Directory UMM :Journals:Journal_of_mathematics:GMJ:
Georgian Mathematical Journal
Volume 10 (2003), Number 1, 63–76
ON THE OSCILLATION OF SOLUTIONS OF FIRST ORDER
DIFFERENTIAL EQUATIONS WITH RETARDED
ARGUMENTS
M. K. GRAMMATIKOPOULOS, R. KOPLATADZE, AND I. P. STAVROULAKIS
Abstract. For the differential equation
m
X
u′ (t) +
pi (t)u(τi (t)) = 0,
i=1
where pi ∈ Lloc (R+ ; R+ ), τi ∈ C(R+ ; R+ ), τi (t) ≤ t for t ∈ R+ , lim τi (t) =
t→+∞
+∞ (i = 1, . . . , m), optimal integral conditions for the oscillation of all solutions are established.
2000 Mathematics Subject Classification: 34C10, 34K11.
Key words and phrases: Retarded arguments, proper solutions, oscillation.
1. Introduction
Consider the differential equation
m
X
u′ (t) +
pi (t)u(τi (t)) = 0,
(1.1)
i=1
where pi ∈ Lloc (R+ ; R+ ), τi ∈ C(R+ ; R+ ), τi (t) ≤ t for t ∈ R+ , lim τi (t) = +∞
t→+∞
(i = 1, . . . , m).
The first systematic study for the oscillation of all solutions of equation (1.1)
for the case of constant coefficients and constant delays was made by Myshkis
[18]. Since then a number of papers have been devoted to this subject. For the
case m=1 the reader is referred to the papers [2–7, 10, 12–14, 16, 18], while for
the case m > 1 to [1, 6, 9, 11, 15, 17]. The difficulties connected with the study
of specific properties of solutions of delay differential equations are emphasized
in the monograph by Hale [8]. In [12] the following statement is proved.
Theorem 1.1. Let m = 1,
Zt
1
lim inf
p1 (s)ds > .
t→+∞
e
τ1 (t)
Then equation (1.1) is oscillatory.
In the case m > 1 there are some difficulties in finding optimal conditions for
the oscillation of solutions of (1.1). In the present paper we make an attempt
at carrying out in this direction. Several sufficient oscillation conditions for the
c Heldermann Verlag www.heldermann.de
ISSN 1072-947X / $8.00 / °
64
M. K. GRAMMATIKOPOULOS, R. KOPLATADZE, AND I. P. STAVROULAKIS
case of several delays are contained in [1, 9, 15, 17]. It is to be pointed out that
the technique used in [17] cannot be applied for equation (1.1).
2. Formulation of the Main Results
Throughout the paper we will assume that pi : R+ → R+ (i = 1, . . . , m)
are locally integrable functions, τi : R+ → R+ (i = 1, . . . , m) are continuous
functions, and
pi (t) ≥ 0, τi (t) ≤ t for t ∈ R+ ,
lim τi (t) = +∞ (i = 1, . . . , m).
t→+∞
(2.1)
Let a ∈ R+ . Denote a0 = inf {τ∗ (t) : t ≥ a}, τ∗ (t) = min {τi (t) : i = 1, . . . , m}.
Definition 2.1. A continuous function u : [a0 , +∞) → R is called a proper
solution of equation (1.1) in [a, +∞) if it is absolutely continuous in each finite
segment contained in [a, +∞) and satisfies (1.1) almost everywhere on [a, +∞)
and sup {|u(s)| : s ≥ t} > 0 for t ≥ a0 .
Definition 2.2. A proper solution of equation (1.1) is said to be oscillatory
if it has a sequence of zeros tending to infinity; otherwise it is said to be nonoscillatory.
Definition 2.3. Equation (1.1) is said to be oscillatory if its every proper
solution is oscillatory.
Theorem 2.1. Let condition (2.1) hold, for some i ∈ {1, . . . , m},
lim inf
t→+∞
Zt
pi (s)ds > 0,
(2.2)
Zt
p(s)ds < +∞
(2.3)
τi (t)
lim sup
t→+∞
σ(t)
and
inf
m Z
X
i=1 t
where
lim inf exp λ
t→+∞
+∞
×
p(t) =
m
X
pi (t),
0
0
p(s)ds
p(ξ)dξ)ds : λ ∈ (0, ∞) > 1,
τZi (s)
pi (s) exp −λ
Zt
σ(t) = inf {τ∗ (s) : s ≥ t ≥ 0} ,
i=1
τ∗ (t) = min {τi (t) : i = 1, . . . , m} .
Then equation (1.1) is oscillatory.
(2.4)
(2.5)
RETARDED DE
65
Remark 2.1. Condition (2.3) is not an essential restriction because if for some
i ∈ {1, . . . , m} ,
Zt
lim sup
pi (s)ds > 1,
t→+∞
τi (t)
then equation (1.1) is oscillatory (see, e.g., [13]).
Theorem 2.2. Let conditions (2.2), (2.3) be fulfilled, p(t) > 0 for sufficiently
large t, and
Zt
lim inf
p(s)ds = αi > 0 (i = 1, . . . , m).
(2.6)
t→+∞
τi (t)
If, moreover, for some t0 ∈ R+ ,
(
Ã
!
)
m
1
1 X
inf
vrai inf
pi (t)eαi λ : λ ∈ (0, +∞) > 1,
λ t≥t0
p(t) i=1
(2.7)
then equation (1.1) is oscillatory.
Theorem 2.3. Let conditions (2.2), (2.3), (2.6) be fulfilled, and p(t) > 0 for
sufficiently large t. Let, moreover, for some t0 ∈ R+ ,
!
Ã
m
1
1 X
αi pi (t) > .
(2.8)
vrai inf
t≥t0
p(t) i=1
e
Then equation (1.1) is oscillatory.
Theorem 2.4. If conditions (2.2), (2.3), (2.6) are fulfilled, and
1
min {αi : i = 1, . . . , m} > ,
e
(2.9)
then equation (1.1) is oscillatory.
Theorem 2.5. Let τi (t) (i = 1, . . . , m) be nondecreasing,
Z∞
| pi (t) − pj (t) | dt < +∞
(i, j = 1, . . . , m),
(2.10)
(i = 1, . . . , m)
(2.11)
0
lim inf
t→+∞
Zt
pi (s)ds = βi > 0
τi (t)
and
min
( m
X eβi λ
i=1
λ
Then equation (1.1) is oscillatory.
: λ ∈ (0, +∞)
)
> 1.
(2.12)
66
M. K. GRAMMATIKOPOULOS, R. KOPLATADZE, AND I. P. STAVROULAKIS
Theorem 2.6. Let conditions (2.10), (2.11) hold, and
m
X
i=1
1
βi > .
e
(2.13)
Then equation (1.1) is oscillatory.
Remark 2.2. It is obvious that Theorem 2.6 coincides with Theorem 1.1 for
the case m = 1.
3. Auxiliary Statements
Lemma 3.1. Let p : R+ → R+ be a summable function in every finite
segment, τ : R+ → R+ be a continuous and nondecreasing function, and
lim τ (t) = +∞. If, moreover,
t→+∞
lim inf
t→+∞
Zt
p(s)ds > 0
(3.1)
τ (t)
and u : [a0 , +∞) → (0, +∞) is a solution of the equation
u′ (t) + p(t)u(τ (t)) = 0,
then there exists λ > 0 such that
t
Z
lim u(t) exp λ p(s)ds = +∞.
t→+∞
(3.2)
(3.3)
0
Proof. First we will show that
lim sup
t→+∞
u(τ (t))
< +∞.
u(t)
(3.4)
By virtue of (3.1) there are c > 0 and t0 ∈ R+ such that
Zt
p(s)ds ≥ c for t ≥ t0 .
