Electronic Journal of Differential Equations, Vol. 2003(2003), No. 47, pp. 1–25.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp ejde.math.swt.edu (login: ftp)
OSCILLATION FOR EQUATIONS WITH POSITIVE AND
NEGATIVE COEFFICIENTS AND DISTRIBUTED DELAY II:
APPLICATIONS
LEONID BEREZANSKY & ELENA BRAVERMAN
Abstract. We apply the results of our previous paper “Oscillation of equations with positive and negative coefficients and distributed delay I: General
results” to the study of oscillation properties of equations with several delays
and positive and negative coefficients
ẋ(t) +
n
X
ak (t)x(hk (t)) −
k=1
m
X
bl (t)x(gl (t)) = 0,
ak (t) ≥ 0, bl (t) ≥ 0,
l=1
integrodifferential equations with oscillating kernels and mixed equations combining two above equations. Comparison theorems, explicit non-oscillation and
oscillation results are presented.
1. Introduction
The study of oscillation properties of delay differential equations with positive
and negative coefficients began in the eighties [12, 14] and was inspired by the study
of equations with oscillating coefficients. This research was later developed in [7, 13],
neutral equations with positive and negative coefficients were studied in [10, 15, 17],
see also recent publications [1, 11, 20]. In [14, 6, 7] the first order equation with
two constant concentrated delays and a positive and a negative coefficient was
studied, while paper [19] considered oscillation of integrodifferential equations with
oscillatory kernels.
In [6] for the equation
ẋ(t) + a(t)x(t − τ ) − b(t)x(t − σ) = 0,
t ≥ t0 ,
(1.1)
where a(t) ≥ 0, b(t) ≥ 0 are continuous functions, τ > σ > 0, the following result
was obtained: Suppose
Z t
b(s)ds ≤ 1, a(t) ≥ b(t − τ + σ),
(1.2)
t−τ +σ
2000 Mathematics Subject Classification. 34K11, 34K15.
Key words and phrases. Oscillation, non-oscillation, distributed delay, equations with
several delays, integrodifferential equations.
c
2003
Southwest Texas State University.
Submitted February 14, 2003. Published April 24, 2003.
L. Berezansky was partially supported by the Israeli Ministry of Absorption.
E. Braverman was partially supported by an URGC Research Grant.
1
2
LEONID BEREZANSKY & ELENA BRAVERMAN
Z
t
[a(s) − b(s − τ + σ)]ds >
lim inf
t→∞
t−τ
EJDE–2003/47
1
.
e
Then all solutions of (1.1) are oscillatory.
In [17] the inequality (1.3) was improved:
Z
Z t
1
1 t
lim inf
[a(s) − b(s − τ + σ)]ds +
b(s − τ )ds > .
t→∞
e t−τ +σ
e
t−τ
(1.3)
(1.4)
Recently numerous publications on the oscillation of delay equations with positive and negative coefficients have appeared (in addition to [1, 4, 5, 11, 16, 20]
see [9] and references therein). However all the publications except [8, 4] consider
equations with constant delays only. Paper [4] deals with a more general case when
the delays are not constant.
Our previous paper [3] gave a general insight into the problem. In [3] we considered the equation with a distributed delay
Z t
Z t
ẋ(t) +
x(s) ds R(t, s) −
x(s) ds T (t, s) = 0, t ≥ 0,
(1.5)
0
0
where both R(t, s) and T (t, s) are nondecreasing in s for each t.
As special cases, (1.5) includes delay differential equations with variable concentrated delays, integrodifferential equations and mixed differential equations. The
basic result of the paper [3] was the relation between the following properties for
(1.5): the existence of a nonoscillatory solution of (1.5), the existence of an eventually positive solution of the corresponding differential inequality and the existence
of a nonnegative solution of some nonlinear integral inequality which is explicitly
constructed by (1.5). Theorems of this kind are well known and widely applied for
delay differential equations with positive coefficients.
In the present paper we apply general results obtained in [3] to specific classes of
equations. Section 2 contains preliminaries and relevant results from paper [3]. In
Section 3 equations with positive and negative coefficients and several concentrated
delays are considered. In Section 4 oscillation and nonoscillation results are deduced
for integrodifferential equations with oscillatory kernels. Section 5 deals with mixed
equations containing both several concentrated delays and an integral term.
2. Preliminaries and General Results
We consider a scalar delay differential equation (1.5) under the following conditions:
(a1) R(t, ·), T (t, ·) are left continuous functions of bounded variation and for
each s their variations on the segment [0, s]
PR (t, s) = varτ ∈[0,s] R(t, τ ), PT (t, s) = varτ ∈[0,s] T (t, τ )
(2.1)
are locally integrable functions in t, R(t, s) = R(t, t+), T (t, s) = T (t, t+),
t < s;
(a2) R(t, ·), T (t, ·) are nondecreasing functions for each t, R(t, s) ≥ T (t, s) for
each t, s;
(a3) For each t1 there exist s1 = s(t1 ) ≤ t1 , r1 = r(t1 ) ≤ t1 , such that R(t, s) = 0
for s < s1 , t > t1 , T (t, s) = 0 for s < r1 , t > t1 ; in addition, functions s(t),
r(t) satisfy
lim s(t) = ∞, lim r(t) = ∞.
t→∞
t→∞
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OSCILLATION FOR EQUATIONS WITH . . .
If (a3) holds then we can introduce the following functions
h(t) = inf s : R(t, s) 6= 0 , g(t) = inf s : T (t, s) 6= 0 ,
s
s
3
(2.2)
such that limt→∞ h(t) = ∞, limt→∞ g(t) = ∞, and (1.5) can be rewritten as
Z t
Z t
ẋ(t) +
x(s) ds R(t, s) −
x(s) ds T (t, s) = 0, t ≥ 0.
(2.3)
h(t)
g(t)
If (a2) and (a3) hold, then obviously h(t) ≤ g(t).
Together with (2.3) we consider for each t0 ≥ 0 an initial-value problem
Z t
Z t
ẋ(t) +
x(s) ds R(t, s) −
x(s) ds T (t, s) = 0, t ≥ t0 .
h(t)
(2.4)
g(t)
x(t) = ϕ(t), t < t0 , x(t0 ) = x0 ,
(2.5)
where ϕ(t) is a Borel measurable bounded function.
Definition. An absolutely continuous on each interval [t0 , c] function x : R → R is
called a solution of problem (2.4), (2.5), if it satisfies (2.4) for almost all t ∈ [t0 , ∞)
and equalities (2.5) for t ≤ t0 .
Definition. For each s ≥ 0 solution X(t, s) of the problem
Z t
Z t
ẋ(t)+
x(s) ds R(t, s)−
x(s) ds T (t, s) = 0, x(t) = 0, t < s, x(s) = 1, (2.6)
h(t)
g(t)
is called a fundamental function of (2.3).
We assume X(t, s) = 0, 0 ≤ t < s.
Definition. We will say that equation (2.3) has a nonoscillatory solution if for
some t0 , ϕ(t) and x0 the solution of (2.4)-(2.5) is eventually positive or eventually
negative. Otherwise all solutions of this equation are oscillatory.
Below we present the results obtained in [3] on oscillation of equation (2.3) with
a distributed delay.
Consider together with (2.3) the following delay differential inequality
Z t
Z t
ẏ(t) +
y(s) ds R(t, s) −
y(s) ds T (t, s) ≤ 0, t ≥ 0.
(2.7)
h(t)
g(t)
The following theorem establishes sufficient nonoscillation conditions.
Lemma 2.1. [3] Suppose (a1)-(a3) hold. Consider the following hypotheses:
(1) There exists t1 ≥ 0 such that for t ≥ t1 the following inequality
Z t
Z
Z t
Z
t
t
u(t) ≥
exp
u(τ )dτ ds R(t, s) −
exp
u(τ )dτ ds T (t, s) (2.8)
h(t)
s
g(t)
s
has a nonnegative locally integrable solution (we assume u(t) = 0 for t <
t1 );
(2) There exists t2 ≥ 0 such that X(t, s) > 0, t ≥ s ≥ t2 ;
(3) Equation (2.3) has a nonoscillatory solution;
(4) Inequality (2.7) has an eventually positive solution.
Then the implication (1) ⇒ (2) ⇒ (3) ⇒ (4) is valid.
Necessary nonoscillation (sufficient oscillation) conditions require some more constraints on R, T . Let
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LEONID BEREZANSKY & ELENA BRAVERMAN
EJDE–2003/47
(a4) for any t R(t, s) − T (t, s − h(t) + g(t)) is nondecreasing in s and
lim sup T (t, t+)[g(t) − h(t)] ≤ l < 1.
t→∞
Lemma 2.2 ([3]). Suppose (a1)–(a4) hold. Then hypotheses (1)–(4) of Theorem
1 are equivalent.
