∞) be defined by
g(t) = f (t)
[
∫ (^) t
t 0
ds
f
2 (s)
]
, t ≥ t 0.
Then (a) g(t) is a solution of Eq. (2); (b) g ∈ G.
2. Conditional stability of the trivial solution
In what follows, we assume that the problem
u
′′ = a(t)u + F(t, u), t ≥ T 0 , (9)
u(T ) = u 0 u
′ (T ) = u 1 , (10)
has a unique solution for any pair of real numbers (u 0 , u 1 ) and any T ≥ t 0 ≥ 0. It is known (see, for
instance, [13]) that (7) implies global existence of solutions of Eq. (1).
Lemma 5. A solution u(t) of Eq. (1) satisfies
lim t→+∞
u(t) = 0 (11)
provided that
lim t→+∞
[
u(t) +
νg
u
′ (t)
]
Proof. Assume that (12) holds. Then (11) follows from the fact that
lim t→+∞
[u(t) exp(νg t)]
′
[exp(νg t)]′^
= lim t→+∞
[
u(t) +
νg
u
′ (t)
]
and l’Hôpital’s rule.
The following result is principal in this work and provides a useful criterion for conditional stability
of the trivial solution of Eq. (1).
Theorem 6. Any bounded solution u(t) of Eq. (1) satisfies (11) and
lim t→+∞
u
′ (t) = 0. (13)
Proof. Step 1. First we note that, by Lemma 3, the derivative of a bounded solution of Eq. (1) is bounded,
i.e., there exists a δ = δ(u) > 0 such that
max[|u(t)|, |u
′ (t)|] < δ. (14)
Suppose, for the sake of contradiction, that either the limit
lim t→+∞
[u(t) + (νg)
− 1 u
′ (t)]
does not exist or differs from zero. Then there exists an ε 0 > 0 and a strictly increasing sequence (tn )n≥ 1
of positive numbers, tn → +∞ as n → +∞, such that
∣ ∣ ∣ ∣
u(tn ) +
νg
u
′ (tn )
ε 0 ,
for all n ≥ 1. Fix an ε satisfying 0 < 2 ε < min[ 1 , (νg )
− 1 , ε 0 δ
− 1 ], and choose an N = N (ε) ≥ 1 large
enough to ensure that g
′ (t) > 0 for all t ≥ T = t (^) N ,
g(T )g
′ (T )
T
ds
g^2 (s)
− ε,
g
2 (T )
T
ds
g^2 (s)
2 νg
− ε,
2 νg
g
2 (T )
T
ds
g^2 (s)
1 − g(T )g
′ (T )
T
ds
g^2 (s)
νg
− ε,
νg
and
I (T ) =
T
g^2 (s)
∫ (^) s
T
h(τ )g
2 (τ )dτ ds <
1 − 2 ε
2 ( 5 + 2 ε)( 1 + (νg)−^1 + ε)
ε 0
δ
Step 2. Define the function p : [T, +∞) → R by
p(t)
def
u 0
g(T )
′ (T )]
∫ (^) t
T
ds
g^2 (s)
Then
| p(t)| ≤
g(T )
[
|u 0 | +
2 νg
|u 1 |
]
and p(t) → K as t → +∞, where
K =
g(T )
u 0
[
1 − g(T )g
′ (T )
T
ds
g^2 (s)
]
[
g
2 (T )
T
ds
g^2 (s)
]}
Writing (15) in the form
K =
g(T )
[
1 − g(T )g
′ (T )
T
ds
g^2 (s)
]
×
[
u 0 + g
2 (T )
T
ds
g^2 (s)
1 − g(T )g
′ (T )
T
ds
g^2 (s)
u 1
]
we deduce that
|K | ≤
g(T )
) [
|u 0 | +
νg
|u 1 |
]
and ∣ ∣ ∣ ∣ ∣
u 0 +
νg
u 1
u 0 + g
2 (T )
T
ds
g^2 (s)
1 − g(T )g
′ (T )
T
ds
g^2 (s)
u 1
νg
− g
2 (T )
+∞
T
ds
g^2 (s)
1 − g(T )g
′ (T )
+∞
T
ds
g^2 (s)
|u 1 |
≤ ε|u 1 | < εδ <
ε 0
u 0 +
νg
u 1
d
dt
[
(U u)(t)
g(t)
]∣
∣ ≤^
g^2 (t)
|u 1 g(T ) − u 0 g
′ (T )| +
∫ (^) t
T
h(τ )
[
|u(τ )|
g(τ )
]
g
2 (τ )dτ
g^2 (T )
|u 1 g(T ) − u 0 g
′ (T )| +
[
|K |
| p(v)|
]
M < +∞,
where u ∈ B, ensures the equicontinuity of set U B. Next, defining l(U u) by
l(U u) = K +
T
g^2 (s)
∫ (^) s
T
F(τ, u(τ ))g(τ )dτ ds,
we have ∣ ∣ ∣ ∣
(U u)(t)
g(t)
− l(U u)
[
g
′ (T )|u 0 | + g(T )|u 1 |
]
t
ds
g
2 (s)
t
g
2 (s)
∫ (^) s
T
|F(τ, u(τ ))|g(τ )dτ ds
≤ N
[∫
+∞
t
ds
g^2 (s)
+∞
t
G(s)ds
]
where N = δ[g(T ) + g
′ (T ) + ( 4 g(T ))
− 1 ( 5 + 2 ε)( 1 + (νg)
− 1
- ε)], which implies that the set U B is
equiconvergent in the sense of [13, p. 349].
Application of the Schauder–Tikhonov theorem yields existence of a fixed point ζ ∈ B of the operator
U. Since the problem (9), (10) has a unique solution, u(t) = ζ(t) for all t ≥ T , this implies
p(t) −
|K |
u(t)
g(t)
≤ p(t) +
|K |
, t ≥ T.
Passing to the limit as t → +∞, we conclude that
lim t→+∞
u(t)
g(t)
= l(u) ∈
[
K −
|K |
, K +
|K |
]
which, by virtue of (16) and Definition 1, contradicts (14).
Remark 7. We conclude the work by commenting on condition (8). Integration by parts and application
of l’Hôpital’s rule yield that (8) holds for a continuously differentiable function h(t) provided that
∫ (^) +∞
t 0
|h
′ (s)|ds < +∞ and 0 ≤ h(t) ≤ H < +∞. (19)
These conditions were exploited by Wintner in [10, pp. 64–65] to ensure asymptotic expansions (5) and
thus are natural for our discussion. The significant role played by the first condition in (19) in asymptotic
integration of Eq. (2) is addressed by Hartman [9, Sections 18, 23].
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