τ (t)
Thus for any t > t0 there exists t∗ > t such that
Zt∗
t
c
p(s)ds = ,
2
Zt
c
p(s)ds ≥ .
2
(3.5)
τ (t∗ )
Without loss of generality we can assume that u(τ (t)) > 0 for t ≥ t0 . In view
of (3.5) from (3.2) we have
u(t) ≥
Zt∗
t
p(s)u(τ (s))ds ≥ u(τ (t∗ ))
Zt∗
t
c
p(s)ds = u(τ (t∗ ))
2
RETARDED DE
67
and
u(τ (t∗ )) ≥
Zt
c
p(s)u(τ (s))ds ≥ u(τ (t)).
2
τ (t∗ )
The last two inequalities result in u(t) ≥ (c2 /4)u(τ (t)). This, in view of the
arbitrariness of t, means that (3.4) is valid. Thus from (3.2) we get
Zt
Zt
4
u(τ (s))
ds ≥ u(t0 ) exp − 2 p(s)ds .
(3.6)
u(t) = u(t0 ) exp − p(s)
u(s)
c
t0
t0
On the other hand, from (3.1) it obviously follows that
Z+∞
p(s)ds = +∞.
t0
Therefore, according to (3.6), there exists λ > 0 such that (3.3) is satisfied. ¤
Lemma 3.2. Let (3.1) be fulfilled, p, q : R+ → R+ be summable functions in
every finite segment, τ, τ0 : R+ → R+ be continuous functions,
lim τ (t) = lim t → +∞τ0 (t) = +∞,
t→+∞
q(t) ≥ p(t), τ0 (t) ≤ τ (t) ≤ t for t ≥ t0 .
(3.7)
If, moreover, v : [t0 , +∞) → (0, +∞) is a solution of the inequality
v ′ (t) + q(t)v(τ0 (t)) ≤ 0,
(3.8)
then equation (3.2) has a solution u : [t1 , +∞) → (0, +∞) satisfying the condition
0 < u(t) ≤ v(t) for t ≥ t1 ,
(3.9)
where t1 ≥ t0 is a sufficiently large number.
Proof. Let v : [t0 , +∞) → (0, +∞) be a solution of inequality (3.8). By (3.1)
and (3.7) there is t1 > t0 such that v(τ0 (t)) > 0 for t > t1 and
Zt
p(s)ds > 0 for t ≥ t1 .
(3.10)
τ (t)
From (3.8) we have
Z+∞
v(t) ≥
q(s)v(τ (s))ds for t ≥ t1 .
t
(3.11)
68
M. K. GRAMMATIKOPOULOS, R. KOPLATADZE, AND I. P. STAVROULAKIS
Denote t∗1 = inf {τ (t) : t ≥ t1 } and consider the sequence of functions ui :
[t∗1 , +∞) → [0, +∞) (i = 1, 2, 3, . . . ) defined by the following equalities:
ui (t) =
+∞
R p(s)u
u1 (t) = v(t) for t ≥ t∗1 ,
i−1 (τ (s))ds
for t ≥ t1
t
v(t) − v(t1 ) + ui (t1 ) for
t∗1
(i = 2, 3, . . . ).
(3.12)
≤ t < t1
On account of the last inequality of (3.7) and conditions (3.10), (3.11) it is clear
that 0 < ui (t) ≤ ui−1 (t) ≤ v(t) (i = 2, 3, . . . ) for t ≥ t1 . Thus 0 ≤ u(t) ≤ v(t)
for t ≥ t1 , where u(t) = limi→+∞ ui (t). Let us show that u(t) > 0 for t ≥ t1 .
Otherwise there is t2 ≥ t1 such that u(t) ≡ 0 for t ≥ t2 and u(t) > 0 for
t ∈ [t∗1 , t2 ). Denote by U the set of points t satisfying τ (t) = t2 , and put
t∗ = min U. Evidently t∗ ≥ t2 . Therefore, by (3.10) and (3.12), we get
Z+∞
Zt∗
u(t2 ) =
p(s)u(τ (s))ds ≥
p(s)u(τ (s))ds > 0.
t2
τ (t∗ )
The obtained contradiction proves that u(t) > 0 for t ≥ t1 . Consequently we
have 0 < u(t) ≤ v(t) for t ≥ t1 .
¤
Lemma 3.3. Let condition (2.1) hold, for some i ∈ {1, . . . , m},
lim inf
t→+∞
Zt
pi (s)ds > 0,
(3.13)
τi (t)
and u : [t0 , +∞) → (0, +∞) be a positive solution of equation (1.1). Then there
exists λ > 0 such that
t
Z
lim u(t) exp λ pi (s)ds = +∞.
(3.14)
t→+∞
0
Proof. It is obvious that u is a solution of the differential inequality
u′ (t) + pi (t)u(τi (t)) ≤ 0 for t ≥ t1 ,
where t1 > t0 is a sufficiently large number. Thus, taking into account (3.13)
and Lemmas 3.1, 3.2, there exists λ > 0 such that (3.14) is fulfilled.
¤
Lemma 3.4. Let t0 ∈ R+ , ϕ, ψ ∈ C([t0 , +∞); (0, +∞)), ψ(t) be non-increasing and
lim ϕ(t) = +∞, lim inf ψ(t)ϕ(t)
e = 0,
t→+∞
t→+∞
where ϕ(t)
e = inf {ϕ(s) : s ≥ t ≥ t0 .} Then there exists an increasing sequence
of points {tk }+∞
k=1 such that tk ↑ +∞ as k ↑ +∞ and
]
ϕ(t
e ≥ ψ(tk )ϕ(t
e k ) for t0 ≤ t ≤ tk .
k ) = ϕ(tk ), ψ(t)ϕ(t)
For the proof of Lemma 3.4 see [11, Lemma 7.1].
RETARDED DE
69
4. Proof of the Main results
Proof of Theorem 2.1. Assume the contrary. Let equation (1.1) have a nonoscillatory proper solution u : [t0 , +∞) → (0, +∞). According to condition
(2.2) and Lemma 3.3, there exists λ > 0 such that
t
Z
(4.1)
lim u(t) exp λ p(s)ds = +∞,
t→+∞
0
where the function p(t) is defined by the first equality of (2.5).
Denote by Λ the set of all λ satisfying condition (4.1), and put λ0 = inf Λ.
Since u(t) is non-increasing, in view of (4.1) it is obvious that λ0 ≥ 0. By the
definition of λ0 and condition (2.4), there exist ε > 0 and λ∗ > λ0 such that
τZi (ξ)
+∞
Z
Zt
m
X
p(s)ds dξ
pi (ξ) exp −λ
lim inf exp λ∗ p(s)ds
t→+∞
i=1 t
0
0
(1+M )ε
> (1 + ε)e
,
Zt
lim u(t) exp λ∗ p(ξ)dξ = +∞,
(4.2)
p(ξ)dξ = 0,
(4.4)
p(s)ds.