To obtain other necessary oscillation conditions we consider the following form
of equation (2.3):
n Z t
m Z t
X
X
ẋ(t) +
x(s) ds Rk (t, s) −
x(s) ds Tl (t, s) = 0, t ≥ 0,
(2.9)
k=1
hk (t)
l=1
gl (t)
where Rk , Tl , hk , gl satisfy the following conditions:
(a1? ) Rk (t, ·), Tl (t, ·) are left continuous functions of bounded variation and for
each s their variations on the segment [0, s] PRk (t, s), PTl (t, s) are locally
integrable functions in t, Rk (t, s) = Rk (t, t+), Tl (t, s) = Tl (t, t+), t < s;
(a2? ) R
are nondecreasing functions for each t,
Pk (t, ·), Tl (t, ·)P
R
(t,
s)
≥
k k
l Tl (t, s) for each t, s;
(a3? ) For each k, l limt→∞ hk (t) = ∞, limt→∞ gl (t) = ∞.
Denote
R(t, s) =
n
X
Rk (t, s), T (t, s) =
k=1
m
X
Tl (t, s),
(2.10)
l=1
h(t) = max hk (t), g(t) = min gl (t).
k
l
(2.11)
Let us also introduce the following additional constraints:
(add1) m = n, Rk (t, s) ≥ Tk (t, s) for each t, s, k = 1, . . . , n; for any t, k, function
Rk (t, s) − Tk (t, s − hk (t) + gk (t)) is nondecreasing in s and
lim sup
t→∞
n
X
Tk (t, t+)[gk (t) − hk (t)] < 1.
k=1
(add2) h(t) ≤ g(t), R(t, s) > T (t, s), R(t, s) − T (t, s − h(t) + g(t)) is nondecreasing
in s for any t and
n
X
lim sup T (t, t+)[g(t) − h(t)] +
Rk (t, t+) h(t) − hk (t)
t→∞
k=1
+
m
X
(2.12)
Tl (t, t+) gl (t) − g(t) < 1.
l=1
Consider together with (2.9) the delay differential inequality
n Z t
m Z t
X
X
ẏ(t) +
y(s) ds Rk (t, s) −
y(s) ds Tl (t, s) ≤ 0, t ≥ 0.
k=1
hk (t)
l=1
(2.13)
gl (t)
The next lemma establishes non-oscillation criteria for (2.9).
Lemma 2.3. [3] Suppose Rk , Tl , hk , gk satisfy (a1? )-(a3? ) and at least one of conditions (add1), (add2) hold. Then the following hypotheses are equivalent:
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OSCILLATION FOR EQUATIONS WITH . . .
5
(1) There exists t1 ≥ 0 such that the inequality
Z
n Z t
X
t
u(t) ≥
exp
u(τ )dτ ds Rk (t, s)
k=1 hk (t)
m Z t
X
s
−
l=1
exp
gl (t)
Z
(2.14)
t
u(τ )dτ ds Tl (t, s),
t ≥ t1 ,
s
has a nonnegative locally integrable solution (we assume u(t) = 0 for t <
t1 );
(2) There exists t2 ≥ 0 such that X(t, s) > 0, t ≥ s ≥ t2 ;
(3) Equation (2.9) has a nonoscillatory solution;
(4) Inequality (2.13) has an eventually positive solution.
Lemmas 2.1–2.3 yield the following comparison result. Let us compare the oscillation properties of the equation
n Z t
m Z t
X
X
ẋ(t) +
x(s) ds Lk (t, s) −
x(s) ds Dl (t, s) = 0,
(2.15)
k=1
h̃k (t)
l=1
g̃l (t)
to the oscillation properties of (2.9).
Lemma 2.4 ([3]).
(1) If (a1? ) − (a3? ) and anyone of the conditions (add1),
(add2) hold for (2.15) (where Rk , Tl are changed by Lk , Dl ), Lk (t, s) ≥
Rk (t, s), Dl (t, s) ≤ Tl (t, s) and (2.15) has a nonoscillatory solution, then
(2.9) has a nonoscillatory solution.
(2) If (a1? ) − (a3? ) and any one of the conditions (add1),(add2) hold for (2.9),
Lk (t, s) ≤ Rk (t, s), Dl (t, s) ≥ Tl (t, s) and all solutions of (2.15) are oscillatory, then all solutions of (2.9) are oscillatory.
Lemma 2.5 describes the asymptotic behavior of nonoscillatory solutions of (2.9).
Lemma 2.5 ([3]). Suppose (a1? )-(a3? ) and anyone of the following conditions
holds:
(1) (add1) is satisfied and for some k
Z ∞
Rk (t, t+) − Tk (t, t+) dt = ∞;
(2.16)
0
(2) (add2) is satisfied and
Z ∞
R(t, t+) − T (t, t+) dt = ∞.
(2.17)
0
Then any nonoscillatory solution x of (2.9) satisfies limt→∞ x(t) = 0.
Consider the following two equations
Z t
x(s) ds R(t, s) − T (t, s − h(t) + g(t))
ẋ(t) +
h(t)
Z
t
+
x(s) exp
g(t)
Z
s
s+h(t)−g(t)
[R(τ, τ +) − T (τ, τ +)]dτ − 1 ds T (t, s) = 0
(2.18)
6
LEONID BEREZANSKY & ELENA BRAVERMAN
EJDE–2003/47
and
t
Z
s
Z
x(s) exp
ẋ(t) +
g(t)
[R(τ, τ +) − T (τ, τ +)]dτ − 1
s−g(t)+h(t)
t
Z
(2.19)
x(s) ds R(t, s − g(t) + h(t)) − T (t, s) = 0.
×ds R(t, s − g(t) + h(t)) +
g(t)
If (a1)-(a4) hold, then equations (2.18),(2.19) contains terms with positive coefficients only. The oscillation properties of these equations will be compared to the
properties of (2.3).
Lemma 2.6 ([3]). Suppose (a1)-(a4) hold for (2.3). If all solutions of either (2.18)
or (2.19) are oscillatory, then all solutions of (2.3) are also oscillatory.
Corollary 2.7. Suppose (a1)-(a4) hold for (2.3) and at least one of the following
four inequalities is satisfied:
nZ t (1) lim inf
R(t, τ ) − T (t, τ − h(t) + g(t)) dτ
t→∞
Z
t
h(t)
+
dτ
g(t)
(2)
τ
Z
exp
t
nZ
lim inf
t→∞
R(t, τ +) − T (t, τ − h(t) + g(t)) dτ
h(t)
t
+
Z
τ
dτ
g(t)
s
Z
o 1
[R(u, u+) − T (u, u+)]du dsT (t, s) >
e
s+h(t)−g(t)
h(τ )
t
nZ
t→∞
τ
Z
dτ
g(t)
lim inf
s
o 1
[R(u, u+) − T (u, u+)]du − 1 ds T (t, s) >
e
s+h(t)−g(t)
h(τ )
Z
(3)
Z
exp
Z
g(τ )
s
[R(u, u+) − T (u, u+)]du − 1
s−g(t)+h(t)
t Z
× ds R(t, s − g(t) + h(t)) +
o 1
R(t, τ − g(t) + h(t)) − T (t, τ ) dτ >
e
g(t)
(4)
t
nZ
lim inf
t→∞
Z
τ
dτ
g(t)
s
Z
g(τ )
[R(u, u+) − T (u, u+)]du
s−g(t)+h(t)
Z
× ds R(t, s − g(t) + h(t)) +
t
1
R(t, τ − g(t) + h(t)) − T (t, τ ) >
e
g(t)
Then all solutions of (2.3) are oscillatory.
Similar results can be obtained for (2.9).
Lemma 2.8 ([3]). Suppose Rk , Tl , hk , gl satisfy (a1? )-(a3? ) and (add1) holds. If
all solutions of either
n Z t
X
ẋ(t) +
x(s) ds Rk (t, s) − Tk (t, s − hk (t) + gk (t))
k=1
t
+
hk (t)
n Z
X
k=1
Z
x(s) exp
gk (t)
× ds Tk (t, s) = 0
s
s+hk (t)−gk (t)
[Rk (τ, τ +) − Tk (τ, τ +)]dτ − 1
(2.20)
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7
or
ẋ(t) +
n Z
X
k=1
t
Z
x(s) exp
gk (t)
s
[Rk (τ, τ +) − Tk (τ, τ +)] × dτ − 1
s−gk (t)+hk (t)
× ds Rk (t, s − gk (t) + hk (t))
t
h
i
+
x(s) ds Rk (t, s − gk (t) + hk (t)) − Tk (t, s) = 0
n Z
X
gk (t)
k=1
(2.21)
are oscillatory, then all solutions of (2.9) are also oscillatory.
