Behavior of solutions of second order differential equations with sublinear damping
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Karsai, J. Graef, J.R. 2021-02-17T06:51:45Z 2021-02-17T06:51:45Z 2005 Behavior of solutions of second order differential equations with sublinear damping / J. Karsai, J.R. Graef // Нелінійні коливання. — 2005. — Т. 8, № 2. — С. 186-200. — Бібліогр.: 11 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/177886 517.9 Supported by the Hungarian National Foundation for Scientific Research Grant No. T 034275. Supported in part by the University of Tennessee at Chattanooga Center of Excellence for Computer Applications. en Інститут математики НАН України Нелінійні коливання Behavior of solutions of second order differential equations with sublinear damping Поведінка розв'язків диференціальних рівнянь другого порядку з сублінійним згасанням Поведение решений дифференциальных уравнений второго порядка с сублинейным угасанием Article published earlier |
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Behavior of solutions of second order differential equations with sublinear damping |
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Behavior of solutions of second order differential equations with sublinear damping Karsai, J. Graef, J.R. |
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Behavior of solutions of second order differential equations with sublinear damping |
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Behavior of solutions of second order differential equations with sublinear damping |
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Behavior of solutions of second order differential equations with sublinear damping |
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Behavior of solutions of second order differential equations with sublinear damping |
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behavior of solutions of second order differential equations with sublinear damping |
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Karsai, J. Graef, J.R. |
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Karsai, J. Graef, J.R. |
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2005 |
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Нелінійні коливання |
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Поведінка розв'язків диференціальних рівнянь другого порядку з сублінійним згасанням Поведение решений дифференциальных уравнений второго порядка с сублинейным угасанием |
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Behavior of solutions of second order differential equations with sublinear damping / J. Karsai, J.R. Graef // Нелінійні коливання. — 2005. — Т. 8, № 2. — С. 186-200. — Бібліогр.: 11 назв. — англ. |
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2025-11-25T22:29:31Z |
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UDC 517 . 9
BEHAVIOR OF SOLUTIONS OF SECOND ORDER
DIFFERENTIAL EQUATIONS WITH SUBLINEAR DAMPING
ПОВЕДIНКА РОЗВ’ЯЗКIВ ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ
ДРУГОГО ПОРЯДКУ З СУБЛIНIЙНИМ ЗГАСАННЯМ
J. Karsai∗
Univ. Szeged, Hungary
e-mail: karsai@dmi.u-szeged.hu
J. R. Graef ∗∗
Univ. Tennessee at Chattanooga
Chattanooga , TN 37403, USA
e-mail: john-graef@utc.edu
The authors investigate the asymptotic behavior of solutions of the damped nonlinear oscillator equation
x′′ + a(t)|x′|αsgn (x′) + f(x) = 0,
where uf(u) > 0 for u 6= 0, a(t) ≥ 0, and α is a positive constant with 0 < α ≤ 1. The case α = 1
has been investigated by a number of other authors. Here, it is shown that the behavior of solutions in the
case of sublinear damping (0 < α < 1) is completely different from that in the case of linear damping
(α = 1). Sufficient conditions for all nonoscillatory solutions to converge to zero and sufficient conditions
for the existence of a nonoscillatory solution that does not converge to zero are given. They also give
sufficient conditions for all solutions to be nonoscillatory. Some open problems for future research are
also indicated.
Вивчається асимптотична поведiнка розв’язкiв нелiнiйного рiвняння з коливанням та згасан-
ням
x′′ + a(t)|x′|αsgn (x′) + f(x) = 0,
де uf(u) > 0 для u 6= 0, a(t) ≥ 0 та α — додатна стала, що задовольняє умову 0 < α ≤ 1.
Випадок α = 1 було розглянуто iншими авторами. Показано, що поведiнка розв’язкiв у випадку
сублiнiйного згасання, тобто для 0 < α < 1, повнiстю вiдрiзняється вiд поведiнки у випадку
лiнiйного згасання (α = 1). Наведено достатнi умови збiжностi до нуля розв’язкiв, що не є
коливними, а також достатнi умови iснування розв’язкiв, що не є коливними i не збiгаються
до нуля. Крiм цього наведено достатнi умови для того, щоб всi розв’язки були неколивними.
Сформульовано кiлька вiдкритих питань для подальшого дослiдження.