(4.5)
(4.3)
t→+∞
0
lim inf exp (λ∗ − ε)
t→+∞
where
M = lim sup
t→+∞
Zt
0
Zt
σ(t)
Due to (4.3) and (4.4) it is clear that the functions ϕ and ψ satisfy the conditions
of Lemma 3.4 where
σ(t)
Z
Zt
ϕ(t) = u(σ(t)) exp λ∗
p(s)ds , ψ(t) = exp −ε p(s)ds
0
0
and the function σ(t) is defined by the last two equalities of (2.5). Therefore,
by Lemma 3.4, there exists an increasing sequence of points {tk }+∞
k=1 such that
Zt
Ztk
e exp −ε p(s)ds for t0 ≤ t ≤ tk , (4.6)
ϕ(t
e k ) exp −ε p(s)ds) ≤ ϕ(t)
0
ϕ(t
e k ) = u(σ(tk )) exp λ∗
0
σ(t
Z k)
0
p(s)ds .
(4.7)
70
M. K. GRAMMATIKOPOULOS, R. KOPLATADZE, AND I. P. STAVROULAKIS
If we take into account the definition of the function σ(t) (see condition (2.5)),
it becomes clear that
τi (s)
Z
p(ξ)dξ : s ≥ t
ρei (t) = inf u(τi (s)) exp λ∗
0
σ(s)
Z
∗
≥ inf u(σ(s)) exp λ
e
(i = 1, . . . , m).
p(ξ)dξ : s ≥ t = ϕ(t)
0
Thus from (1.1) we get
u(σ(tk )) ≥
m
X
i=1
≥
tk
σ(tk )
t
m Zk
X
i=1
Z+∞
pi (s)u(τi (s))ds +
pi (s)u(τi (s))ds
Ztk
σ(tk )
m Z
X
pi (s) exp −λ∗
+
≥
t
m Zk
X
i=1
σ(tk )
m Z
X
pi (s) exp −λ∗
+∞
+
i=1 t
k
pi (s) exp −λ∗
whence, in view of (4.6), we find
u(σ(tk )) ≥
m
X
i=1
×
Ztk
exp ε
σ(tk )
+
m
X
i=1
m Z
X
+∞
= ϕ(t
e k)
i=1 t
k
Zs
0
0
τZi (s)
p(ξ)dξ ρei (s)ds
e
p(ξ)dξ ϕ(s)ds
0
τZi (s)
e
p(ξ)dξ ϕ(s)ds
0
ϕ(t
e k ) exp −ε
τZi (s)
pi (s) exp −λ∗
i=1 t
k
p(ξ)dξ ρei (s)ds
0
+∞
τZi (s)
Ztk
0
p(ξ)dξ
p(ξ)dξ pi (s) exp −λ∗
τZi (s)
0
p(ξ)dξ ds
τZi (s)
Z+∞
ϕ(t
e k)
pi (s) exp −λ∗
p(ξ)dξ ds
tk
pi (s) exp −λ∗
τZi (s)
0
0
p(ξ)dξ ds − ϕ(t
e k ) exp −ε
Ztk
0
p(s)ds
RETARDED DE
×
t
m Zk
X
i=1
σ(tk )
= ϕ(t
e k)
m
X
i=1
exp ε
Zs
0
τZi (ξ)
Z+∞
p(ξ)dξ d
p(ξ1 dξ1 )dξ
pi (ξ) exp −λ∗
0
s
Ztk
exp −ε
71
σ(tk )
p(s)ds
Z+∞
σ(tk )
pi (s) exp −λ∗
τZi (s)
p(ξ)dξ ds.
0
By (4.7), for sufficiently large k we obtain
+∞
σ(tk )
τZi (s)
Z
Z
m
X
p(ξ)dξ
p(ξ)dξ ds ≤ 1
pi (s) exp −λ∗
e−(1+M )ε
exp λ∗
i=1
Consequently,
0
lim inf exp λ∗
t→+∞
Zt
0
0
σ(tk )
p(ξ)dξ
m Z
X
+∞
i=1 t
pi (s) exp −λ∗
τZi (s)
p(ξ)dξ ds ≤ e(1+M )ε .
0
This contradicts inequality (4.2) and the proof of the theorem is complete. ¤
Proof of Theorem 2.2. It suffices to show that conditions (2.6) and (2.7) imply
inequality (2.4). Indeed, by (2.6) and (2.7) there exist ε > 0 and t1 > t0 such
that
Zt
p(s)ds > αi − ε for t ≥ t1 (i = 1, . . . , m)
(4.8)
τi (t)
and for any λ ∈ (0, +∞),
m
1 X
pi (t)eλ(αi −ε) ≥ (1 + ε)λ for t ≥ t1 .
p(t) i=1
(4.9)
According to (4.8) and (4.9), for any λ ∈ (0, +∞) we find
t
τZi (s)
+∞
Z
m Z
X
p(ξ)dξ ds
exp λ p(s)ds
pi (s) exp −λ
0
≥ exp λ
i=1 t
Zt
0
p(s)ds
≥ λ(1 + ε) exp λ
Zt
0
= 1 + ε for t ≥ t1 .
0
Z+∞X
m
t
i=1
pi (s)eλ(αi −ε) exp −λ
p(s)ds
Z+∞
t
exp −λ
Zs
0
−
Zs
0
p(ξ)dξ ds
p(ξ)dξ p(s)ds
Therefore condition (2.4) holds and the proof of the theorem is complete.
¤
72
M. K. GRAMMATIKOPOULOS, R. KOPLATADZE, AND I. P. STAVROULAKIS
Proof of Theorem 2.3. From (2.8), using the inequality ex ≥ ex, clearly follows
(2.7). This completes the proof.
¤
Proof of Theorem 2.4. It is enough to show that (2.9) yields (2.4). Indeed,
according to (2.9) there exist t1 ∈ R+ and ε > 0 such that
Zt
1+ε
for t ≥ t1
e
p(s)ds ≥
(i = 1, . . . , m).
τi (t)
Thus for any λ ∈ (0, +∞) we have
t
τZi (s)
+∞
Z
Z
m
X
p(ξ)dξ ds
pi (s) exp −λ
exp λ p(s)ds
0
≥e
(1+ε)λ
e
exp λ
i=1 t
Zt
0
p(s)ds
Z+∞
t
p(s) exp −λ
e(1 + ε)λ
= 1 + ε.
λe
≥
0
τZi (s)
0
p(ξ)dξ ds
Consequently, (2.4) is satisfied.
¤
Proof of Theorem 2.5. Below we will assume that
lim sup
t→+∞
Zt
pi (s)ds ≤ 1 (i = 1, . . . , m).
τi (t)
Otherwise it is easy to show that (1.1) is oscillatory. Thus, by virtue of (2.10),
condition (2.3) is satisfied. Therefore it is enough to show that inequality (2.4)
holds. Due to (2.11) and (2.12) there exist t1 ∈ R+ and ε ∈ (0, βi ) such that
Zt
pi (s)ds > βi − ε fort ≥ t1 (i = 1, . . . , m)
(4.10)
τi (t)
and for any λ ∈ (0, +∞),
m
X
e(βi −ε)λ
i=1
λ
> 1 + ε.
(4.11)
Put
τ (s)
¯
m ¯ Zi
X
¯
¯
¯
qi (ξ)dξ ¯¯ for s ≥ t ≥ t1
pi (t) − p1 (t) = qi (t), η(t, s) = exp −λ
¯
i=1
t
RETARDED DE
73
(i = 1, . . . , m) and
ψ(t, λ) = exp λ
Zt
m Z
X
p(s)ds
0
+∞
pi (s) exp −λ
i=1 t
τZi (s)
0
p(ξ)dξ ds.