Corollary 2.9. Suppose (a1? )-(a3? ) and (add1) hold for (2.9) and at least one of
the following inequalities is satisfied:
Z
Z t
n
X
t (1) lim inf
Rk (t, τ ) − T (t, τ − hk (t) + gk (t)) dτ +
dτ
t→∞
Z
hk (t)
k=1
τ
×
exp
Z
1
[Rk (u, u+) − Tk (u, u+)]du − 1 ds Tk (t, s) >
e
s+hk (t)−gk (t)
hk (τ )
(2)
lim inf
t→∞
Z
Z
n
X
Z
+
(3)
t
Rk (t, τ +) − Tk (t, τ − hk (t) + gk (t)) dτ
τ
Z
dτ
gk (t)
lim inf
t→∞
hk (τ )
Z
n
X
k=1
hk (t)
k=1
t
gk (t)
s
s
1
[Rk (u, u+) − Tk (u, u+)]du dsTk (t, s) >
e
s+hk (t)−gk (t)
t
Z
τ
dτ
gk (t)
exp
gk (τ )
Z
s
[Rk (u, u+) − Tk (u, u+)]
s−gk (t)+hk (t)
× du − 1 ds Rk (t, s − gk (t) + hk (t))
Z t
1
+
Rk (t, τ − gk (t) + hk (t)) − Tk (t, τ ) dτ >
e
gk (t)
(4)
lim inf
t→∞
Z
n
X
k=1
t
gk (t)
Z
τ
Z
s
[Rk (u, u+) − Tk (u, u+)]du
dτ
gk (τ )
s−gk (t)+hk (t)
Z
× ds Rk (t, s − gk (t) + hk (t)) +
t
1
Rk (t, τ − gk (t) + hk (t)) − Tk (t, τ ) >
e
gk (t)
Then all solutions of (2.3) are oscillatory.
Let us proceed with nonoscillation conditions.
Lemma 2.10 ([3]). Suppose (a1)-(a4) hold for (2.3) and there exists λ, 0 < λ < 1,
such that
Z g(t)
1 1
lim sup
[R(s, s+) − λT (s, s+)] ds < ln ,
(2.22)
e λ
t→∞
h(t)
Z t
1
lim sup
[R(s, s+) − λT (s, s+)]ds < .
(2.23)
e
t→∞
h(t)
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LEONID BEREZANSKY & ELENA BRAVERMAN
EJDE–2003/47
Then (2.3) has a nonoscillatory solution.
Lemma 2.11 ([3]). Suppose n = m, conditions (a1? )-(a3? ), (add1) and the following inequality
n Z t
X
1
1
lim sup
Rk (s, s+) − Tk (s, s+) ds <
(2.24)
e
e
t→∞
hk (t)
k=1
hold. Then (2.9) has a nonoscillatory solution.
3. Equations with Concentrated Delays
Let us study oscillation properties of a delay differential equation with several
variable concentrated delays
n
m
X
X
ẋ(t) +
ak (t)x(hk (t)) −
bl (t)x(gl (t)) = 0, t ≥ 0.
(3.1)
k=1
l=1
This equation is a special case of (1.5) when we assume
n
m
X
X
R(t, s) =
ak (t)χ[hk (t),∞) (s), T (t, s) =
bl (t)χ[gl (t),∞) (s),
k=1
(3.2)
l=1
where χ[c,d] is a characteristic function of segment [c, d].
An initial value problem, definitions of a solution, the fundamental solution,
oscillatory and nonoscillatory solutions for equation (3.1) are the same as for (2.9).
The hypotheses of Lemma 2.1 are satisfied for the delay equation (3.1) if the
following conditions hold:
(C1) ak (t) ≥ 0, bl (t) ≥ 0 are Lebesgue measurable essentially locally bounded
functions;
(C2) For any t ≥ s ≥ 0
n
m
X
X
ak (t)χ[hk (t),∞) (s) ≥
bl (t)χ[gl (t),∞) (s).
k=1
l=1
(C3) hk (t), gl (t) : [0, ∞) → R are Lebesgue measurable functions, hk (t) ≤
t, gl (t) ≤ t, limt→∞ hk (t) = ∞, limt→∞ gl (t) = ∞.
Consider the inequality
n
m
X
X
ẏ(t) +
ak (t)y(hk (t)) −
bl (t)y(gl (t)) ≤ 0, t ≥ 0.
(3.3)
k=1
l=1
The following proposition is an immediate consequence of Lemma 2.1.
Proposition 3.1. Suppose (C1)–(C3) hold. Consider the following hypotheses:
(1) There exists t1 ≥ 0 such that the inequality
Z
Z
n
m
X
t
X
t
u(t) ≥
ak (t) exp
u(s)ds −
bl (t) exp
u(s)ds , t ≥ t1 (3.4)
k=1
hk (t)
l=1
gl (t)
has a nonnegative locally integrable solution (we assume u(t) = 0 for t <
t1 );
(2) There exists t2 ≥ 0 such that the fundamental function X(t, s) > 0, t ≥ s ≥
t2 ;
(3) Equation (3.1) has a nonoscillatory solution;
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9
(4) Inequality (3.3) has an eventually positive solution.
Then the implications 1) ⇒ 2) ⇒ 3) ⇒ 4) are valid.
Oscillation criteria for equations with several concentrated delays can be obtained as corollaries of Lemma 2.3. To this end denote
m
n
X
X
bl (t).
h(t) = max hk (t), g(t) = min gl (t), a(t) =
ak (t), b(t)) =
k
l
l=1
k=1
Proposition 3.2 provides oscillation criteria.
Proposition 3.2. Suppose (C1), (C3) and anyone of the following conditions
(C4) m = n, ak ≥ bk , hk (t) ≤ gk (t) for k = 1, . . . , n, t ≥ 0,
n
X
lim sup
bk (t)[gk (t) − hk (t)] < 1
t→∞
k=1
(C5) h(t) ≤ g(t), a(t) ≥ b(t),
n
m
X
X
ak (t)[h(t) − hk (t)] +
bl (t)[gl (t) − g(t)] < 1
lim sup b(t)[g(t) − h(t)] +
t→∞
k=1
l=1
hold. Then all four hypotheses of Proposition 3.1 are equivalent.
Remark. It is to be noted that (C5) is a special case of (C4), if we rewrite the left
hand side of (3.1) as a sum of (n + m + 1) positive and (n + m + 1) negative terms:
n
m
X
X
ẋ(t) +
ak (t)x(hk (t)) −
bl (t)x(gl (t))
k=1
= ẋ(t) +
l=1
n
X
ak (t)x(hk (t)) −
−
bl (t)x(g(t)) +
l=1
ak (t)x(h(t)) +
k=1
k=1
m
X
n
X
m
X
n
X
ak (t)x(h(t))
k=1
bl (t)x(g(t)) −
l=1
m
X
bl (t)x(gl (t))
l=1
= ẋ(t) + a(t)x(h(t)) − b(t)x(g(t)) +
n
X
ak (t) x(hk (t)) − x(h(t))
k=1
+
m
X
bl (t) x(g(t)) − x(gl (t))
l=1
We proceed with comparison results which are deduced from Lemma 2.4. To
this end consider the equation
n
m
X
X
ẋ(t) +
ãk (t)x(h̃k (t)) −
b̃l (t)x(g̃l (t)) = 0, t ≥ 0.
(3.5)
k=1
l=1
Proposition 3.3.
(1) Suppose (C1), (C3) and either (C4) or (C5) hold for
(3.5), where ak , bl , hk , gl are changed by ãk , b̃l , h̃k , g̃l , respectively. If ãk (t) ≥
ak (t), b̃l (t) ≤ bl (t), h̃k (t) ≤ hk (t), g̃l (t) ≥ gl (t) and (3.5) has a nonoscillatory solution, then (3.1) also has a nonoscillatory solution.
(2) Suppose (C1), (C3) and either (C4) or (C5) hold. If ãk (t) ≤ ak (t), b̃l (t) ≥
bl (t), h̃k (t) ≥ hk (t), g̃l (t) ≤ gl (t) and all solutions of (3.5) are oscillatory,
then all solutions of (3.1) are also oscillatory.