1. Introduction. In this paper, we investigate the asymptotic behavior of the solutions of the
damped nonautonomous nonlinear differential equation
x′′ + a(t)|x′|αsgn (x′) + f(x) = 0, (1.1)
∗ Supported by the Hungarian National Foundation for Scientific Research Grant No. T 034275.
∗∗ Supported in part by the University of Tennessee at Chattanooga Center of Excellence for Computer Appli-
cations.
c© J. Karsai, J. R. Graef, 2005
186 ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2
BEHAVIOR OF SOLUTIONS OF SECOND ORDER DIFFERENTIAL EQUATIONS WITH SUBLINEAR DAMPING 187
where a : [0,∞) → [0,∞) and f : (−∞,∞) → (−∞,∞) are continuous, uf(u) > 0 for u 6= 0,
and α is a positive constant with 0 < α ≤ 1. The literature on the attractivity behavior in the
case of linear damping, i.e., for the equation
x′′ + a(t)x′ + f(x) = 0, (1.2)
as well as for the related equation
x′′ + q(t)f(x) = 0
is quite large since these are classical problems in the study of ordinary differential equations.
There are a great number of generalizations and extensions to equations with different kinds
of nonlinearities especially for α > 1. Attractivity criteria in the case α > 1 are often based on
the fact that |u|α < |u| for α > 1 and |u| < 1. Such estimates do not hold if 0 < α < 1, and
this partially explains why this case has not been as extensively studied.
In [1, 2], the authors investigated autonomous equations with sublinear damping. As a speci-
al case of more general results, we obtained that every solution of (1.1) with 0 < α < 1 and
a(t) ≡ a, i.e., the equation
x′′ + a|x′|αsgn (x′) + f(x) = 0, (1.3)
is nonoscillatory. On the other hand, it can easily be proved that for α > 1, every solution
of (1.3) is oscillatory. As we will see, if the equation is nonautonomous, the situation is more
complicated and correspondingly more interesting.
In the case of the nonlinear equation (1.1), the damping effect is a result of two factors,
namely, the size of the coefficient a(t) and the value of α. We will refer to the damping as being
linear, sublinear, or superlinear depending on whether we have α = 1, 0 < α < 1, or α > 1,
respectively. For equation (1.1) with a linear damping term, i.e., equation (1.2), it is known (see
[3]; also see Lemma 1.1 below) that a solution either oscillates or is eventually monotonic. As a
consequence, in this case, if the damping coefficient a(t) is too small, then all solutions oscillate
and there are some solutions that do not tend to zero as t → ∞; this has been referred to as
under damping. On the other hand, if a(t) is too large, then all solutions are nonoscillatory and
there are solutions that do not approach zero. This situation is known as over damping. Between
these two extreme cases, we have what is called small damping if all solutions oscillate and tend
to zero as t → ∞, and large damping if the solutions are nonoscillatory and tend to zero as
t → ∞. Although this classification of damping phenomena was originally applied to the linear
equation with linear damping, i.e., equation (1.2) with f(x) = x (see [4]), it can also be applied
to the nonlinear equation (1.1) as well. But it should be kept in mind that for equation (1.1), the
total damping effect depends on both the size of a(t) and the value of α.
To illustrate these various behaviors, consider the Figures 1 – 5 that represent the solutions
of equation (1.1) in the case of under damping, small damping, constant damping, large dam-
ping, and over damping in the superlinear (α = 2), linear (α = 1), and sublinear (α = 0, 5)
cases, respectively. All these simulations were produced taking f(x) = x.
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2
188 J. KARSAI, J. R. GRAEF
α = 2 α = 1 α = 0, 5
Fig. 1. Under damping: x′′ +
1
(1 + t)2
|x′|αsgn (x′) + x = 0.
α = 2 α = 1 α = 0, 5
Fig. 2. Small damping: x′′ +
1
1 + t
|x′|αsgn (x′) + x = 0.
α = 2 α = 1 α = 0, 5
Fig. 3. Constant damping: x′′ + |x′|αsgn (x′) + x = 0.
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2
BEHAVIOR OF SOLUTIONS OF SECOND ORDER DIFFERENTIAL EQUATIONS WITH SUBLINEAR DAMPING 189
α = 2 α = 1 α = 0, 5
Fig. 4. Large damping: x′′ + (1 + t)|x′|αsgn (x′) + x = 0.
