Oscillatory solutions of some perturbed second order differential equations

0.12 MB·English

by  Octavian G. Mustafa

Oscillatory solutions of some perturbed second
order differential equations
0 Octavian G. Mustafa
1
Faculty of Mathematics, D.A.L.,
0
2
University of Craiova, Romania

n
e-mail: [email protected]
a
J

6

] Abstract We discuss the occurrence of oscillatory solutions which decay to
A
0 as s → +∞ for a class of perturbed second order ordinary differential equa-
C
tions. As opposed to other results in the recent literature, the perturbation
.
h
is as small as desired in terms of its improper integrals and it is indepen-
t
a
dent of the coefficients of the non-oscillatory unperturbed equation. This
m
class of equations reveals thus a new pathology in the theory of perturbed
[
oscillations.
1
2000 MSC: 34C10; 34E10
v
1 Keywords: Oscillatory solution; Perturbed differential equation; Induced
3
oscillation
9
0
.
1
1 Introduction
0
0
1
We are concerned in this note with a class of perturbed differential equations
:
v
of second order
i
X
r h′′ +p(s,h,h′) = q(s), s ≥ s > 0, (1)
a 0
andtheoccurrenceofoscillatorysolutionsto(1)whichdecayto0ass → +∞
when the perturbation q oscillates.
Though the oscillation literature is vast, most of the investigations are
generalizations of the perspective given by the fundamental papers due to A.
Kartsatos [3, 4], that is, if the unperturbed equation
h′′ +p(s,h,h′) = 0, s ≥ s , (2)
0
1
Induced oscillations to ODE’s 2
is oscillatory and the function Q, where Q′′ = q, is itself oscillatory and
decaying to 0 when s → +∞ then the perturbed equation (1) is oscillatory.
For recent advancements in this respect, the reader can consult [8].
The case where the equation (2) is non-oscillatory is much more compli-
cated, however, when the function Q is strongly oscillatory in some sense, see
e.g. [5], the perturbed equation (1) will possess both oscillatory solutions,
vanishing at infinity, and non-oscillatory solutions. In terms of the improper
integrals used for transforming the differential equations into integral equa-
tions and in the averaging process of the former ones, this strong oscillation
of Q means that the integrals involved have extra large sizes in comparison
with the similar integrals of the coefficients of the unperturbed equation (2).
The following problem is, to the best of our knowledge, still open: which
are (if any) the conditions to be imposed on an oscillatory perturbation q
that will produce oscillatory, vanishing at infinity, solutions to the equation
(1) whose unperturbed part (2) is not only non-oscillatory but its coefficients
are also independent of q ?
Inthisnote,wepresentaclassofsecond-orderdifferentialequationswhich
arenon-oscillatorybutwillproducesuchoscillatorysolutionswhenperturbed
by a small, oscillatory perturbation whose size is independent of the sizes of
the coefficients of the equations.
2 Induced oscillations to equation (1)
Consider the following perturbed equation
h q(s)
h′′ +p(s) h′ − + = 0, s ≥ s > 0, (3)
0
s s
(cid:18) (cid:19)
where the coefficients p,q : [s ,+∞) → R are continuous and such that
0
p(s) ≥ 0, s ≥ s , p,q ∈ L1((s ,+∞),R). (4)
0 0
The equation has been employed recently in studying a class of reaction-
diffusion equations by means of the comparison method, see [2] and its refer-
ences. It is also connected with the Lie theory of integration, see [7, Example
2.62, p. 149].
According to the classical asymptotic integration theory [1], the condi-
tions (4) imply that, given c 6= 0, both the equation (3) and its unperturbed
part have solutions which behave asymptotically as
h(s) = c·s+o(s) when s → +∞, (5)
Induced oscillations to ODE’s 3
so, obviously, they are non-oscillatory.
In the next sections additional restrictions will be imposed on q that are
independent of p and will produce an oscillatory solution to the equation (3)
which vanishes at infinity.
It is useful to remark that the only place with respect to the integration
formula (5) where oscillations can “hide” is given by c = 0. The hypotheses
(4), however, yield also the existence of non-oscillatory solutions to (3) that
obey the description (5) for c = 0. See in this respect [6, Proposition 1].