10
LEONID BEREZANSKY & ELENA BRAVERMAN
EJDE–2003/47
To apply Proposition 3.3 consider the autonomous delay equation
ẏ +
n
X
ci y(t − δi ) −
i=1
m
X
dl y(t − σl ) = 0, t ≥ 0,
(3.6)
l=1
where the following conditions hold:
(A1) ci ≥ 0, dl ≥ 0, δi ≥ 0, σl ≥ 0
and one of the two following conditions satisfied:
Pn
(A2) n = P
m, ci ≥ di , δi ≥P
σi , i=1 di (δi − σi ) < 1;
n
m
(A3) c = i=1 ci ≥ d = l=1 dl , δ = min δi ≥ σ = max σl ,
d(δ − σ) +
n
X
ci (δi − δ) +
i=1
m
X
dl (σ − σl ) < 1;
l=1
Corollary 3.1.
(1) Suppose (C1)-(C3), (A1) and at least one of (A2), (A3)
hold. If ak (t) ≤ ck , hk (t) ≥ t − δk , bl (t) ≥ dl , gl (t) ≤ t − σl and (3.6) has a
nonoscillatory solution, then (3.1) also has a nonoscillatory solution.
(2) Suppose (C1), (C3), (A1) and at least one of (C4), (C5) hold. If ak (t) ≥
ck , hk (t) ≤ t − δk , bl (t) ≤ dl , gl (t) ≥ t − σl and all solutions of (3.6) are
oscillatory, then all solutions of (3.1) are also oscillatory.
Lemma 2.5 immediately implies the following result on the asymptotic behaviour
of solutions.
Proposition 3.4. Suppose(C1),(C3) and one of the following two conditions is
satisfied
R∞
(1) (C4) holds and there exists such k that 0 [ak (t) − bk (t)]dt = ∞.
R∞
R ∞ Pn
Pm
(2) (C5) holds and 0 [a(t) − b(t)]dt = 0
k=1 ak (t) −
l=1 bl (t) dt = ∞.
Then any nonoscillatory solution of (3.1) satisfies limt→∞ y(t) = 0.
Proposition 3.5. Suppose the hypotheses (C1), (C3), (C4) hold. If all solutions
of anyone of the following equations are oscillatory
ẋ(t) +
n
X
[ak (t) − bk (t)]x(hk (t))
k=1
+
n
X
k=1
ẋ(t) +
Z
bk (t) exp
(3.7)
gk (t)
[ak (s) − bk (s)]ds − 1 x(gk (t)) = 0,
hk (t)
Z
n
X
ak (t) − bk (t) + ak (t) exp
gk (t)
[ak (s) − bk (s)]ds − 1 x(gk (t)) = 0,
hk (t)
k=1
(3.8)
ẋ(t) +
n
X
[ak (t) − bk (t)]x(hk (t)) +
k=1
n
X
Z
gk (t)
[ak (s) − bk (s)]ds = 0,
bk (t)x(gk (t))
hk (t)
k=1
(3.9)
ẋ(t) +
n X
k=1
Z
gk (t)
ak (t) − bk (t) + ak (t)
[ak (s) − bk (s)]ds x(gk (t)) = 0,
hk (t)
then all solutions of (3.1) are also oscillatory.
(3.10)
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OSCILLATION FOR EQUATIONS WITH . . .
11
Remark. Since (C5) is a special case of (C4), similar equations can be presented
if (C1),(C3),(C5) are satisfied.
Corollary 3.2. Suppose (C1), (C3), (C4) are satisfied and at least one of the
following conditions holds:
(1)
lim inf
t→∞
n
nX
[ak (t) − bk (t)](t − hk (t))
k=1
+
n
X
Z
bk (t) exp
hk (t)
k=1
(2)
lim inf
t→∞
n
nX
gk (t)
o 1
[ak (s) − bk (s)]ds − 1 (t − gk (t)) >
e
[ak (t) − bk (t)](t − hk (t))
k=1
+
n
X
Z
hk (t)
k=1
(3)
lim inf
t→∞
n Z
nX
k=1
o 1
[ak (s) − bk (s)]ds >
e
t
ak (s) − bk (s)
gk (t)
Z
gk (s)
o 1
[ak (τ ) − bk (τ )]dτ − 1 ds >
e
hk (s)
Z
Z
n
t
gk (s)
nX
o 1
lim inf
ak (s) − bk (s) + ak (s)
[ak (τ ) − bk (τ )]dτ ds > ,
t→∞
e
gk (t)
hk (s)
+ ak (s) exp
(4)
gk (t)
bk (t)(t − hk (t))
k=1
then all solutions of (3.1) are oscillatory.
Corollary 3.3. Suppose (A1), (A2) and at least one of the following conditions
hold:
Pn (1)
(ak − bk)δk + bk e(ak −bk )(δk−σk ) − 1 σk > 1/e;
Pnk=1
(2)
(a − bk ) δk + bk σk (δk − σk ) > 1/e;
Pnk=1 k
k)
(3)
a − bk + ak e(ak −bk )(δk −σ
− 1 σk > 1/e;
Pnk=1 k
(4)
k=1 (ak − bk ) 1 + ak (δk − σk ) σk > 1/e.
Then all solutions of (3.6) are oscillatory.
Now we proceed with explicit nonoscillation conditions.
Proposition 3.6. Suppose either (C1), (C3), (C4) hold and the following inequality is satisfied
n Z t
X
1
1
lim sup
ak (s) − bk (s) ds < .
e
e
t→∞
hk (t)
k=1
or (C1), (C3), (C5) hold and
n Z
n
1 X t
lim sup 1 −
ak (s) ds
e
t→∞
k=1 hk (t)
Z t X
m
m Z
o 1
n
1X
1 X t
+
ak (s) −
bk (s) ds + 1 −
bl (s) ds < .
e
e
e
h(t)
g(t)
k=1
l=1
Then (3.1) has a nonoscillatory solution.
l=1
12
LEONID BEREZANSKY & ELENA BRAVERMAN
EJDE–2003/47
Denote
H(t) = min hk (t),
G(t) = max gl (t).
k
l
Proposition 3.7. Suppose there exist ã(t), b̃(t), such that
b̃(t) ≤ b(t) ≤ a(t) ≤ ã(t),
where
a(t) =
n
X
ak (t), b(t) =
m
X
k=1
bl (t),
l=1
there exist finite limits
t
Z
Z
B11 = lim
t→∞
ã(s) ds,
t→∞
H(t)
Z t
B21 = lim
t→∞
t
B12 = lim
ã(s) ds,
B22 = lim
t→∞
G(t)
b̃(s) ds,
(3.11)
b̃(s) ds,
(3.12)
H(t)
Z t
G(t)
and (C1)–(C3) hold. Suppose, in addition, that the system
ln x1 > x1 B11 − x2 B12
(3.13)
ln x2 < x1 B21 − x2 B22
(3.14)
has a positive solution (x1 , x2 ) such that eventually x1 ã(t) ≥ x2 b̃(t). Then (3.1)
has a nonoscillatory solution.
Proof. Consider the function u(t) = x1 ã(t) − x2 b̃(t). Then u(t) is nonnegative and
the system (3.13)-(3.14) yields
x1 > exp{x1 B11 − x2 B12 },
x2 < exp{x1 B21 − x2 B22 }.
Thus by definitions (3.11)-(3.12) there exists t1 > 0, such that for t ≥ t1
Z t
Z t
Z
t
x1 ≥ exp x1
ã(s) ds − x2
b̃(s) ds = exp
u(s) ds
H(t)
t
H(t)
t
Z
−x2 ≥ − exp x1
H(t)
Z
ã(s) ds − x2
G(t)
b̃(s) ds = − exp
t
Z
G(t)
u(s) ds
G(t)
After multiplying the first inequality by ã(t), the second one by b̃(t) and summation
we have
u(t) = x1 ã(t) − x2 b̃(t)
Z
Z
t
≥ ã(t) exp
u(s) ds − b̃(t) exp
H(t)
t
≥ a(t) exp
Z
u(s) ds − b(t) exp
=
≥
k=1
n
X
k=1
ak (t) exp
Z
u(s) ds
G(t)
Z
t
u(s) ds −
H(t)
ak (t) exp
u(s) ds
G(t)
t
H(t)
n
X
t
Z
t
hk (t)
u(s) ds −
m
X
bl (t) exp
l=1
m
X
t
Z
u(s) ds
G(t)
bl (t) exp
l=1
By Proposition 3.1, (3.1) has a nonoscillatory solution.