According to (4.10), (4.11), for any λ ∈ (0, +∞) we have
Zt
m Zt
X
ψ(t, λ) = exp λ
qi (s))ds exp λm p1 (s)ds
i=1 0
×
+∞
m Z
X
i=1 t
(qi (s) + p1 (s)) exp −λ
≥ exp λm
Zt
0
p1 (s)ds
m
X
× exp −λm
Zs
0
0
τZi (s)
qi (ξ)dξ exp −λm
i=1 0
Z+∞
i=1 t
Zt
− exp λm
m
X
0
mp1 (s) exp −λm
Z+∞X
m
p1 (s)ds
t
τZi (s)
0
τZi (s)
0
p1 (ξ)dξ ds
p1 (ξ)dξ η(t, s)ds
|qi (s)|η(t, s)
i=1
Zs
p1 (ξ)dξ exp λm
p1 (ξ)dξ ds.
τi (s)
Therefore, if we take into account the condition lim η(t, s) = 1, then by (2.3)
t→+∞
and (2.10) we obtain for any λ ∈ (0, +∞),
lim inf ψ(t, λ) ≥ lim inf exp λm
t→+∞
t→+∞
Zt
0
p1 (s)ds
+∞
Zs
m Z
X
×
mp1 (s) exp −λm p1 (ξ)dξ e(βi −ε)λ ds
i=1 t
0
−lim sup eλm(1+M )
t→+∞
= lim inf exp λm
t→+∞
Zt
0
×
Z+∞X
m
t
|qi (ξ)|η(t, ξ)dξ
i=1
Z+∞
Zs
p1 (s)ds
mp1 (s) exp −λm p1 (ξ)dξ
0
t
m
X
i=1
e(βi −ε)λ =
m
X
e(βi −ε)λ
i=1
λ
.
74
M. K. GRAMMATIKOPOULOS, R. KOPLATADZE, AND I. P. STAVROULAKIS
Consequently, according to (4.11) inequality (4.2) evidently holds. The proof is
complete.
¤
The validity of Theorem 2.6 easily follows from Theorem 2.5 if we take into
consideration the inequality ex ≥ ex.
Remark 4.1. As it is noted in the Introduction, several sufficient conditions
for the oscillation of equation (1.1) for τi (t) = t − τi (i = 1, . . . , m), where τi
(i = 1, . . . , m) are positive constants, are established in [1,15,17], while a nonintegral condition is given in [9] for τi (t) = t − Ti (t), where Ti are continuous
and positive-valued functions on [0,∞). However, as the following example
indicates, even in the case of constant coefficients and constant delays none of
the conditions in the said papers [1,9,15,17] is satisfied, while the conditions of
Theorem 2.5 are satisfied.
Example 4.1. Consider the equation
u′ (t) + u(t − τ ) + u(t − (1/e − τ )) = 0,
(4.12)
where τ ∈ (0, 1e ), τ 6= 1/2e. It is easy to see that none of the conditions in
[1,9,15,17] is satisfied. However we will show that the conditions of Theorem
2.5 are satisfied. To this end it suffices to show the validity of the inequality
!
)
(Ã
1
eτ λ e( e −τ )λ
+
: λ ∈ (0, ∞) > 1.
(4.13)
min
λ
λ
Since
min
½
¾
eτ λ
: λ ∈ (0, ∞) = τ e,
λ
τλ
min
(
)
1
e( e −τ )λ
: λ ∈ (0, ∞) = 1 − τ e
λ
( 1 −τ )λ
and the functions eλ , e e λ attain their minima at different points, it is clear
that (4.13) is valid. According to Theorem 2.5 all solutions of equation (4.12)
oscillate.
Acknowledgement
The research was supported by the Greek Ministry of Development in the
framework of Bilateral S&T Cooperation between the Hellenic Republic and
the Republic of Georgia.
References
1. O. Arino, I. Gyori, and A. Jawhari, Oscillation criteria in delay equations. J. Differential Equations 53(1984), 115–123.
2. J. Chao, Oscillation of linear differential equations with deviating arguments. Theory
Practice Math. 1(1991), 32–41.
3. A. Elbert and I. P. Stavroulakis, Oscillations of first order differential equations
with deviating arguments. Recent trents in differential equations, 163–178, World Sci.
Ser. Appl. Anal., 1, World Sci. Publishing, River Edge, NJ, 1992.
RETARDED DE
75
4. A. Elbert and I. P. Stavroulakis, Oscillation and nonoscillation criteria for delay
differential equations. Proc. Amer. Math. Soc. 123(1995), No. 5, 1503–1510.
5. L. H. Erbe and B. G. Zhang, Oscillation for first order linear differential equations
with deviating arguments. Differential Integral Equations 1(1988), 305–314.
6. N. Fukagai and T. Kusano, Oscillation theory of first order functional-differential
equation with deviating arguments. Ann. Mat. Pura Appl. 136(1984), No. 4, 95–117.
7. M. K. Grammatikopoulos and M. R. Kulenovic, Oscillatory and asymptotic properties of first order differential equations and inequalities with a deviating argument.
Math. Nachr. 123(1985), 7–21.
8. J. K. Hale, Theory of functional differential equations. Springer-Verlag, New York,
1977.
9. B. R. Hunt and J. A. Yorke, When all solutions of x′ = −Σqi (t)x(t − Ti (t)) oscillate.
J. Differential Equations 53(1984), 139–145.
10. R. G. Koplatadze, On zeros of first order differential equations with retarded argument.
(Russian) Trudy Inst. Prikl. Mat. I. N. Vekua 14(1983), 128–134.
11. R. Koplatadze, On oscillatory properties of solutions of functional differential equations. Mem. Differential Equations Math. Phys. 3(1994), 1–177.
12. R. G. Koplatadze and T. A. Chanturia, On oscillatory and monotone solutions
of first order differential equations with retarded argument. (Russian) Differentsial’nye
Uravneniya 18(1982), No. 8, 1463–1465.
13. R. Koplatadze and G. Kvinikadze, On the oscillation of solutions of first order delay
differential inequalities and equations. Georgian Math. J. 1(1994), No. 6, 675–685.
14. G. Ladas, V. Lakshmikantham, and L. S. Papadakis, Oscillations of higher-order
retarded differential equations generated by retarded argument. Delay and Functional
Differential Equations and Their Applications, 219-231, Academic Press, New York, 1972.
15. G. Ladas and I. P. Stavroulakis, Oscillations caused by several retarded and advanced
arguments. J. Differential Equations 44(1982), No. 1, 134–152.
16. B. Li, Oscillations of delay differantial equations with variable coefficients. J. Math. Anal.
Appl. 192(1995), 312–321.
17. B. Li, Oscillations of first order delay differential equations. Proc. Amer. Math. Soc.
124(1996), 3729–3737.
18. A. D. Myshkis, Linear differential equations with the delayed argument. (Russian)
Nauka, Moscow, 1972.