3 Oscillation lemmas
Consider the following ordinary differential equation
dz
+p(s)z +q(s) = 0, s ≥ s > 0, (6)
0
ds
where the coefficients p,q : [s ,+∞) → R are continuous and
0
p(s) ≥ 0, s ≥ s .
0
Lemma 1 Assume that p,q ∈ L1((s ,+∞),R) and there exists the un-
0
bounded from above, increasing sequence (a ) of numbers from [s ,+∞)
m m≥1 0
such that
q(s) < 0, s ∈ (a ,a ), q(s) > 0, s ∈ (a ,a ),
2m 2m+1 2m+1 2m+2
and
a2m+1 a2m+2
|q(s)|ds > 3 q(s)ds, (7)
Za2m Za2m+1
and
a2m+2 +∞
q(s)ds > 2 |q(s)|ds. (8)
Za2m+1 Za2m+2
Then, the equation (6) has an oscillatory solution z such that lim z(s) =
s→+∞
0 and
z(a ) < 0, z(a ) > 0
2m 2m+1
for all the m’s great enough.
Induced oscillations to ODE’s 4
Proof. The function z : [s ,+∞) → R given by the formula
0
s +∞ τ
z(s) = exp − p(τ)dτ · q(τ)exp p(ξ)dξ dτ, (9)
(cid:18) Zs0 (cid:19) Zs (cid:18)Zs0 (cid:19)
where s ≥ s , is a solution of equation (6).
0
Take m ≥ 1 great enough to verify
0
3 +∞
ln > p(s)ds, m ≥ m . (10)
0
2
Za2m
In particular, we have the estimates
a2m+2
3 > 2exp p(s)ds
(cid:18)Za2m (cid:19)
which implies that
a2m a2m+2
3exp p(s)ds > 2exp p(s)ds , (11)
(cid:18)Zs0 (cid:19) (cid:18)Zs0 (cid:19)
and respectively
+∞
2 > exp p(s)ds
(cid:18)Za2m+1 (cid:19)
which implies that
a2m+1 +∞
2exp p(s)ds > exp p(s)ds . (12)
(cid:18)Zs0 (cid:19) (cid:18)Zs0 (cid:19)
Now, by means of (7), (11), we deduce that
a2m+1 s
|q(s)|exp p(τ)dτ ds
Za2m (cid:18)Zs0 (cid:19)
a2m+1 a2m
≥ |q(s)|ds·exp p(τ)dτ
Za2m (cid:18)Zs0 (cid:19)
a2m+2 a2m
≥ 3 q(s)ds·exp p(τ)dτ
Za2m+1 (cid:18)Zs0 (cid:19)
a2m+2 a2m+2
> 2exp p(τ)dτ · q(s)ds
(cid:18)Zs0 (cid:19) Za2m+1
a2m+2 s
≥ 2 q(s)exp p(τ)dτ ds
Za2m+1 (cid:18)Zs0 (cid:19)
Induced oscillations to ODE’s 5
and, by means of (8), (12), that
a2m+2 s
q(s)exp p(τ)dτ ds
Za2m+1 (cid:18)Zs0 (cid:19)
a2m+2 a2m+1
≥ q(s)ds·exp p(τ)dτ
Za2m+1 (cid:18)Zs0 (cid:19)
+∞ a2m+1
> 2 |q(s)|ds·exp p(τ)dτ
Za2m+2 (cid:18)Zs0 (cid:19)
+∞ +∞
≥ exp p(τ)dτ · |q(s)|ds
(cid:18)Zs0 (cid:19) Za2m+2
+∞ s
≥ |q(s)|exp p(τ)dτ ds.