Z
t
u(s) ds .
gl (t)
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Example 1. Consider the equation
n
n
X
t X bk
t ak
x
−
x
= 0, t ≥ t0 > 0,
ẋ(t) +
t
µk
t
νk
k=1
13
(3.15)
k=1
where ak ≥ bk ≥ 0, µk ≥ νk > 1. We apply Corollary 3.2 (Ineq. 4):
Z
n Z t
X
ak − bk
ak s/νk ak − bk dτ ds
lim inf
+
t→∞
s
s s/µk
τ
k=1 t/νk
n Z t
X
ak s
s ak − bk
= lim inf
+
ln
−
(ak − bk ) ds
t→∞
s
s
νk
µk
k=1 t/νk
Z
n
t
X
1 ak µk = lim inf
(ak − bk )
+
ln
ds
t→∞
s
νk
t/νk s
k=1
= lim inf
t→∞
=
n
X
n
X
(ak − bk ) 1 + ak ln
k=1
(ak − bk ) 1 + ak ln
k=1
Thus if
n
X
t µk ln t − ln
νk
νk
µk ln νk
νk
(ak − bk ) 1 + ak ln
k=1
µk 1
ln νk > ,
νk
e
then all solutions of (3.15) are oscillatory. For nonoscillation results, we apply
Proposition 3.6.
n
n Z t
X
X
ak
1 bk 1 t −
ds = lim sup
ak − bk ln t − ln
lim sup
s
e s
e
µk
t→∞
t→∞
t/µk
k=1
k=1
=
n
X
k=1
Pn
1
e bk
1 ak − bk ln µk
e
Thus if k=1 ak −
ln µk < 1/e, then (3.15) has a nonoscillatory solution.
Example 2. Consider the equation
n
X
ak
t b t
ẋ(t) +
x
− x
= 0, t ≥ t0 > 0,
(3.16)
t
µk
t ν
k=1
Pn
where k=1 ak ≥ b ≥ 0, µk ≥ ν > 1. Unlike in Example 1, here the number of
positive terms is not equal to the number of negative terms. This equation can be
rewritten in one of the following forms:
Pn
Pn ak t b t Pn
(1) ẋ(t) + k=1 atk x µtk − k=1 atk x νt +
k=1 t x ν − t x ν = 0
Pn
Pn
(2) ẋ(t) + k=1 atk x µtk − k=1 btk x νt = 0, where b1 = b, bk = 0, k > 1
and a1 P
≥ b.
Pn
Pn
n
(3) ẋ(t) + k=1 atk x µtk − k=1 λkt b x νt = 0, where λk > 0,
k=1 λk = 1
and ak ≥ λk b.
A computation similar to Example 1 yields that if anyone of the following three
conditions holds
Pn
1
(1)
k=1 ak − b ln ν > e
14
LEONID BEREZANSKY & ELENA BRAVERMAN
EJDE–2003/47
Pn
(2) a1 ≥ b and (a1 − b) 1 + a1 ln µν1 ln ν + k=2 ak 1 +ak ln µνk ln ν > 1/e
Pn
(3) For each k ak ≥ λk b and k=1 ak − λk b 1 + ak ln µνk ln ν > 1/e.
then all solutions of (3.16) are oscillatory.
If anyone of the following three conditions holds
Pn
Pn
1
(1)
ak − b ln ν < 1/e,
k +
k=1 ak 1 − e ln µ
k=1
Pn
(2) a1 ≥ b and a1 − 1e b ln µ1 + k=2 ak ln µk < 1e ,
Pn
(3) For each k ak ≥ λk b and k=1 ak − 1e λk b ln µk < 1/e,
then (3.16) has a nonoscillatory solution.
4. Integrodifferential Equations
In this section we will study the following integrodifferential equation
Z t
Z t
ẋ(t) +
K(t, s)x(s) ds −
M (t, s)x(s) ds = 0,
0
(4.1)
0
(4.1) is a special case of (1.5) if we assume
Z s
Z
R(t, s) =
K(t, ζ) dζ, T (t, s) =
0
s
M (t, ζ) dζ.
(4.2)
0
The hypotheses of Lemma 2.1 are satisfied for the delay equation (4.1) if the following conditions hold:
(I1) K(t, s), M (t, s) are Lebesgue integrable over each finite square [0, b] × [0, b]
functions;
(I2) There exist finite functions h(t), g(t) such that h(t) = inf{s|K(t, s) ≥ 0},
g(t) = inf{s|M (t, s) ≥ 0} and limt→∞ h(t) = ∞, limt→∞ g(t) = ∞;
(I3) For each t, s K(t, s) ≥ 0, M (t, s) ≥ 0 and
Z s
Z s
K(t, τ ) dτ ≥
M (t, τ ) dτ.
h(s)
g(s)
For t0 ≥ 0 consider an initial-value problem
Z t
Z t
ẋ(t) +
K(t, s)x(s) ds −
M (t, s)x(s) ds = 0,
h(t)
t > t0 .
(4.3)
g(t)
x(t) = ϕ(t),
t < t0 , x(t0 ) = x0 .
(4.4)
As in the general case we will say that (4.1) has a nonoscillatory solution if for some
t0 ≥ 0, ϕ(t) and x0 the solution of (4.3)-(4.4) is eventually positive or eventually
negative. Otherwise all solutions of (4.1) are oscillatory.
Consider in addition the inequality
Z t
Z
Z t
Z
t
t
u(t) ≥
K(t, s) exp
u(τ )dτ ds −
M (t, s) exp
u(τ )dτ ds. (4.5)
h(t)
s
g(t)
s
Lemma 2.1 immediately implies the following proposition.
Proposition 4.1. Suppose (I1)-(I3) hold. Consider the following hypotheses
(1) There exists t1 ≥ 0 such that inequality (4.5) has a nonnegative locally
integrable solution;
(2) There exists t2 ≥ 0 such that X(t, s) > 0, t ≥ s ≥ t2 , where X(t, s) is a
fundamental function of (4.1);
(3) (4.1) has a nonoscillatory solution;
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15
(4) Inequality
t
Z
t
Z
K(t, s)y(s) ds −
ẏ(t) +
M (t, s)y(s) ds ≤ 0
t0
(4.6)
t0
has an eventually positive solution.
Then the implications 1) ⇒ 2) ⇒ 3) ⇒ 4) are valid.
Now let us apply Lemma 2.2 to (4.1). Let us introduce the following additional
assumption
(I4) K(t, s) ≥ M (t, s − h(t) + g(t)) for each t, s and
Z t
lim sup[g(t) − h(t)]
M (t, s) ds < 1.
t→∞
g(t)
Proposition 4.2. Suppose (I1)–(I4) hold. Then all four hypotheses of Proposition
4.1 are equivalent.
In addition to (4.1) consider the integrodifferential equation
n Z t
m Z t
X
X
ẋ(t) +
Ki (t, s)x(s) ds −
Ml (t, s)x(s) ds = 0,
i=1
hi (t)
(4.7)
gl (t)
l=1
where the following conditions hold:
(I1? ) Ki (t, s), Ml (t, s) are Lebesgue integrable over each finite square [0, b]×[0, b]
functions;
(I2? ) There exist finite functions hi (t) = inf{s|Ki (t, s) ≥ 0},
gl (t) = inf{s|Ml (t, s) ≥ 0} and limt→∞ hi (t) = ∞, limt→∞ gl (t) = ∞;
(I3? ) For each t, s, i, l Ki (t, s) ≥ 0, Ml (t, s) ≥ 0,
n Z t
m Z t
X
X
Ki (t, τ ) dτ ≥
Ml (t, τ ) dτ
hi (t)
i=1
l=1
gl (t)
and one of two following conditions is satisfied:
(I4? ) m = n, Ki (t, s) ≥ Mi (t, s) for each t, s, i = 1, . . . , n, for any t, i Ki (t, s) ≥
Mi (t, s − hi (t) + gi (t)) and
Z t
n
X
lim sup
[gi (t) − hi (t)]
Mi (t, s) ds < 1.
t→∞
gi (t)
i=1
(I5? ) For each i, l, t, s h(t) ≤ g(t), where h, g are defined in (2.11) and
K(t, s) =
m
X
i=1
Ki (t, s) ≥
l
X
Ml (t, s) = M (t, s),
l=1
we have K(t, s) ≥ M (t, s − h(t) + g(t)) and
Z t
Z
n
h
X
lim sup (g(t) − h(t))
M (t, s) ds +
(h(t) − hi (t))
t→∞
+
m
X
l=1
g(t)
Z
t
(gl (t) − g(t))
i=1
t
hi (t)
Ki (t, s) ds
(4.8)
i
Ml (t, s) ds < 1.
gl (t)
Proposition 4.3. Suppose (I1? )–(I3? ) and one of (I4? ),(I5? ) hold. Then the following hypotheses are equivalent:
16
LEONID BEREZANSKY & ELENA BRAVERMAN
(1) Inequality
n Z t
m Z
X
X
ẏ +
Ki (t, s)y(s) ds −
i=1
hi (t)
l=1
EJDE–2003/47
t
Ml (t, s)y(s) ds ≤ 0,
t ≥ 0,
(4.9)
gl (t)
has an eventually positive solution;
(2) There exists t2 ≥ 0 such that X(t, s) > 0, t ≥ s ≥ t2 , where X(t, s) is a
fundamental function of (4.7);
(3) (4.7) has a nonoscillatory solution.