α = 2 α = 1 α = 0, 5
Fig. 5. Over damping: x′′ + (1 + t)2|x′|αsgn (x′) + x = 0.
Our interest in this paper is an examination of the impact of the nonlinear damping term on
the resulting large and over damping effects that result in equation (1.1). To further illustrate the
interaction between a(t) and α, consider the following examples. Observe that x1(t) = 1+
1
1 + t
is an over damped solution of each of the equations
x′′ +
(
4 + 5 t + 4 t2 + t3
)
1 + t
x′ + x = 0,
x′′ + (1 + t)3
(
4 + 5 t + 4 t2 + t3
)
x′
3 + x = 0,
x′′ +
(
4 + 5 t + 4 t2 + t3
)
(1 + t)
5
3
sgn (x′) |x′|
2
3 + x = 0,
x′′ +
(
4 + 5 t + 4 t2 + t3
)
(1 + t)2
sgn (x′)
√
|x′|+ x = 0,
and
x′′ +
(
4 + 5 t + 4 t2 + t3
)
(1 + t)
7
3
sgn (x′) |x′|
1
3 + x = 0.
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2
190 J. KARSAI, J. R. GRAEF
To see the large damping effect, observe that x2(t) =
1
1 + t
is a solution of each of the equations
x′′ +
(
3 + 2 t + t2
)
1 + t
x′ + x = 0,
x′′ + (1 + t)3
(
3 + 2 t + t2
)
x′
3 + x = 0,
x′′ +
(
3 + 2 t + t2
)
(1 + t)
5
3
sgn (x′) |x′|
2
3 + x = 0,
x′′ +
(
3 + 2 t + t2
)
(1 + t)2
sgn (x′)
√
|x′|+ x = 0,
and
x′′ +
(
3 + 2 t + t2
)
(1 + t)
7
3
sgn (x′) |x′|
1
3 + x = 0.
Notice that the smaller the value of α in the damping term a(t)|x′|αsgn (x′), the smaller the
“degree"of a(t) that is needed to result in an over damping effect. In addition, while in the
case of linear damping, nonoscillatory solutions do not exist for 0 < a(t) ≤ a0, where a0
is small enough (e.g., a(t) < 2 for f(x) = x), the last example above shows that there can
exist monotone solutions in the sublinear damping case even if limt→∞ a(t) = 0. On the other
hand, Figures 4 and 5 show that solutions can oscillate in the case of superlinear damping with
limt→∞ a(t) = ∞, while in the linear and sublinear cases, over damped behavior is observed.
The following type of lemma has proved to be fundamental in the study of these kinds of
problems.
Lemma 1.1. Let x(t) be a solution of (1.1) with x′(t1) = x′(t2) = 0 and x′(t) 6= 0 for
t ∈ (t1, t2). Then, there exists t̃ ∈ (t1, t2) such that x(t̃) = 0. Consequently, every solution of
(1.1) is either oscillatory or eventually monotonic.
The proof is analogous to the proof of Lemma 1 in [5], and so in the interest of space, we
omit the details. The next lemma is independent of the value of α > 0 and guarantees that
nonoscillatory solutions have to eventually decrease in absolute value. A proof of this lemma
can be found in [6] where equation (1.1) with α > 1 is considered.
Lemma 1.2. Suppose that f is nondecreasing. Let x(t) be a solution of (1.1) such that x(t1) 6=
6= 0 and x(t)x′(t) ≥ 0 for all t ∈ (t1, t2). Then t2 − t1 < |x′(t1)/f(x(t1))|.
At times, we will make use of the energy function
V (t) =
x′2(t)
2
+ F (x(t)), where F (x) :=
∫
hx
0f(u) du. (1.4)
It is easy to see that along solutions of (1.1), we have
V ′(t) = −a(t)|x′(t)|α+1 ≤ 0 (1.5)
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2
BEHAVIOR OF SOLUTIONS OF SECOND ORDER DIFFERENTIAL EQUATIONS WITH SUBLINEAR DAMPING 191
so that the energy along solutions is nonincreasing. Our results in the remainder of this paper
will require that the function f be nondecreasing. This ensures that limx→∞ F (x) = ∞ and
guarantees that every solution is continuable and is bounded on [0,∞).