(Received 26.06.2002)
Authors’ addresses:
M. K. Grammatikopoulos and I. P. Stavroulakis
Department of Mathematics
University of Ioannina
451 10 Ioannina
Greece
E-mail: [email protected]
[email protected]
76
M. K. GRAMMATIKOPOULOS, R. KOPLATADZE, AND I. P. STAVROULAKIS
R. Koplatadze
A. Razmadze Mathematical Institute
Georgian Academy of Sciences
1, M. Aleksidze St., Tbilisi 380093
Georgia
E-mail: [email protected]
Volume 10 (2003), Number 1, 63–76
ON THE OSCILLATION OF SOLUTIONS OF FIRST ORDER
DIFFERENTIAL EQUATIONS WITH RETARDED
ARGUMENTS
M. K. GRAMMATIKOPOULOS, R. KOPLATADZE, AND I. P. STAVROULAKIS
Abstract. For the differential equation
m
X
u′ (t) +
pi (t)u(τi (t)) = 0,
i=1
where pi ∈ Lloc (R+ ; R+ ), τi ∈ C(R+ ; R+ ), τi (t) ≤ t for t ∈ R+ , lim τi (t) =
t→+∞
+∞ (i = 1, . . . , m), optimal integral conditions for the oscillation of all solutions are established.
2000 Mathematics Subject Classification: 34C10, 34K11.
Key words and phrases: Retarded arguments, proper solutions, oscillation.
1. Introduction
Consider the differential equation
m
X
u′ (t) +
pi (t)u(τi (t)) = 0,
(1.1)
i=1
where pi ∈ Lloc (R+ ; R+ ), τi ∈ C(R+ ; R+ ), τi (t) ≤ t for t ∈ R+ , lim τi (t) = +∞
t→+∞
(i = 1, . . . , m).
The first systematic study for the oscillation of all solutions of equation (1.1)
for the case of constant coefficients and constant delays was made by Myshkis
[18]. Since then a number of papers have been devoted to this subject. For the
case m=1 the reader is referred to the papers [2–7, 10, 12–14, 16, 18], while for
the case m > 1 to [1, 6, 9, 11, 15, 17]. The difficulties connected with the study
of specific properties of solutions of delay differential equations are emphasized
in the monograph by Hale [8]. In [12] the following statement is proved.
Theorem 1.1. Let m = 1,
Zt
1
lim inf
p1 (s)ds > .
t→+∞
e
τ1 (t)
Then equation (1.1) is oscillatory.
In the case m > 1 there are some difficulties in finding optimal conditions for
the oscillation of solutions of (1.1). In the present paper we make an attempt
at carrying out in this direction. Several sufficient oscillation conditions for the
c Heldermann Verlag www.heldermann.de
ISSN 1072-947X / $8.00 / °
64
M. K. GRAMMATIKOPOULOS, R. KOPLATADZE, AND I. P. STAVROULAKIS
case of several delays are contained in [1, 9, 15, 17]. It is to be pointed out that
the technique used in [17] cannot be applied for equation (1.1).
2. Formulation of the Main Results
Throughout the paper we will assume that pi : R+ → R+ (i = 1, . . . , m)
are locally integrable functions, τi : R+ → R+ (i = 1, . . . , m) are continuous
functions, and
pi (t) ≥ 0, τi (t) ≤ t for t ∈ R+ ,
lim τi (t) = +∞ (i = 1, . . . , m).
t→+∞
(2.1)
Let a ∈ R+ . Denote a0 = inf {τ∗ (t) : t ≥ a}, τ∗ (t) = min {τi (t) : i = 1, . . . , m}.
Definition 2.1. A continuous function u : [a0 , +∞) → R is called a proper
solution of equation (1.1) in [a, +∞) if it is absolutely continuous in each finite
segment contained in [a, +∞) and satisfies (1.1) almost everywhere on [a, +∞)
and sup {|u(s)| : s ≥ t} > 0 for t ≥ a0 .
Definition 2.2. A proper solution of equation (1.1) is said to be oscillatory
if it has a sequence of zeros tending to infinity; otherwise it is said to be nonoscillatory.
Definition 2.3. Equation (1.1) is said to be oscillatory if its every proper
solution is oscillatory.
Theorem 2.1. Let condition (2.1) hold, for some i ∈ {1, . . . , m},
lim inf
t→+∞
Zt
pi (s)ds > 0,
(2.2)
Zt
p(s)ds < +∞
(2.3)
τi (t)
lim sup
t→+∞
σ(t)
and
inf
m Z
X
i=1 t
where
lim inf exp λ
t→+∞
+∞
×
p(t) =
m
X
pi (t),
0
0
p(s)ds
p(ξ)dξ)ds : λ ∈ (0, ∞) > 1,
τZi (s)
pi (s) exp −λ
Zt
σ(t) = inf {τ∗ (s) : s ≥ t ≥ 0} ,
i=1
τ∗ (t) = min {τi (t) : i = 1, . . . , m} .
Then equation (1.1) is oscillatory.
(2.4)
(2.5)
RETARDED DE
65
Remark 2.1. Condition (2.3) is not an essential restriction because if for some
i ∈ {1, . . . , m} ,
Zt
lim sup
pi (s)ds > 1,
t→+∞
τi (t)
then equation (1.1) is oscillatory (see, e.g., [13]).
Theorem 2.2. Let conditions (2.2), (2.3) be fulfilled, p(t) > 0 for sufficiently
large t, and
Zt
lim inf
p(s)ds = αi > 0 (i = 1, . . . , m).
(2.6)
t→+∞
τi (t)
If, moreover, for some t0 ∈ R+ ,
(
Ã
!
)
m
1
1 X
inf
vrai inf
pi (t)eαi λ : λ ∈ (0, +∞) > 1,
λ t≥t0
p(t) i=1
(2.7)
then equation (1.1) is oscillatory.
Theorem 2.3. Let conditions (2.2), (2.3), (2.6) be fulfilled, and p(t) > 0 for
sufficiently large t. Let, moreover, for some t0 ∈ R+ ,
!
Ã
m
1
1 X
αi pi (t) > .
(2.8)
vrai inf
t≥t0
p(t) i=1
e
Then equation (1.1) is oscillatory.
Theorem 2.4. If conditions (2.2), (2.3), (2.6) are fulfilled, and
1
min {αi : i = 1, . . . , m} > ,
e
(2.9)
then equation (1.1) is oscillatory.
Theorem 2.5. Let τi (t) (i = 1, . . . , m) be nondecreasing,
Z∞
| pi (t) − pj (t) | dt < +∞
(i, j = 1, . . . , m),
(2.10)
(i = 1, . . . , m)
(2.11)
0
lim inf
t→+∞
Zt
pi (s)ds = βi > 0
τi (t)
and
min
( m
X eβi λ
i=1
λ
Then equation (1.1) is oscillatory.
: λ ∈ (0, +∞)
)
> 1.
(2.12)
66
M. K. GRAMMATIKOPOULOS, R. KOPLATADZE, AND I. P. STAVROULAKIS
Theorem 2.6. Let conditions (2.10), (2.11) hold, and
m
X
i=1
1
βi > .
e
(2.13)
Then equation (1.1) is oscillatory.
Remark 2.2. It is obvious that Theorem 2.6 coincides with Theorem 1.1 for
the case m = 1.
3. Auxiliary Statements
Lemma 3.1. Let p : R+ → R+ be a summable function in every finite
segment, τ : R+ → R+ be a continuous and nondecreasing function, and
lim τ (t) = +∞. If, moreover,
t→+∞
lim inf
t→+∞
Zt
p(s)ds > 0
(3.1)
τ (t)
and u : [a0 , +∞) → (0, +∞) is a solution of the equation
u′ (t) + p(t)u(τ (t)) = 0,
then there exists λ > 0 such that
t
Z
lim u(t) exp λ p(s)ds = +∞.
t→+∞
(3.2)
(3.3)
0
Proof. First we will show that
lim sup
t→+∞
u(τ (t))
< +∞.
u(t)
(3.4)
By virtue of (3.1) there are c > 0 and t0 ∈ R+ such that
Zt
p(s)ds ≥ c for t ≥ t0 .