(4) There exists t3 ≥ 0 such that for t ≥ t3 the inequality
Z
n Z t
X
t
u(t) ≥
Ki (t, s) exp
u(τ )dτ ds
i=1 hi (t)
m Z t
X
−
l=1
s
Ml (t, s) exp
Z
gl (t)
(4.10)
t
u(τ )dτ ds
s
has a nonnegative locally integrable solution.
We proceed with comparison results which are deduced from Lemma 2.4. To
this end consider the equation
n Z t
m Z t
X
X
ẋ(t) +
K̃i (t, s)x(s) ds −
M̃l (t, s)x(s) ds = 0.
(4.11)
i=1
h̃k (t)
g̃l (t)
l=1
For (4.11) hi ,gl are changed by h̃i , g̃l , respectively.
Proposition 4.4.
(1) Suppose (I1? )–(I3? ) and at least one of (I4? ),(I5? ) hold,
where Ki ,Ml ,h,g are changed by K̃i , M̃l , h̃, g̃, respectively. If K̃i (t, s) ≥
Ki (t, s), M̃l (t, s) ≤ Ml (t, s) and (4.11) has a nonoscillatory solution, then
(4.7) also has a nonoscillatory solution.
(2) Suppose (I1? )-(I3? ) and at least one of (I4? ),(I5? ) hold. If K̃i (t, s) ≤
Ki (t, s), M̃l (t, s) ≥ Ml (t, s) and all solutions of (4.11) are oscillatory, then
all solutions of (4.7) are also oscillatory.
Corollary 4.1.
(1) Suppose (I1? )–(I3? ), (A1) and at least one of (A2), (A3)
hold. If for each t, i, l
Z t
Z t
Ki (t, s) ds ≤ ci ,
Ml (t, s) ds ≥ dl
hi (t)
gl (t)
and (3.6) has a nonoscillatory solution, then (4.7) also has a nonoscillatory
solution.
(2) Suppose (I1? ), (I3? ), (A1) and at least on of (I4? ), (I5? ) hold. If for each
t, i, l
Z
Z
t
t
Ki (t, s) ds ≥ ci ,
hi (t)
Ml (t, s) ds ≤ dl
gl (t)
and all solutions of (3.6) are oscillatory, then all solutions of (4.7) are also
oscillatory.
Lemma 2.5 immediately implies the following result on the asymptotic behaviour
of solutions.
Proposition 4.5. Suppose (I1? )–(I3? ) is satisfied and anyone of the following
conditions holds:
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17
R∞ Rt
(1) (I4? ) is satisfied and for some l, 0
(Kl (t, s) − Ml (t, s)) ds dt = ∞
t0
R∞ Rt
(2) (I5? ) is satisfied and 0
(K(t, s) − M (t, s)) ds dt = ∞.
t0
Then any nonoscillatory solution y of (4.7) satisfies limt→∞ y(t) = 0.
Lemma 2.8 yields oscillation conditions for (4.7).
Proposition 4.6. Suppose hypotheses (I1? )–(I4? ) hold. If all solutions of anyone
of the following two equations are oscillatory
n Z t
n Z t
X
X
ẋ(t) +
Kl (t, s) − Ml (t, s − hl (t) + gl (t)) x(s) ds +
Ml (t, s)x(s)
hl (t)
l=1
× exp
Z
l=1
s
τ
Z
dτ
s+hl (t)−gl (t)
gl (t)
Kl (τ, ζ) − Ml (τ, ζ) dζ − 1 ds = 0 ,
hl (τ )
(4.12)
ẋ(t) +
n Z
X
× exp
Kl (t, s + hl (t) − gl (t))x(s)
gl (t)
l=1
t
s
Z
+
Kl (τ, ζ) − Ml (τ, ζ) dζ − 1
dτ
s+hl (t)−gl (t)
n Z
X
τ
Z
(4.13)
hl (τ )
t
Kl (t, s + hl (t) − gl (t)) − Ml (t, s) x(s) ds = 0 ,
gl (t)
l=1
then all solutions of (4.7) are also oscillatory.
Remark. Since (I5? ) is a special case of (I4? ), similar equations can be presented
if (I1? )-(I3? ),(I5? ) are satisfied.
The oscillation conditions for integrodifferential equations with nonnegative kernels [2] lead to the following result.
Corollary 4.2. Suppose the hypotheses (I1)-(I4) hold for (4.1) and at least one of
the following inequalities is valid:
Z s
nZ t Z s
1) lim inf
K(t, τ ) dτ −
M (t, τ − h(t) + g(t)) dτ ds
t→∞
h(t)
h(s)
t
Z
+
Z
g(s)
s
τ
dτ
exp
g(t)
Z
h(τ )
2)
lim inf
t→∞
t
Z
h(t)
s
M (t, τ − h(t) + g(t)) dτ ds
g(s)
Z
Z
τ
Z
s
h(τ )
t→∞
nZ
t
g(t)
Z
τ
dτ
g(τ )
exp
Z
u
K(u, τ ) dτ
s+h(t)−g(t)
Z u
−
lim inf
Z
dτ
g(t)
3)
K(u, τ )dτ
h(u)
s
K(t, τ ) dτ −
+
u
o 1
M (u, τ )dτ du − 1 M (t, s)ds >
e
g(u)
Z
h(s)
t
Z
s+h(t)−g(t)
Z u
−
nZ
h(u)
o 1
M (u, τ ) dτ du M (t, s) ds >
e
g(u)
s
s−g(t)+h(t)
Z
u
K(u, τ )dτ
h(u)
18
LEONID BEREZANSKY & ELENA BRAVERMAN
Z
u
EJDE–2003/47
M (u, τ )dτ du − 1 K(t, s − g(t) + h(t))ds
−
g(u)
Z
t
+
Z
g(t)
4)
s
Z
o 1
M (t, τ − h(t) + g(t)) dτ ds >
e
g(s)
t
u
h(s)
lim inf
nZ
t→∞
s
K(t, τ − g(t) + h(t)) dτ −
Z
τ
dτ
g(t)
g(τ )
Z u
−
Z
s
hZ
s−g(t)+h(t)
K(u, τ )dτ
h(u)
i M (u, τ )dτ du K(t, s − g(t) + h(t))ds
g(u)
Z
t
+
g(t)
Z
s
Z
K(t, τ − g(t) + h(t)) dτ −
h(s)
s
o 1
M (t, τ − h(t) + g(t)) dτ ds >
e
g(s)
Then all solutions of (4.1) are oscillatory.
Lemma 2.11 implies the following nonoscillation results for (4.7).
Proposition 4.7. Suppose (I1? )–(I4? ) and the following inequality
Z
Z
n Z t
X
s
1
1 s
lim sup
Ml (s, τ ) dτ ds <
Kl (s, τ ) dτ −
e gl (s)
e
t→∞
hl (t)
hl (s)
(4.14)
l=1
hold. Then (4.7) has a nonoscillatory solution.
Proposition 4.8. Suppose (I1? )–(I3? ),(I5? ) and the following inequality
Z
Z s
Z t
Z
n
s
X
1 t
lim sup
1−
ds
Kl (s, τ ) dτ
ds
K(s, τ ) dτ
e hl (t)
t→∞
hl (s)
0
0
l=1
Z
Z
Z s
1
1 s
1 t
−
M (s, τ ) dτ + 1 −
ds
Ml (s, τ ) dτ <
e 0
e gl (t)
e
gl (s)
hold. Then (4.7) has a nonoscillatory solution.
Similar to Proposition 3.7 the following result can be obtained. Let H(t) =
mini hi (t), G(t) = maxl gl (t).
Proposition 4.9. Suppose there exist K̃(t, s), M̃ (t, s), such that
where K(t, s) =
and are finite:
Z t
B11 = lim
t→∞
Z
s
ds
H(t)
Z t
B21 = lim
t→∞
M̃ (t, s) ≤ M (t, s) ≤ K(t, s) ≤ K̃(t, s),
Pm
i=1 Ki (t, s), M (t, s) =
l=1 Ml (t, s), the following limits exist
Pn
H(s)
Z s
ds
G(t)
Z
t
K̃(s, τ ) dτ, B12 = lim
t→∞
H(t)
Z t
K̃(s, τ ) dτ, B22 = lim
G(s)
t→∞
Z
s
ds
M̃ (s, τ ) dτ, (4.15)
H(s)
Z s
ds
G(t)
M̃ (s, τ ) dτ,
(4.16)
G(s)
and (I1)–(I4) hold for K̃(t, s), M̃ (t, s). Suppose, in addition, that the system
ln x1 > x1 B11 − x2 B12
(4.17)
ln x2 < x1 B21 − x2 B22
(4.18)
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OSCILLATION FOR EQUATIONS WITH . . .