The remainder of the paper is devoted primarily to the study of equation (1.1) with a
sublinear damping term, i.e., with 0 < α < 1. The next section contains our results on the
asymptotic behavior of nonoscillatory solutions of equation (1.1). We give sufficient conditions
for all nonoscillatory solutions to converge to zero and sufficient conditions for the existence of
a nonoscillatory solution that does not converge to zero. In the last section, we give sufficient
conditions for all solutions to be nonoscillatory.
2. Properties of nonoscillatory solutions. Necessary and sufficient conditions to guarantee
that every nonoscillatory solution of (1.1) with a linear damping term (α = 1) tends to zero
were given by Smith [4].
Theorem 2.1 ([4], Theorem 1). Assume that a(t) ≥ a0 > 0. Then every nonoscillatory
solution of the equation
x′′ + a(t)x′ + x = 0 (2.1)
tends to zero as t → ∞ if and only if
∞∫
0
e−H(s)
s∫
0
eH(u)duds = ∞, (2.2)
where H(t) =
∫ t
0
a(s)ds.
The proof of this theorem is based on the fact that by integrating equation (2.1), the deri-
vative of a solution can be written in the form
x′(t) = exp
− t∫
T
a(s)ds
x′(T )−
t∫
0
x(s) exp
s∫
0
a(u)du
ds
.
For each nonoscillatory solution of (1.1), we can also obtain a similar expression. Let the soluti-
on x(t) be monotonic and satisfy x(t)x′(t) ≤ 0 on the interval [T,∞). Then, equation (1.1) can
be written in the form
x′′(t) + a(t)|x′(t)|α−1x′(t) + f(x(t)) = 0.
Observe that the function |u|α−1 (u > 0) is increasing for α > 1, and is decreasing and has a
singularity at u = 0 if 0 < α < 1. Integrating this equation, we obtain
x′(t) = exp
− t∫
T
a(s)|x′(s)|α−1ds
×
×
x′(T )−
t∫
T
f(x(s)) exp
s∫
T
a(u)|x′(u)|α−1du
ds
. (2.3)
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2
192 J. KARSAI, J. R. GRAEF
This expression is the basis for generalizations of Smith’s result to more general equations, such
as
x′′ + g(t, x, x′)x′ + f(x) = 0;
see, for example [3, 5, 7 – 11].
By deriving appropriate estimates for x′(t), we will be able to obtain results for the case of
sublinear damping.
Theorem 2.2. Assume that 0 < α ≤ 1, f is nondecreasing, and
∞∫
0
exp
− τ∫
0
L
a
B
1−α
α
τ∫
0
exp
s∫
0
L
a
B
1−α
α
ds dτ = ∞ (2.4)
for every constant L > 0, where
B(t) := exp
− t∫
0
a(s)ds
t∫
0
exp
s∫
0
a(u)du
ds.
Then limt→∞ x(t) = 0 for every nonoscillatory solution x(t) of (1.1).
Proof. To the contrary, assume that x(t) is solution of (1.1) that is nonoscillatory on some
interval [T1,∞), T1 ≥ 0, and that x(t) does not tend to zero as t → ∞. In view of Lemmas 1.1
and 1.2, we can assume that x(t) > 0 and x′(t) < 0 on [T1,∞). Moreover, since x(t) is bounded,
it must be the case that x′(t) → 0 as t → ∞, so choose T ≥ T1 such that |x′(t)| < 1 for t > T .
Letting δ = inft≥T f(x(t)) and replacing T in (2.3) by 2T , we have
x′(t) ≤ −δ exp
− t∫
2T
a(s)|x′(s)|α−1ds
t∫
2T
exp
s∫
2T
a(u)|x′(u)|α−1du
ds =
= −δ
t∫
2T
exp
− t∫
s
a(u)|x′(u)|α−1du
ds.
An integration from 2T to t yields
0 < x(t) ≤ x(2T )− δ
t∫
2T
τ∫
2T
exp
− τ∫
s
a(u)|x′(u)|α−1du
ds dτ. (2.5)
We wish to obtain a lower estimate for the integral on the right-hand side of the above inequali-
ty so that condition (2.4) will imply that the right-hand side of (2.5) tends to −∞ as t → ∞.
This will contradict the fact that x(t) is positive and complete the proof of the theorem.