Za2m+2 (cid:18)Zs0 (cid:19)
Finally,
a2m +∞ τ
z(a )·exp p(τ)dτ = q(τ)exp p(ξ)dξ dτ
2m
(cid:18)Zs0 (cid:19) Za2m (cid:18)Zs0 (cid:19)
a2m+1 a2m+2 +∞ τ
= + + q(τ)exp p(ξ)dξ dτ
(cid:18)Za2m Za2m+1 Za2m+2(cid:19) (cid:18)Zs0 (cid:19)
a2m+1 a2m+2 +∞
≤ − |q(τ)|+ q(τ)+ |q(τ)|
(cid:18) Za2m Za2m+1 Za2m+2 (cid:19)
τ
×exp p(ξ)dξ dτ
(cid:18)Zs0 (cid:19)
a2m+1 a2m+2 τ
< − |q(τ)|+2 q(τ) exp p(ξ)dξ dτ
(cid:18) Za2m Za2m+1 (cid:19) (cid:18)Zs0 (cid:19)
< 0
and respectively
a2m+1 +∞ τ
z(a )exp p(τ)dτ = q(τ)exp p(ξ)dξ dτ
2m+1
(cid:18)Zs0 (cid:19) Za2m+1 (cid:18)Zs0 (cid:19)
a2m+2 +∞ τ
= + q(τ)exp p(ξ)dξ dτ
(cid:18)Za2m+1 Za2m+2(cid:19) (cid:18)Zs0 (cid:19)
a2m+2 +∞ τ
≥ q(τ)− |q(τ)| exp p(ξ)dξ dτ
(cid:18)Za2m+1 Za2m+2 (cid:19) (cid:18)Zs0 (cid:19)
> 0.
(cid:3)
The proof is complete.
Induced oscillations to ODE’s 6
Lemma 2 Assume that the hypotheses of Lemma 1 hold. Suppose also that
a2m+1 a2 +∞
(s−a )|q(s)|ds > 6· 2m+1 |q(s)|ds (13)
2m
a
Za2m 2m Za2m+1
and that
a2m+2 a2 +∞
(s−a )q(s)ds > 4· 2m+2 |q(s)|ds (14)
2m+1
a
Za2m+1 2m+1 Za2m+2
for all the m’s great enough.
Then, the function h : [s ,+∞) → R with the formula
0
+∞ z(τ)
h(s) = −s dτ, s ≥ s , (15)
τ2 0
Zs
where z is the oscillatory solution (9), is itself oscillatory, that is
h(a ) > 0, h(a ) < 0
2m 2m+1
for all the m’s great enough.
Proof. Recall the definition (10) of m .
0
Step 1. We shall establish that h(a ) > 0.
2m
As before,
+∞ z(s) a2m+1 a2m+2 +∞ z(s)
ds = + + ds. (16)
s2 s2
Za2m (cid:18)Za2m Za2m+1 Za2m+2(cid:19)
Notice that the third term of the decomposition can be estimated by
+∞ z(s) +∞ |z(s)|
ds ≤ ds
s2 s2
(cid:12)Za2m+2 (cid:12) Za2m+2
(cid:12)(cid:12) (cid:12)(cid:12) +∞ 1 +∞ +∞
(cid:12) (cid:12) ≤ exp p(τ)dτ |q(τ)|dτds
s2
Za2m+2 (cid:18)Zs (cid:19)Zs
+∞ ds +∞ +∞
≤ ·exp p(τ)dτ · |q(τ)|dτ
s2
Za2m+2 (cid:18)Za2m+2 (cid:19) Za2m+2
3/2 +∞ 2 +∞
< |q(τ)|dτ < |q(τ)|dτ,
a a
2m+2 Za2m+2 2m+1 Za2m+2
where m ≥ m .
0
Induced oscillations to ODE’s 7
The first term of the decomposition (16) can be decomposed further into
a2m+1 z(s) a2m+1 1 s
ds = exp − p(τ)dτ
s2 s2
Za2m Za2m (cid:18) Zs0 (cid:19)
a2m+1 +∞ τ
× + q(τ)exp p(ξ)dξ dτds.
(cid:18)Zs Za2m+1(cid:19) (cid:18)Zs0 (cid:19)
The second part of this decomposition is estimated by
a2m+1 1 s +∞ τ
exp − p(τ)dτ q(τ)exp p(ξ)dξ dτds
s2
(cid:12)Za2m (cid:18) Zs0 (cid:19)Za2m+1 (cid:18)Zs0 (cid:19) (cid:12)
(cid:12)(cid:12) a2m+1 ds +∞ +∞ (cid:12)(cid:12)
(cid:12)≤ ·exp p(τ)dτ · |q(τ)|dτ (cid:12)
s2
Za2m (cid:18)Za2m (cid:19) Za2m+1
3/2 +∞ 2 +∞
< |q(τ)|dτ < |q(τ)|dτ,
a a
2m Za2m+1 2m Za2m+1
where m ≥ m , while the first part is a negative quantity.