19
has a positive solution (x1 , x2 ) such that eventually
Z t
Z t
x1
K̃(t, s) ds ≥ x2
M̃ (t, s) ds.
h(t)
g(t)
Then (4.7) has a nonoscillatory solution.
Example 3. Consider the integrodifferential equation
Z t
ẋ(t) +
L(t, s)x(s) ds = 0.
(4.19)
0
Let α > 0,
(
α sin(s − t), 0 ≤ t − s ≤ 2π,
L(t, s) =
0,
otherwise,
1
(|L(t, s)| + L(t, s)) = α sin(s − t)χ[t−2π,t−π] (s),
2
1
M (t, s) = L− (t, s) = (|L(t, s)| − L(t, s)) = −α sin(s − t)χ[t−π,t] (s).
2
Then h(t) = t − 2π, g(t) = t − π, M (t, s + π) = K(t, s) and
Z t
lim sup [g(t) − h(t)]
M (t, s) ds
K(t, s) = L+ (t, s) =
t→∞
g(t)
Z t
= lim sup [g(t) − h(t)]
α(− sin(s − t)) ds
t→∞
t−π
= lim sup πα cos 0 − cos(−π) = 2πα.
t→∞
Thus the hypothesis (I4) holds for (4.19) if α < 1/(2π). Let us proceed to nonoscillation conditions for this equation. To this end we will apply Proposition 4.7. We
have
Z t
Z t−π
K(t, s) ds =
α sin(s − t) ds = 2α,
Z
h(t)
t
t−2π
Z t
M (t, s) ds = −
g(t)
α sin(s − t) ds = 2α,
t−π
which after the substitution in (4.14) yields
Z
Z t hZ s
i
1
1 s
M (s, τ ) dτ ds = [t − h(t)]2α 1 −
lim sup
K(s, τ ) dτ −
e g(s)
e
t→∞
h(t)
h(s)
e−1
1
= 2π2α
< ,
e
e
which is satisfied when 4πα(e − 1) < 1. Consequently, if α <
has a nonoscillatory solution.
Example 4. Let 0 < α < β. Consider equation (4.19) with


α sin(s − t), 0 ≤ t − s ≤ π,
L(t, s) = β sin(s − t), π ≤ t − s ≤ 2π,


0,
otherwise,
1
4π(e−1) ,
then (4.19)
20
LEONID BEREZANSKY & ELENA BRAVERMAN
EJDE–2003/47
Then K(t, s) = β sin(s − t)χ[t−2π,t−π] (s), M (t, s) = α sin(s − t)χ[t−π,t] (s) and
Z t
lim sup [g(t) − h(t)]
M (t, s) ds = 2πα.
t→∞
g(t)
1
The hypothesis (I4) holds for (4.19) if in addition α < 2π
. Similarly to Example 1,
if 4π(βe − α) < 1, we have
Z t Z s
Z
α
1 s
K(s, τ ) dτ −
M (s, τ ) dτ ds = 2π2 β −
e g(s)
e
h(t)
h(s)
4π
1
=
(βe − α) < .
e
e
1
Consequently, if βe − α < 4π , then (4.19) has a nonoscillatory solution.
For this kernel we can also obtain oscillation conditions. Since
Z s
Z s
K(t, τ ) dτ −
M (t, τ − h(t) + g(t)) dτ = 2(β − α)
h(s)
g(s)
for t − π ≤ τ ≤ t, we have
Z u
Z
K(u, τ ) dτ −
h(u)
u
M (u, τ ) dτ = 2(β − α).
g(u)
Then after substituting these results into the first formula in Corollary 4.2 we have
Z s
i
nZ t hZ s
K(t, τ ) dτ −
M (t, τ − h(t) + g(t)) dτ ds
lim inf
t→∞
h(t)
Z τ
t
Z
+
h(s)
dτ
g(t)
Z u
exp
g(s)
nZ
h(τ )
s
s+h(t)−g(t)
Z
u
K(u, τ ) dτ
h(u)
o
o
M (u, τ ) dτ du − 1 M (t, s) ds
−
g(u)
t
Z
hZ
≥
t−π
s
Z
s
i
M (t, τ − h(t) + g(t)) dτ ds
K(t, τ ) dτ −
h(s)
g(s)
Z
t
+ 2(β − α)
Z
τ
e2π(β−α) − 1 ds T (t, s)
dτ
g(t)
h(τ )
= 2π(β − α) + e2π(β−α) − 1 2π.
Thus if 2π(β − α + exp{2π(β − α)} − 1) > 1/e, then all solutions of (4.19) are
oscillatory.
5. Mixed Equations
In this section we will consider mixed equations
n
m
r Z t
X
X
X
ak (t)x(hk (t)) −
bl (t)x(gl (t)) +
Ki (t, s)x(s)ds
ẋ(t) +
−
Mj (t, s)x(s)ds = 0,
j=1
i=1
l=1
k=1
p Z t
X
0
(5.1)
t ≥ t0 ≥ 0,
0
with the initial conditions
x(t) = ϕ(t),
t < t0 , x(t0 ) = x0 .
(5.2)
EJDE–2003/47
OSCILLATION FOR EQUATIONS WITH . . .
21
We assume that (C1)–(C3) and (I1? )–(I3? ) hold. To avoid confusion, in (I2? )
instead of hk , gl the following functions are introduced:
h̃i = inf{s|Ki (t, s) ≥ 0},
g̃j = inf{s|Mj (t, s) ≥ 0}.
Consider in addition the following hypotheses:
(M1) m = n, ak ≥ bk , hk (t) ≤ gk (t) for k, t, t ≥ 0, r=p, Ki (t, s) ≥ Mi (t, s −
h̃i (t) + g̃i (t)) and
Z t
n
r
nX
o
X
lim sup
bk (t)[gk (t) − hk (t)] +
[g̃i (t) − h̃i (t)]
Mi (t, s)ds < 1.
t→∞
g̃i (t)
i=1
k=1
Pm
Pn
≤ h̃(t) ≤ g̃(t) ≤
(M2) h(t) ≤ g(t), a(t) = k=1 ak (t)P
≥ b(t) = l=1 bl (t), h̃i (t)P
r
p
g̃j (t), for each i, j, t, K(t, s) = i=1 Ki (t, s) ≥ M (t, s) = j=1 Mj (t, s) for
each t, s and
n
m
n
X
X
lim sup b(t)[g(t) − h(t)] +
ak (t)[h(t) − hk (t)] +
bl (t)[gl (t) − g(t)]
t→∞
k=1
t
Z
+ (g̃(t) − h̃(t))
M (t, s) ds +
g̃(t)
+
m
X
n
X
Z
(h̃(t) − h̃i (t))
Ki (t, s) ds
h̃i (t)
i=1
Z
t
l=1
t
o
Ml (t, s) ds < 1.
(g̃l (t) − g̃(t))
g̃l (t)
l=1
Obviously two other combinations of (C4), (C5) with (I4? ), (I5? ) can be considered.
Proposition 5.1. Suppose (C1)–(C3), (I1? )–(I3? ) and at least one of hypotheses
(M1), (M2) hold. Then the following hypotheses are equivalent.
(1) There exists t1 ≥ 0 such that for t ≥ t1 the inequality
u(t) ≥
n
X
ak (t) exp
hk (t)
k=1
+
r Z
X
i=1
m
X
u(s)ds −
bl (t) exp
t
Z
t
Ki (t, s) exp
h̃i (t)
Z
u(τ )dτ ds −
s
t
u(s)ds
gk (t)
l=1
t
Z
p Z
X
t
Mj (t, s) exp
g̃l (t)
j=1
Z
t
u(τ )dτ ds
s
has a nonnegative locally integrable solution (we assume u(t) = 0 for t <
t1 );
(2) There exists t2 ≥ 0 such that the fundamental function of (5.1) X(t, s) > 0,
t ≥ s ≥ t2 ;
(3) Equation (5.1) has a nonoscillatory solution;
(4) The inequality
ẏ(t) +
n
X
k=1
+
r Z
X
i=1
0
t
ak (t)y(hk (t)) −
m
X
bl (t)y(gl (t))
l=1
p Z t
X
Ki (t, s)y(s) ds −
j=1
has an eventually positive solution.
0
(5.3)
Mj (t, s)x(s) ds ≤ 0
22
LEONID BEREZANSKY & ELENA BRAVERMAN
EJDE–2003/47
The comparison result for equation (5.1) combines Propositions 3.3 and 4.4.