To obtain this lower bound on the above integral, we must derive an upper estimate for
a(t)|x′|α−1(t) on [T, t). Now since 0 < α ≤ 1, this in turn means we need a lower estimate for
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2
BEHAVIOR OF SOLUTIONS OF SECOND ORDER DIFFERENTIAL EQUATIONS WITH SUBLINEAR DAMPING 193
|x′(t)|. Let y(t) = |x′(t)| = −x′(t); then from (1.1), we obtain
y(t) = y(T )−
t∫
T
a(s)yα(s) ds +
t∫
T
f(x(s)) ds,
and since |y(t)| < 1 and 0 < α ≤ 1, this implies
z(t) = yα(t) ≥ y(T )−
t∫
T
a(s)z(s) ds + δ(t− T ).
By a standard differential inequalities argument, we have
z(t) ≥ exp
− t∫
T
a(s)ds
y(T ) + δ
t∫
T
exp
s∫
T
a(u)du
ds
≥
≥ δ exp
− t∫
T
a(s)ds
t∫
T
exp
s∫
T
a(u)du
ds = δB(t). (2.6)
Thus,
a(t)|x′(t)|α−1 ≤ a(t)
(
1
δB(t)
) 1−α
α
= k
a(t)
B(t)
1−α
α
for t > T , where k = δ(α−1)/α. It follows that
t∫
2T
τ∫
2T
exp
− τ∫
s
a(u)|x′(u)|α−1du
ds dτ ≥
≥
t∫
2T
τ∫
2T
exp
−k
τ∫
s
k
a(u)
B(u)
1−α
α
du
ds dτ.
Note that taking the integral on [2T,∞), we avoided the singularity of 1/B(t) at t = T . This
completes the proof of the theorem.
The following result is the counterpart to Theorem 2.2; it gives conditions for the existence
of a nonoscillatory solution that does not tend to zero.
Theorem 2.3. Assume that 0 < α ≤ 1, 0 < a0 ≤ a(t), and f is nondecreasing. If
∞∫
0
exp
− τ∫
0
L
a
B1−α
τ∫
0
exp
s∫
0
L
a
B1−α
ds dτ < ∞ (2.7)
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2
194 J. KARSAI, J. R. GRAEF
for every constant L > 0, where B(t) is defined in Theorem 2.2, then there exists a nonoscillatory
solution of (1.1) such that limt→∞ x(t) 6= 0.
Proof. We are going to construct a solution of (1.1) that does not tend to zero. Consider the
solution with the initial conditions x(T ) > 0, F (x(T )) = 1/2 and x′(T ) = 0, where T will be
chosen later. Assuming that the statement is not true, this solution is either oscillatory, or it is
monotone decreasing on [T,∞) and limt→∞ x(t) = 0. Let K := supt≥T |f(x(t))| < ∞; note
that the value of K depends only on the initial conditions since the energy function defined in
(1.4) is nonincreasing. Let [T, T̄ ) be the maximal interval to the right of T on which x(t) > 0.
Note that we may have T̄ = ∞. Now, x′(t) is negative on some interval to the right of T , so
choose T1 > T such that x(t) > x(T )/2 for t ∈ [T, T1]. Taking T1 in (2.3) in place of T , for
t ∈ [T1, T̄ ), we then have
x′(t) = exp
− t∫
T1
a(s)|x′(s)|α−1ds
×
×
x′(T1)−
t∫
T1
f(x(s)) exp
s∫
T1
a(u)|x′(u)|α−1du
ds
. (2.8)
Next observe that y2(t) = [x′(t)]2 ≤ 2V (t) ≤ 2V (T ) = 2F (x(T )) = 1, so y(t) ≤ 1 for
t ≥ T . Let z(t) = |x′(t)| = −x′(t); then from (1.1), we have
z′ = −a(t)zα(t) + f(x(t)) ≤ −a(t)z(t) + K ≤ −a(t)z(t) + K,
from which it follows that
z(t) ≤ K exp
− t∫
T
a(s)ds
t∫
T
exp
s∫
T
a(u)du
ds ≤ KB(t).
We then have
a(t)|x′|α−1(t) ≥ a(t)
z1−α(t)
≥ K1
a(t)
B1−α(t)
for t ≥ T , where K1 = Kα−1. We can then bound the first term in (2.8) as follows:
x′(T1) exp
− t∫
T1
a(s)|x′(s)|α−1ds
≥ −|x′(T1)| exp
− t∫
T1
K1
a(s)
B1−α(s)
ds
.