0
The second term of the decomposition (16) is estimated by
a2m+2 z(s) a2m+2 ds +∞ +∞
ds ≤ ·exp p(τ)dτ · |q(τ)|dτ
s2 s2
(cid:12)Za2m+1 (cid:12) Za2m+1 (cid:18)Za2m+1 (cid:19) Za2m+1
(cid:12)(cid:12) 3/2 +∞ (cid:12)(cid:12) 2 +∞
<(cid:12) (cid:12)|q(τ)|dτ < |q(τ)|dτ,
a a
2m+1 Za2m+1 2m+1 Za2m+1
where m ≥ m .
0
Collecting all the estimates, we deduce that
+∞ z(s)
ds
s2
Za2m
a2m+1 1 s a2m+1 τ
< exp − p(τ)dτ q(τ)exp p(ξ)dξ dτds
s2
Za2m (cid:18) Zs0 (cid:19)Zs (cid:18)Zs0 (cid:19)
2 2 +∞ 2 +∞
+ + |q(τ)|dτ + |q(τ)|dτ
a a a
(cid:18) 2m 2m+1(cid:19)Za2m+1 2m+1 Za2m+2
a2m+1 1 s a2m+1 τ
< exp − p(τ)dτ q(τ)exp p(ξ)dξ dτds
s2
Za2m (cid:18) Zs0 (cid:19)Zs (cid:18)Zs0 (cid:19)
6 +∞
+ |q(τ)|dτ.
a
2m Za2m+1
Finally, notice that
a2m+1 1 s a2m+1 τ
exp − p(τ)dτ q(τ)exp p(ξ)dξ dτds
s2
(cid:12)Za2m (cid:18) Zs0 (cid:19)Zs (cid:18)Zs0 (cid:19) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Induced oscillations to ODE’s 8
a2m+1 1 s a2m+1 τ
= exp − p(τ)dτ |q(τ)|exp p(ξ)dξ dτds
s2
Za2m (cid:18) Zs0 (cid:19)Zs (cid:18)Zs0 (cid:19)
a2m+1 1 s a2m+1 s
≥ exp − p(τ)dτ |q(τ)|dτ exp p(ξ)dξ ds
s2
Za2m (cid:18) Zs0 (cid:19)Zs (cid:18)Zs0 (cid:19)
a2m+1 1 a2m+1 a2m+1 d 1 a2m+1
≥ |q(τ)|dτds = − |q(τ)|dτds
s2 ds s
Za2m Zs Za2m (cid:18) (cid:19)Zs
1 a2m+1 a2m+1 |q(s)|
= |q(τ)|dτ − ds
a s
2m Za2m Za2m
a2m+1 s−a 1
2m
= |q(s)|ds·
s a
Za2m 2m
1 a2m+1
> · (s−a )|q(s)|ds
2m
a a
2m+1 2m Za2m
1 a2 +∞ a +∞
> ·6 2m+1 |q(s)|ds = 6 2m+1 |q(s)|ds
a a a a2
2m+1 2m 2m Za2m+1 2m Za2m+1
6 +∞
> |q(s)|ds.
a
2m Za2m+1
In conclusion, since q is negative valued in (a ,a ), we get that
2m 2m+1
+∞ z(s)
ds
s2
Za2m
a2m+1 1 s a2m+1 τ
< − exp − p(τ)dτ |q(τ)|exp p(ξ)dξ dτds
s2
Za2m (cid:18) Zs0 (cid:19)Zs (cid:18)Zs0 (cid:19)
6 +∞
+ |q(τ)|dτ
a
2m Za2m+1
< 0.
The first step is complete.
Step 2. We shall establish that h(a ) < 0.
2m+1
First,
+∞ z(s) a2m+2 +∞ z(s)
ds = + ds. (17)
s2 s2
Za2m+1 (cid:18)Za2m+1 Za2m+2(cid:19)
The second term of the decomposition is estimated by
+∞ z(s)
ds
s2
(cid:12)Za2m+2 (cid:12)
(cid:12)(cid:12) +∞ ds (cid:12)(cid:12) +∞ +∞
(cid:12)≤ ·e(cid:12)xp p(τ)dτ · |q(s)|ds
s2
Za2m+2 (cid:18)Za2m+2 (cid:19) Za2m+2
Induced oscillations to ODE’s 9
2 +∞
< |q(s)|ds,
a
2m+2 Za2m+2
where m ≥ m .