Consider the comparison equation
n
m
X
X
ẋ(t) +
ãk (t)x(h̄k (t)) −
b̃l (t)x(ḡl (t))
k=1
+
r Z
X
t
K̃i (t, s)x(s) ds −
l=1
p Z t
X
0
i=1
(5.4)
M̃j (t, s)x(s) ds = 0.
0
j=1
Proposition 5.2. 1) Suppose (C1)-(C3),(I1? )-(I3? ) and either (M1) or (M2) hold
for (5.4), where ak , bl , hk , gl ,Ki ,Ml are changed by ãk , b̃l , h̄k , ḡl , K̃i , M̃l , respectively.
If ãk (t) ≥ ak (t), b̃l (t) ≤ bl (t), h̄k (t) ≤ hk (t), ḡl (t) ≥ gl (t), K̃i (t, s) ≥ Ki (t, s),
M̃l (t, s) ≤ Ml (t, s) and (5.4) has a nonoscillatory solution, then (5.1) also has a
nonoscillatory solution.
2) Suppose (C1)-(C3),I1? )-(I3? ) and either (M1) or (M2) hold for (5.1). If ãk (t) ≤
ak (t), b̃l (t) ≥ bl (t), h̄k (t) ≥ hk (t), ḡl (t) ≤ gl (t), K̃i (t, s) ≤ Ki (t, s), M̃l (t, s) ≥
Ml (t, s) and all solutions of (5.4) are oscillatory, then all solutions of (5.1) are
also oscillatory.
Proposition 5.3.R Suppose
(C1)–(C3),
(I1? )–(I3? ), (M1) hold and either there
∞
exists such k that 0 ak (t) − bk (t) dt = ∞ or there exists such i that
Z ∞hZ t
Z t
i
Ki (t, s) ds −
Mi (t, s) ds dt = ∞.
0
h̃i (t)
g̃i (t)
Then any nonoscillatory solution of (5.1) tends to zero at infinity.
Remark. Similar result can be obtained if (C1)–(C3), I1? )–(I3? ), (M2) are satisfied.
Proposition 5.4 presents oscillation conditions for (5.1).
Proposition 5.4. Suppose (C1)–(C3), (I1? )–(I3? ), (M1) and the following inequality hold
n
nX
lim inf
[ak (t) − bk (t)](t − hk (t))
t→∞
+
+
n
X
Z
bk (t) exp
hZ
h̃i (t)
r Z
X
t
[ak (s) − bk (s)]ds − 1 (t − gk (t))
s
Z
s
Ki (t, τ ) dτ −
h̃i (s)
Z
τ
dτ
i=1 g̃i (t)
Z u
−
gk (t)
hk (t)
k=1
r Z t
X
i=1
+
k=1
h̃i (τ )
i
Mi (t, τ − h̃i (t) + g̃i (t)) dτ ds
g̃i (s)
exp
nZ
s
s+h̃i (t)−g̃i (t)
hZ
u
Ki (u, τ )dτ
h̃i (u)
i o
o 1
Mi (u, τ )dτ du − 1 Mi (t, s)ds >
e
g̃i (u)
Then all solutions of (5.1) are oscillatory.
Remark. Similarly to inequality 1) in Corollary 2.9 other oscillation conditions
for (5.1) can be deduced using inequalities 2)-4) of this corollary.
Proposition 5.5 present nonoscillation conditions.
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OSCILLATION FOR EQUATIONS WITH . . .
23
Proposition 5.5. Suppose (C1)–(C3), (I1? )–(I3? ), (M1) and the following inequality holds
n Z t
nX
1
lim sup
ak (s) − bk (s) ds
e
t→∞
k=1 hk (t)
Z
Z
Z
r
t
s
h
io 1
X
1 s
+
ds
Ki (s, τ ) dτ −
Mi (s, τ ) dτ
< .
e g̃i (s)
e
h̃i (s)
i=1 h̃i (t)
Then (5.1) has a nonoscillatory solution.
Remark. Similar results are obtained (see Propositions 3.6, 4.7 and 4.8) if (M2)
is satisfied instead of (M1).
As a final example, consider the following equation of the mixed type
Z t
n
X
ẋ(t) +
ak (t)x(hk (t)) −
K(t, s)x(s) ds = 0,
(5.5)
0
k=1
under the following conditions:
(m1) ak ≥ 0 are Lebesgue measurable bounded functions, K is Lebesgue integrable over each finite square [0, b] × [0, b];
(m2) There exists finite function g(t) = inf{s|K(t, s) > 0} and limt→∞ g(t) = ∞;
limt→∞ hk (t) = ∞ P
for each k;
n
s
(m3) For any t ≥ s ≥ 0, k=1 ak (t)χ[hk (t),∞)(s)≥Rg(s)
K(t,τ ) dτ .
Consider also the following hypothesis
Pn
(m4) There exist constants c1 , . . . , cn , k=1 ck = 1 such that
Z s−hk (t)+g(t)
ak (t)χ[hk (t),∞) (s) ≥ ck
K(t, τ ) dτ
s−hk (t)
and
lim sup
t→∞
n
X
ck (g(t) − hk (t))
Z
t
K(t, s) ds < 1.
g(t)
k=1
Proposition 5.6. Suppose (m1)–(m3) hold. Consider the following hypotheses
(1) There exists t1 ≥ 0 such that the inequality
Z
Z t
Z
n
X
t
t
u(t) ≥
ak (t) exp
u(s)ds −
K(t, s) exp
u(τ )dτ , t ≥ t1
k=1
hk (t)
g(t)
s
has a nonnegative locally integrable solution (we assume u(t) = 0 for t <
t1 );
(2) There exists t2 ≥ 0 such that the fundamental function of (5.5) X(t, s) > 0,
t ≥ s ≥ t2 ;
(3) Equation (5.5) has a nonoscillatory solution;
(4) The inequality
Z t
n
X
ẏ(t) +
ak (t)y(hk (t)) −
K(t, s)y(s) ds ≤ 0
(5.6)
k=1
0
has an eventually positive solution.
Then the implications 1) ⇒ 2) ⇒ 3) ⇒ 4) are valid. If in addition (m4) holds then
hypotheses 1)–4) are equivalent.
24
LEONID BEREZANSKY & ELENA BRAVERMAN
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To deduce a comparison result we introduce the equation
Z t
n
X
ẋ(t) +
bk (t)x(h̃k (t)) −
M (t, s)x(s) ds = 0.
(5.7)
0
k=1
Proposition 5.7. 1) Suppose (m1)-(m4) hold, where ak , hk , K are changed by
bk , h̃k , M , respectively. If bk (t) ≥ ak (t), h̃k (t) ≤ hk (t), M (t, s) ≥ K(t, s) for each
t, s, k and (5.7) has a nonoscillatory solution, then (5.5) also has a nonoscillatory
solution.
2) Suppose (m1)-(m4) hold. If bk (t) ≤ ak (t), h̃k (t) ≥ hk (t), M (t, s) ≤ K(t, s) for
each t, s, k and all solutions of (5.7) are oscillatory, then all solutions of (5.5) are
also oscillatory.
Proposition 5.8. Suppose (m1)–(m4) hold and
Z ∞hX
Z t
n
i
ak (t) −
K(t, s) ds dt = ∞.
0
g(t)
k=1
Then any nonoscillatory solution x of (5.5) satisfies limt→∞ x(t) = 0.
Note that Corollary 2.9, 1) implies the following result.
Proposition 5.9. Suppose (m1)–(m4) and the following inequality hold
Z s−hk (t)+g(t)
Z t
n Z t
n
nX
X
lim inf
ak (s) − ck
ck
K(t, τ ) dτ ds +
dτ
t→∞
Z
τ
×
hk (τ )
k=1
hk (t)
exp
Z
s−hk (t)
s
ak (u) −
s+hk (t)−g(t)
k=1
Z
g(t)
u
o 1
K(u, ζ) dζ du − 1 K(t, s)ds > .
e
g(u)
Then all solutions of (5.5) are oscillatory.
Remark. Similarly inequalities 2)-4) in Corollary 2.9 can be rewritten for (5.5).
Proposition 5.10. Suppose (m1)–(m4) and the inequality
Z
n Z t
nX
o 1
ck s
K(s, τ ) dτ ds <
lim sup
ak (s) −
e g(s)
e
t→∞
hk (t)
k=1
holds. Then (5.5) has a nonoscillatory solution.
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Leonid Berezansky
Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105,
Israel
E-mail address: brznsky@cs.bgu.ac.il
Elena Braverman
Department of Mathematics and Statistics, University of Calgary, 2500 University
Drive N. W., Calgary, Alberta, Canada, T2N 1N4
Fax: (403)-282-5150, phone: (403)-220-3956
E-mail address: maelena@math.ucalgary.ca