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BEHAVIOR OF SOLUTIONS OF SECOND ORDER DIFFERENTIAL EQUATIONS WITH SUBLINEAR DAMPING 195
For the second term in (2.8), we have
− exp
− t∫
T1
a(s)|x′(s)|α−1ds
t∫
T1
f(x(s)) exp
s∫
T1
a(u)|x′(u)|α−1du
ds ≥
≥ −
t∫
T1
f(x(s)) exp
− t∫
s
a(u)|x′(u)|α−1du
ds ≥
≥ −
t∫
T1
f(x(s)) exp
− t∫
s
K1
a(u)
B1−α(u)
du
ds ≥
≥ −K
t∫
T1
exp
− t∫
s
K1
a(u)
B1−α(u)
du
ds.
Applying the above estimates to (2.8) and then integrating from T1 to t, we obtain
x(t) ≥ x(T1)− |x′(T1)|
t∫
T1
exp
− s∫
T1
K1
a(u)
B1−α(u)
du
ds−
−K
t∫
T1
s∫
T1
exp
− s∫
u
K1
a(w)
B1−α(w)
dw
du ds. (2.9)
Now l’Hôpital’s rule shows that 1/B(t) is bounded below away from zero, and since a(t) is as
well, we see that the first integral in (2.9) converges. Hence, T can be chosen such that
|x′(T1)|
t∫
T1
exp
− s∫
T1
K1
a(u)
B1−α(u)
du
ds < F−1(1
2)/8 = x(T )/8.
The last term in (2.9) can be estimated from above by
t∫
T1
s∫
T1
exp
− s∫
u
K1
a(w)
B1−α(w)
dw
du ds ≤
≤ K
t∫
T1
τ∫
T1
exp
− τ∫
s
K1
a(u)
B1−α(u)
du
ds dτ.
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196 J. KARSAI, J. R. GRAEF
The integral on the right is equivalent to (2.7) and so it converges; for sufficiently large T , it can
be bounded above by x(T )/8 = F−1
(1
2
)
/8, where F−1 is the inverse function of F on [0,∞).
Thus, we are able to choose T > 0 so that (2.9) yields
x(t) ≥ x(T1)− x(T )/8− x(T )/8 = x(T )/4 = F−1
(
1
2
)
/4,
and we see that the solution x(t) does not tend to zero.
Remark 2.1. Notice that if α = 1, then Theorems 2.2 and 2.3 taken together become equi-
valent to Theorem 2.1 of Smith above.
Applying our results above can be difficult due to complicated structure of the expression
exp
− t∫
0
h(τ)dτ
t∫
0
exp
τ∫
0
h(s)ds
dτ
which appears nested within both conditions (2.4) and (2.7). The following two lemmas will aid
in formulating some criteria that are somewhat easier to apply than Theorems 2.2 and 2.3.
Lemma 2.1. Assume that h(t) > 0 is nondecreasing. Then
lim
t→∞
h(t) exp
− t∫
0
h(τ) dτ
t∫
0
exp
τ∫
0
h(s) ds
dτ ≥ 1.
The proof of the above lemma as well as the following lemma can be found in [6].
Lemma 2.2. Assume that lim inft→∞ h(t) > 0 and limt→∞
h′(t)
h(t)
= 0. Then
lim
t→∞
h(t) exp
− t∫
0
h(τ) dτ
t∫
0
exp
τ∫
0
h(s) ds
dτ = 1.
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BEHAVIOR OF SOLUTIONS OF SECOND ORDER DIFFERENTIAL EQUATIONS WITH SUBLINEAR DAMPING 197
Now assume that a(t) ≤ ā(t), where ā(t) is nondecreasing. From Lemma 2.1, we have
B(t) = exp
− t∫
0
a(s)ds
t∫
0
exp
s∫
0
a(u)du
ds ≥
≥ exp
− t∫
0
ā(s)ds
t∫
0
exp
s∫
0
ā(u)du
ds ≥
≥ exp
− t∫
0
ā(s)ds
t∫
0
ā(s)
ā(t)
exp
s∫
0
ā(u)du
ds =
=
1− exp
− t∫
0
ā(s)ds
/
ā(t) ≥ k/ā(t),
where the last inequality holds for t ≥ T for some k > 0 and some sufficiently large T > 0.