0
The first term of the decomposition (17) is decomposed further into
a2m+2 z(s) a2m+2 1 s
ds = exp − p(τ)dτ
s2 s2
Za2m+1 Za2m+1 (cid:18) Zs0 (cid:19)
a2m+2 +∞ τ
× + q(τ)exp p(ξ)dξ dτds.
(cid:18)Zs Za2m+2(cid:19) (cid:18)Zs0 (cid:19)
The second part of this decomposition is estimated by
a2m+2 1 s +∞ τ
exp − p(τ)dτ q(τ)exp p(ξ)dξ dτds
s2
(cid:12)Za2m+1 (cid:18) Zs0 (cid:19)Za2m+2 (cid:18)Zs0 (cid:19) (cid:12)
(cid:12)(cid:12) a2m+2 ds +∞ +∞ (cid:12)(cid:12)
(cid:12)≤ ·exp p(τ)dτ · |q(s)|ds (cid:12)
s2
Za2m+1 (cid:18)Za2m+1 (cid:19) Za2m+2
2 +∞
< |q(s)|ds,
a
2m+1 Za2m+2
where m ≥ m .
0
Finally, notice that
a2m+2 1 s a2m+2 τ
exp − p(τ)dτ q(τ)exp p(ξ)dξ dτds
s2
Za2m+1 (cid:18) Zs0 (cid:19)Zs (cid:18)Zs0 (cid:19)
a2m+2 1 s a2m+2 s
≥ exp − p(τ)dτ q(τ)dτ exp p(ξ)dξ ds
s2
Za2m+1 (cid:18) Zs0 (cid:19)Zs (cid:18)Zs0 (cid:19)
a2m+2 1 a2m+2
≥ q(τ)dτds
s2
Za2m+1 Zs
a2m+2 1
≥ (s−a )q(s)ds·
2m+1
a a
Za2m+1 2m+2 2m+1
a +∞ 4 +∞
2m+2
> 4 |q(s)|ds > |q(s)|ds.
a2 a
2m+1 Za2m+2 2m+1 Za2m+2
In conclusion, since q is positive valued in (a ,a ), we get that
2m+1 2m+2
+∞ z(s)
ds
s2
Za2m+1
a2m+2 1 s a2m+2 τ
> exp − p(τ)dτ q(τ)exp p(ξ)dξ dτds
s2
Za2m+1 (cid:18) Zs0 (cid:19)Zs (cid:18)Zs0 (cid:19)
Induced oscillations to ODE’s 10
4 +∞
− |q(s)|ds
a
2m+1 Za2m+2
> 0.
(cid:3)
The second step is complete.
Observe that
d h(s) z(s)
= , s ≥ s .
ds s s2 0
(cid:18) (cid:19)
Concluding this section, the function h given by
h(s) +∞ h(τ) ′ +∞ z(τ)
= − dτ = − dτ
s τ τ2
Zs (cid:18) (cid:19) Zs
is an oscillatory solution of the equation (3). Since, according to Lemma 1,
lim z(s) = 0, an application of the L’Hoˆpital rule yields
s→+∞
+∞ z(τ)
lim h(s) = lim s dτ = 0.
s→+∞ s→+∞ τ2
Zs
4 Example
We start with an auxiliary result.
Lemma 3 Set C > 0. There exists ε ∈ (0,1) small enough such that
εm2 +∞
C · > kεk2, m ≥ 1. (18)
m
k=m+1
X
Proof. The inequality can be recast as
+∞
C
> (m+p)ε(m+p)2−m2, m ≥ 1. (19)
m
p=1
X
Notice that (m+p)2 −m2 = p2 +2mp ≥ 2m+p for all m,p ≥ 1.
Since ε ∈ (0,1), we deduce that
+∞ +∞
(m+p)ε(m+p)2−m2 < εm ·ε (m+p)εm+p−1
p=1 p=1
X X
+∞
ε
< εm ·ε qεq−1 = εm · ,
(1−ε)2
q=1
X