τ (t)
Thus for any t > t0 there exists t∗ > t such that
Zt∗
t
c
p(s)ds = ,
2
Zt
c
p(s)ds ≥ .
2
(3.5)
τ (t∗ )
Without loss of generality we can assume that u(τ (t)) > 0 for t ≥ t0 . In view
of (3.5) from (3.2) we have
u(t) ≥
Zt∗
t
p(s)u(τ (s))ds ≥ u(τ (t∗ ))
Zt∗
t
c
p(s)ds = u(τ (t∗ ))
2
RETARDED DE
67
and
u(τ (t∗ )) ≥
Zt
c
p(s)u(τ (s))ds ≥ u(τ (t)).
2
τ (t∗ )
The last two inequalities result in u(t) ≥ (c2 /4)u(τ (t)). This, in view of the
arbitrariness of t, means that (3.4) is valid. Thus from (3.2) we get
Zt
Zt
4
u(τ (s))
ds ≥ u(t0 ) exp − 2 p(s)ds .
(3.6)
u(t) = u(t0 ) exp − p(s)
u(s)
c
t0
t0
On the other hand, from (3.1) it obviously follows that
Z+∞
p(s)ds = +∞.
t0
Therefore, according to (3.6), there exists λ > 0 such that (3.3) is satisfied. ¤
Lemma 3.2. Let (3.1) be fulfilled, p, q : R+ → R+ be summable functions in
every finite segment, τ, τ0 : R+ → R+ be continuous functions,
lim τ (t) = lim t → +∞τ0 (t) = +∞,
t→+∞
q(t) ≥ p(t), τ0 (t) ≤ τ (t) ≤ t for t ≥ t0 .
(3.7)
If, moreover, v : [t0 , +∞) → (0, +∞) is a solution of the inequality
v ′ (t) + q(t)v(τ0 (t)) ≤ 0,
(3.8)
then equation (3.2) has a solution u : [t1 , +∞) → (0, +∞) satisfying the condition
0 < u(t) ≤ v(t) for t ≥ t1 ,
(3.9)
where t1 ≥ t0 is a sufficiently large number.
Proof. Let v : [t0 , +∞) → (0, +∞) be a solution of inequality (3.8). By (3.1)
and (3.7) there is t1 > t0 such that v(τ0 (t)) > 0 for t > t1 and
Zt
p(s)ds > 0 for t ≥ t1 .
(3.10)
τ (t)
From (3.8) we have
Z+∞
v(t) ≥
q(s)v(τ (s))ds for t ≥ t1 .
t
(3.11)
68
M. K. GRAMMATIKOPOULOS, R. KOPLATADZE, AND I. P. STAVROULAKIS
Denote t∗1 = inf {τ (t) : t ≥ t1 } and consider the sequence of functions ui :
[t∗1 , +∞) → [0, +∞) (i = 1, 2, 3, . . . ) defined by the following equalities:
ui (t) =
+∞
R p(s)u
u1 (t) = v(t) for t ≥ t∗1 ,
i−1 (τ (s))ds
for t ≥ t1
t
v(t) − v(t1 ) + ui (t1 ) for
t∗1
(i = 2, 3, . . . ).
(3.12)
≤ t < t1
On account of the last inequality of (3.7) and conditions (3.10), (3.11) it is clear
that 0 < ui (t) ≤ ui−1 (t) ≤ v(t) (i = 2, 3, . . . ) for t ≥ t1 . Thus 0 ≤ u(t) ≤ v(t)
for t ≥ t1 , where u(t) = limi→+∞ ui (t). Let us show that u(t) > 0 for t ≥ t1 .
Otherwise there is t2 ≥ t1 such that u(t) ≡ 0 for t ≥ t2 and u(t) > 0 for
t ∈ [t∗1 , t2 ). Denote by U the set of points t satisfying τ (t) = t2 , and put
t∗ = min U. Evidently t∗ ≥ t2 . Therefore, by (3.10) and (3.12), we get
Z+∞
Zt∗
u(t2 ) =
p(s)u(τ (s))ds ≥
p(s)u(τ (s))ds > 0.
t2
τ (t∗ )
The obtained contradiction proves that u(t) > 0 for t ≥ t1 . Consequently we
have 0 < u(t) ≤ v(t) for t ≥ t1 .
¤
Lemma 3.3. Let condition (2.1) hold, for some i ∈ {1, . . . , m},
lim inf
t→+∞
Zt
pi (s)ds > 0,
(3.13)
τi (t)
and u : [t0 , +∞) → (0, +∞) be a positive solution of equation (1.1). Then there
exists λ > 0 such that
t
Z
lim u(t) exp λ pi (s)ds = +∞.
(3.14)
t→+∞
0
Proof. It is obvious that u is a solution of the differential inequality
u′ (t) + pi (t)u(τi (t)) ≤ 0 for t ≥ t1 ,
where t1 > t0 is a sufficiently large number. Thus, taking into account (3.13)
and Lemmas 3.1, 3.2, there exists λ > 0 such that (3.14) is fulfilled.
¤
Lemma 3.4. Let t0 ∈ R+ , ϕ, ψ ∈ C([t0 , +∞); (0, +∞)), ψ(t) be non-increasing and
lim ϕ(t) = +∞, lim inf ψ(t)ϕ(t)
e = 0,
t→+∞
t→+∞
where ϕ(t)
e = inf {ϕ(s) : s ≥ t ≥ t0 .} Then there exists an increasing sequence
of points {tk }+∞
k=1 such that tk ↑ +∞ as k ↑ +∞ and
]
ϕ(t
e ≥ ψ(tk )ϕ(t
e k ) for t0 ≤ t ≤ tk .
k ) = ϕ(tk ), ψ(t)ϕ(t)
For the proof of Lemma 3.4 see [11, Lemma 7.1].
RETARDED DE
69
4. Proof of the Main results
Proof of Theorem 2.1. Assume the contrary. Let equation (1.1) have a nonoscillatory proper solution u : [t0 , +∞) → (0, +∞). According to condition
(2.2) and Lemma 3.3, there exists λ > 0 such that
t
Z
(4.1)
lim u(t) exp λ p(s)ds = +∞,
t→+∞
0
where the function p(t) is defined by the first equality of (2.5).
Denote by Λ the set of all λ satisfying condition (4.1), and put λ0 = inf Λ.
Since u(t) is non-increasing, in view of (4.1) it is obvious that λ0 ≥ 0. By the
definition of λ0 and condition (2.4), there exist ε > 0 and λ∗ > λ0 such that
τZi (ξ)
+∞
Z
Zt
m
X
p(s)ds dξ
pi (ξ) exp −λ
lim inf exp λ∗ p(s)ds
t→+∞
i=1 t
0
0
(1+M )ε
> (1 + ε)e
,
Zt
lim u(t) exp λ∗ p(ξ)dξ = +∞,
(4.2)
p(ξ)dξ = 0,
(4.4)
p(s)ds.