Thus,
a(t)/B
1−α
α (t) ≤ k1ā(t)1/α
for t ≥ T , where k1 = k
α−1
α . Repeating the above argument, we obtain the following corollary
to Theorem 2.2.
Corollary 2.1. Assume that 0 < α ≤ 1, f is nondecreasing, 0 ≤ a(t) < ā(t), ā(t) is
nondecreasing, and
∞∫
0
1
ā1/α(t)
dt = ∞. (2.10)
Then limt→∞ x(t) = 0 for every nonoscillatory solution x(t) of (1.1).
Similar arguments, together with an application of Lemma 2.2 using a lower estimate on
a(t), yield an upper estimate for the integral in (2.7). As a consequence, we have the following
corollary to Theorem 2.3.
Corollary 2.2. Assume that 0 < α ≤ 1, f is nondecreasing, 0 < a0 ≤ a(t) ≤ a(t), a(t) is
differentiable, limt→∞ a′(t)/a(t) = 0, and
∞∫
0
1
a2−α(t)
dt < ∞. (2.11)
Then there exists a nonoscillatory solution x(t) of (1.1) such that limt→∞ x(t) = 0.
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198 J. KARSAI, J. R. GRAEF
Proof. First note that an upper estimate for the integral B(t) can be obtained by replacing
a(t) by a(t). Hence, from Lemma 2.2 with h(t) ≡ a(t), we have
B(t) = exp
− t∫
0
a(s)ds
t∫
0
exp
s∫
0
a(u)du
ds ≤
≤ exp
− t∫
0
a(s)ds
t∫
0
exp
s∫
0
a(u)du
ds ≤ K
a(t)
for some K > 0 and t ≥ t1 with t1 sufficiently large. This implies that the function a(t)/B1−α(t)
in condition (2.7) can be estimated from below by
a(t)
B1−α(t)
≥ a2−α(t)
K1−α
.
If we again apply Lemma 2.2 this time with h(t) ≡ a2−α(t), we see that there exists a number
K1 such that
∞∫
0
exp
− τ∫
0
L
a
B1−α
τ∫
0
exp
s∫
0
L
a
B1−α
ds dτ ≤ K1
∞∫
0
1
a2−α(τ)
dτ,
and this completes the proof of the corollary.
Applying Corollaries 2.1 and 2.2 to the case c1t
γ ≤ a(t) ≤ c2t
σ with 0 < γ ≤ σ and c1 < c2,
we obtain that if σ ≤ α, then every nonoscillatory solution of (1.1) tends to zero as t → ∞,
and if γ(2 − α) > 1, then there exists a nonoscillatory solution that does not tend to zero.
These results verify our preliminary observations that over damping appears for much smaller
functions a(t) in the case of sublinear damping than it does for the case of linear damping.
Remark 2.2. Lemmas 2.1 and 2.2 show that conditions (2.4) and (2.7) are essentially equi-
valent to (2.10) and (2.11) above, respectively, in case a(t) is nondecreasing. It remains an open
question as to whether these conditions can be improved.
Remark 2.3. In Lemma 1.2, and consequently in Theorems 2.2 and 2.3 as well as Corollaries
2.1 and 2.2, the condition that f be nondecreasing can be replaced by
f(u) is bounded away from zero if u is bounded away from zero.
3. Sufficient conditions for nonoscillation. Although Lemma 1.1 allows for the existence of
both oscillatory and nonoscillatory solutions, our preliminary examples with f(x) = x and our
results in [1, 2] suggest that, under some mild additional conditions, all solutions of (1.1) may in
fact be nonoscillatory. The following theorem holds.
Theorem 3.1. Assume that a(t) ≥ a0 > 0 and lim(x,y)→(0,0) f ′(x)|y|2(1−α) = 0 on the
set {(x, y) : Cf(x) = |y|αsgn (y)} for some C > 1/a0. Then every solution x(t) of (1.1) is
nonoscillatory and
lim sup
t→∞
∣∣∣∣ [x′(t)]αf(x(t))
∣∣∣∣ ≤ 1
a0
.
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BEHAVIOR OF SOLUTIONS OF SECOND ORDER DIFFERENTIAL EQUATIONS WITH SUBLINEAR DAMPING 199
For the proof of Theorem 3.1, we need the following lemma that is somewhat interesting in
its own right; its conclusion is independent of the form of the nonlinearity f(x) as well as the
value of α > 0.
Lemma 3.1. If a(t) ≥ a0 > 0, then limt→∞ V (t) = 0 for every oscillatory solution of (1.1).
Proof. Let x(t) be an oscillatory solution of (1.1) such that limt→∞ V (t) = ε > 0. Then,
there exists an infinite sequence of intervals [tn, sn] such that tn < sn < tn+1, x(tn) = 0,
F (x(sn)) = ε/2, and x(t) > 0 and x′(t) > 0 on [tn, sn] for n = 1, 2, . . . . From the definition of
V (t), we obtain that ε/2 ≤ [x′(t)]2 ≤ 2K on the intervals [tn, sn], where K = V (t1). Hence,
x(sn) = F−1(ε/2) <
√
2K (sn − tn).
Now, [x′(t)]2 ≥ ε/2 on [tn, sn] implies |x′(t)|α+1 ≥ (ε/2)(α+1)/2, so an integration of V ′(t)
yields
V (t) = V (t1)−
t∫
t1
a(u)|x′(u)|α+1du ≤ V (t1)−
∑
sn<t
sn∫
tn
a(u)|x′(u)|α+1du ≤
≤ V (t1)−
∑
sn<t
a0(ε/2)(α+1)/2(sn − tn) → −∞
as t → ∞. This contradicts the nonnegativity of V (t) and completes the proof of the lemma.
Proof of Theorem 3.1. Assume that there exists a nontrivial oscillatory solution x(t) of (1.1).
By Lemma 3.1, limt→∞ V (t) = 0 along this solution. Let ε > 0 be given and choose T ≥ 0
such that V (t) ≤ ε for t ≥ T . The trajectory (x(t), y(t)) = (x(t), x′(t)) passes infinitely many
times through the section defined by
ΓC = {(x, y) : x ≥ 0, y ≤ 0, y2/2 + F (x) ≤ ε, Cf(x) ≥ |y|α}.
We will show that if ε is small enough, the trajectory cannot leave this region. This means that
the solution is nonoscillatory, which contradicts our assumption.
Clearly, ΓC is closed and is bounded by the curves G1 = {(x, y) : y = 0}, G2 = {(x, y) :
y2/2 + F (x) = ε}, and G3 = {(x, y) : Cf(x) = |y|α}. Since the tangent vector at y = 0
is (0,−f(x)), the trajectory enters ΓC through G1. The energy is nonincreasing along x(t), so
the trajectory cannot leave ΓC through G2. Finally, consider the curve Cf(x) = |y|α. Let the
numbers sn, n = 1, 2, . . . , be given by T < s1 < . . . < sn < sn+1, Cf(x(sn)) = |y(sn)|α,
x(sn) > 0, and y(sn) < 0. The tangent vector to the trajectory is (y(sn), a(sn)|y(sn)|α −
−f(x(sn)), and the normal vector to the curve G3 is (−Cf ′(x(sn))/α|y(sn)|α−1,−1). Using
the notation x(sn) = xn, y(sn) = yn, and a(sn) = an, their scalar product is
−C
f ′(xn)
α|yn|α−1
yn − an|yn|α + f(xn) = Cf(xn)
(
C
α
f ′(xn)|yn|2(1−α) − an +
1
C
)
. (3.1)
Since, C > 1/a0 and limn→∞ f ′(xn)|yn|2(1−α) = 0, it follows that T can be chosen large enough
so that (3.1) is negative, i.e., the tangent vector to the trajectory at every intersection point
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200 J. KARSAI, J. R. GRAEF
(x(sn), y(sn)) is directed towards the interior of ΓC . This shows the monotonicity of the soluti-
ons. Since the trajectories are trapped in the region ΓC , the second part of the theorem is also
proved.
Observe that the hypotheses of the above theorem do not explicitely use the assumption
0 < α < 1; the nonlinearity of f(x) is also involved (see the autonomous case in [1, 2]. We easily
can see that if f(x) = |x|βsgn (x), the continuity condition takes the simple form β > α/(2−α),
which clearly holds if, for example, 0 < α < 1 and β ≥ 1.
Finally, note that Fig. 2 and the examples in the introduction show that for 0 < α < 1, there
can exist nonoscillatory solutions of equation (1.1) with a(t) not bounded away from zero, for
example, if a(t) = 1/tω with ω > 0. The extension of our nonoscillation result to this case
remains an open problem.
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Received 02.12.2004
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