(4.5)
(4.3)
t→+∞
0
lim inf exp (λ∗ − ε)
t→+∞
where
M = lim sup
t→+∞
Zt
0
Zt
σ(t)
Due to (4.3) and (4.4) it is clear that the functions ϕ and ψ satisfy the conditions
of Lemma 3.4 where
σ(t)
Z
Zt
ϕ(t) = u(σ(t)) exp λ∗
p(s)ds , ψ(t) = exp −ε p(s)ds
0
0
and the function σ(t) is defined by the last two equalities of (2.5). Therefore,
by Lemma 3.4, there exists an increasing sequence of points {tk }+∞
k=1 such that
Zt
Ztk
e exp −ε p(s)ds for t0 ≤ t ≤ tk , (4.6)
ϕ(t
e k ) exp −ε p(s)ds) ≤ ϕ(t)
0
ϕ(t
e k ) = u(σ(tk )) exp λ∗
0
σ(t
Z k)
0
p(s)ds .
(4.7)
70
M. K. GRAMMATIKOPOULOS, R. KOPLATADZE, AND I. P. STAVROULAKIS
If we take into account the definition of the function σ(t) (see condition (2.5)),
it becomes clear that
τi (s)
Z
p(ξ)dξ : s ≥ t
ρei (t) = inf u(τi (s)) exp λ∗
0
σ(s)
Z
∗
≥ inf u(σ(s)) exp λ
e
(i = 1, . . . , m).
p(ξ)dξ : s ≥ t = ϕ(t)
0
Thus from (1.1) we get
u(σ(tk )) ≥
m
X
i=1
≥
tk
σ(tk )
t
m Zk
X
i=1
Z+∞
pi (s)u(τi (s))ds +
pi (s)u(τi (s))ds
Ztk
σ(tk )
m Z
X
pi (s) exp −λ∗
+
≥
t
m Zk
X
i=1
σ(tk )
m Z
X
pi (s) exp −λ∗
+∞
+
i=1 t
k
pi (s) exp −λ∗
whence, in view of (4.6), we find
u(σ(tk )) ≥
m
X
i=1
×
Ztk
exp ε
σ(tk )
+
m
X
i=1
m Z
X
+∞
= ϕ(t
e k)
i=1 t
k
Zs
0
0
τZi (s)
p(ξ)dξ ρei (s)ds
e
p(ξ)dξ ϕ(s)ds
0
τZi (s)
e
p(ξ)dξ ϕ(s)ds
0
ϕ(t
e k ) exp −ε
τZi (s)
pi (s) exp −λ∗
i=1 t
k
p(ξ)dξ ρei (s)ds
0
+∞
τZi (s)
Ztk
0
p(ξ)dξ
p(ξ)dξ pi (s) exp −λ∗
τZi (s)
0
p(ξ)dξ ds
τZi (s)
Z+∞
ϕ(t
e k)
pi (s) exp −λ∗
p(ξ)dξ ds
tk
pi (s) exp −λ∗
τZi (s)
0
0
p(ξ)dξ ds − ϕ(t
e k ) exp −ε
Ztk
0
p(s)ds
RETARDED DE
×
t
m Zk
X
i=1
σ(tk )
= ϕ(t
e k)
m
X
i=1
exp ε
Zs
0
τZi (ξ)
Z+∞
p(ξ)dξ d
p(ξ1 dξ1 )dξ
pi (ξ) exp −λ∗
0
s
Ztk
exp −ε
71
σ(tk )
p(s)ds
Z+∞
σ(tk )
pi (s) exp −λ∗
τZi (s)
p(ξ)dξ ds.
0
By (4.7), for sufficiently large k we obtain
+∞
σ(tk )
τZi (s)
Z
Z
m
X
p(ξ)dξ
p(ξ)dξ ds ≤ 1
pi (s) exp −λ∗
e−(1+M )ε
exp λ∗
i=1
Consequently,
0
lim inf exp λ∗
t→+∞
Zt
0
0
σ(tk )
p(ξ)dξ
m Z
X
+∞
i=1 t
pi (s) exp −λ∗
τZi (s)
p(ξ)dξ ds ≤ e(1+M )ε .
0
This contradicts inequality (4.2) and the proof of the theorem is complete. ¤
Proof of Theorem 2.2. It suffices to show that conditions (2.6) and (2.7) imply
inequality (2.4). Indeed, by (2.6) and (2.7) there exist ε > 0 and t1 > t0 such
that
Zt
p(s)ds > αi − ε for t ≥ t1 (i = 1, . . . , m)
(4.8)
τi (t)
and for any λ ∈ (0, +∞),
m
1 X
pi (t)eλ(αi −ε) ≥ (1 + ε)λ for t ≥ t1 .
p(t) i=1
(4.9)
According to (4.8) and (4.9), for any λ ∈ (0, +∞) we find
t
τZi (s)
+∞
Z
m Z
X
p(ξ)dξ ds
exp λ p(s)ds
pi (s) exp −λ
0
≥ exp λ
i=1 t
Zt
0
p(s)ds
≥ λ(1 + ε) exp λ
Zt
0
= 1 + ε for t ≥ t1 .
0
Z+∞X
m
t
i=1
pi (s)eλ(αi −ε) exp −λ
p(s)ds
Z+∞
t
exp −λ
Zs
0
−
Zs
0
p(ξ)dξ ds
p(ξ)dξ p(s)ds
Therefore condition (2.4) holds and the proof of the theorem is complete.
¤
72
M. K. GRAMMATIKOPOULOS, R. KOPLATADZE, AND I. P. STAVROULAKIS
Proof of Theorem 2.3. From (2.8), using the inequality ex ≥ ex, clearly follows
(2.7). This completes the proof.
¤
Proof of Theorem 2.4. It is enough to show that (2.9) yields (2.4). Indeed,
according to (2.9) there exist t1 ∈ R+ and ε > 0 such that
Zt
1+ε
for t ≥ t1
e
p(s)ds ≥
(i = 1, . . . , m).
τi (t)
Thus for any λ ∈ (0, +∞) we have
t
τZi (s)
+∞
Z
Z
m
X
p(ξ)dξ ds
pi (s) exp −λ
exp λ p(s)ds
0
≥e
(1+ε)λ
e
exp λ
i=1 t
Zt
0
p(s)ds
Z+∞
t
p(s) exp −λ
e(1 + ε)λ
= 1 + ε.
λe
≥
0
τZi (s)
0
p(ξ)dξ ds
Consequently, (2.4) is satisfied.
¤
Proof of Theorem 2.5. Below we will assume that
lim sup
t→+∞
Zt
pi (s)ds ≤ 1 (i = 1, . . . , m).
τi (t)
Otherwise it is easy to show that (1.1) is oscillatory. Thus, by virtue of (2.10),
condition (2.3) is satisfied. Therefore it is enough to show that inequality (2.4)
holds. Due to (2.11) and (2.12) there exist t1 ∈ R+ and ε ∈ (0, βi ) such that
Zt
pi (s)ds > βi − ε fort ≥ t1 (i = 1, . . . , m)
(4.10)
τi (t)
and for any λ ∈ (0, +∞),
m
X
e(βi −ε)λ
i=1
λ
> 1 + ε.
(4.11)
Put
τ (s)
¯
m ¯ Zi
X
¯
¯
¯
qi (ξ)dξ ¯¯ for s ≥ t ≥ t1
pi (t) − p1 (t) = qi (t), η(t, s) = exp −λ
¯
i=1
t
RETARDED DE
73
(i = 1, . . . , m) and
ψ(t, λ) = exp λ
Zt
m Z
X
p(s)ds
0
+∞
pi (s) exp −λ
i=1 t
τZi (s)
0
p(ξ)dξ ds.
According to (4.10), (4.11), for any λ ∈ (0, +∞) we have
Zt
m Zt
X
ψ(t, λ) = exp λ
qi (s))ds exp λm p1 (s)ds
i=1 0
×
+∞
m Z
X
i=1 t
(qi (s) + p1 (s)) exp −λ
≥ exp λm
Zt
0
p1 (s)ds
m
X
× exp −λm
Zs
0
0
τZi (s)
qi (ξ)dξ exp −λm
i=1 0
Z+∞
i=1 t
Zt
− exp λm
m
X
0
mp1 (s) exp −λm
Z+∞X
m
p1 (s)ds
t
τZi (s)
0
τZi (s)
0
p1 (ξ)dξ ds
p1 (ξ)dξ η(t, s)ds
|qi (s)|η(t, s)
i=1
Zs
p1 (ξ)dξ exp λm
p1 (ξ)dξ ds.
τi (s)
Therefore, if we take into account the condition lim η(t, s) = 1, then by (2.3)
t→+∞
and (2.10) we obtain for any λ ∈ (0, +∞),
lim inf ψ(t, λ) ≥ lim inf exp λm
t→+∞
t→+∞
Zt
0
p1 (s)ds
+∞
Zs
m Z
X
×
mp1 (s) exp −λm p1 (ξ)dξ e(βi −ε)λ ds
i=1 t
0
−lim sup eλm(1+M )
t→+∞
= lim inf exp λm
t→+∞
Zt
0
×
Z+∞X
m
t
|qi (ξ)|η(t, ξ)dξ
i=1
Z+∞
Zs
p1 (s)ds
mp1 (s) exp −λm p1 (ξ)dξ
0
t
m
X
i=1
e(βi −ε)λ =
m
X
e(βi −ε)λ
i=1
λ
.
74
M. K. GRAMMATIKOPOULOS, R. KOPLATADZE, AND I. P. STAVROULAKIS
Consequently, according to (4.11) inequality (4.2) evidently holds. The proof is
complete.
¤
The validity of Theorem 2.6 easily follows from Theorem 2.5 if we take into
consideration the inequality ex ≥ ex.
Remark 4.1. As it is noted in the Introduction, several sufficient conditions
for the oscillation of equation (1.1) for τi (t) = t − τi (i = 1, . . . , m), where τi
(i = 1, . . . , m) are positive constants, are established in [1,15,17], while a nonintegral condition is given in [9] for τi (t) = t − Ti (t), where Ti are continuous
and positive-valued functions on [0,∞). However, as the following example
indicates, even in the case of constant coefficients and constant delays none of
the conditions in the said papers [1,9,15,17] is satisfied, while the conditions of
Theorem 2.5 are satisfied.
Example 4.1. Consider the equation
u′ (t) + u(t − τ ) + u(t − (1/e − τ )) = 0,
(4.12)
where τ ∈ (0, 1e ), τ 6= 1/2e. It is easy to see that none of the conditions in
[1,9,15,17] is satisfied. However we will show that the conditions of Theorem
2.5 are satisfied. To this end it suffices to show the validity of the inequality
!
)
(Ã
1
eτ λ e( e −τ )λ
+
: λ ∈ (0, ∞) > 1.
(4.13)
min
λ
λ
Since
min
½
¾
eτ λ
: λ ∈ (0, ∞) = τ e,
λ
τλ
min
(
)
1
e( e −τ )λ
: λ ∈ (0, ∞) = 1 − τ e
λ
( 1 −τ )λ
and the functions eλ , e e λ attain their minima at different points, it is clear
that (4.13) is valid. According to Theorem 2.5 all solutions of equation (4.12)
oscillate.
Acknowledgement
The research was supported by the Greek Ministry of Development in the
framework of Bilateral S&T Cooperation between the Hellenic Republic and
the Republic of Georgia.
References
1. O. Arino, I. Gyori, and A. Jawhari, Oscillation criteria in delay equations. J. Differential Equations 53(1984), 115–123.
2. J. Chao, Oscillation of linear differential equations with deviating arguments. Theory
Practice Math. 1(1991), 32–41.
3. A. Elbert and I. P. Stavroulakis, Oscillations of first order differential equations
with deviating arguments. Recent trents in differential equations, 163–178, World Sci.
Ser. Appl. Anal., 1, World Sci. Publishing, River Edge, NJ, 1992.
RETARDED DE
75
4. A. Elbert and I. P. Stavroulakis, Oscillation and nonoscillation criteria for delay
differential equations. Proc. Amer. Math. Soc. 123(1995), No. 5, 1503–1510.
5. L. H. Erbe and B. G. Zhang, Oscillation for first order linear differential equations
with deviating arguments. Differential Integral Equations 1(1988), 305–314.
6. N. Fukagai and T. Kusano, Oscillation theory of first order functional-differential
equation with deviating arguments. Ann. Mat. Pura Appl. 136(1984), No. 4, 95–117.
7. M. K. Grammatikopoulos and M. R. Kulenovic, Oscillatory and asymptotic properties of first order differential equations and inequalities with a deviating argument.
Math. Nachr. 123(1985), 7–21.
8. J. K. Hale, Theory of functional differential equations. Springer-Verlag, New York,
1977.
9. B. R. Hunt and J. A. Yorke, When all solutions of x′ = −Σqi (t)x(t − Ti (t)) oscillate.
J. Differential Equations 53(1984), 139–145.
10. R. G. Koplatadze, On zeros of first order differential equations with retarded argument.
(Russian) Trudy Inst. Prikl. Mat. I. N. Vekua 14(1983), 128–134.
11. R. Koplatadze, On oscillatory properties of solutions of functional differential equations. Mem. Differential Equations Math. Phys. 3(1994), 1–177.
12. R. G. Koplatadze and T. A. Chanturia, On oscillatory and monotone solutions
of first order differential equations with retarded argument. (Russian) Differentsial’nye
Uravneniya 18(1982), No. 8, 1463–1465.
13. R. Koplatadze and G. Kvinikadze, On the oscillation of solutions of first order delay
differential inequalities and equations. Georgian Math. J. 1(1994), No. 6, 675–685.
14. G. Ladas, V. Lakshmikantham, and L. S. Papadakis, Oscillations of higher-order
retarded differential equations generated by retarded argument. Delay and Functional
Differential Equations and Their Applications, 219-231, Academic Press, New York, 1972.
15. G. Ladas and I. P. Stavroulakis, Oscillations caused by several retarded and advanced
arguments. J. Differential Equations 44(1982), No. 1, 134–152.
16. B. Li, Oscillations of delay differantial equations with variable coefficients. J. Math. Anal.
Appl. 192(1995), 312–321.
17. B. Li, Oscillations of first order delay differential equations. Proc. Amer. Math. Soc.
124(1996), 3729–3737.
18. A. D. Myshkis, Linear differential equations with the delayed argument. (Russian)
Nauka, Moscow, 1972.
(Received 26.06.2002)
Authors’ addresses:
M. K. Grammatikopoulos and I. P. Stavroulakis
Department of Mathematics
University of Ioannina
451 10 Ioannina
Greece
E-mail: [email protected]
[email protected]
76
M. K. GRAMMATIKOPOULOS, R. KOPLATADZE, AND I. P. STAVROULAKIS
R. Koplatadze
A. Razmadze Mathematical Institute
Georgian Academy of Sciences
1, M. Aleksidze St., Tbilisi 380093
Georgia
E-mail: [email protected]