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

u′(t) + ∑_{i=1}^{m} p_{i}(t)u(τ_{i}(t)) = 0,

where p_{i} ∈ L_{loc}(R_{+}; R_{+}), τ_{i} ∈ C(R_{+}; R_{+}), τ_{i}(t) ≤ t for t ∈ R^{+}, lim_{t→+∞} τ_{i}(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

u′(t) + ∑_{i=1}^{m} p_{i}(t)u(τ_{i}(t)) = 0,  (1.1)

where p_{i} ∈ L_{loc}(R_{+}; R_{+}), τ_{i} ∈ C(R_{+}; R_{+}), τ_{i}(t) ≤ t for t ∈ R_{+}, lim_{t→+∞} τ_{i}(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,

lim inf_{t→+∞} ∫_{τ_{1}(t)}^{t} p_{1}(s)ds > \frac{1}{e}.

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

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

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 p_{i} : R_{+} → R_{+} (i = 1, …, m) are locally integrable functions, τ_{i} : R_{+} → R_{+} (i = 1, …, m) are continuous functions, and

p_{i}(t) ≥ 0, τ_{i}(t) ≤ t for t ∈ R_{+}, lim_{t→+∞} τ_{i}(t) = +∞ (i = 1, …, m). (2.1)

Let a ∈ R_{+}. Denote a_{0} = inf {τ_{∗}(t) : t ≥ a}, τ_{∗}(t) = min {τ_{i}(t) : i = 1, …, m}.

Definition 2.1. A continuous function u : [a_{0}, +∞) → 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 ≥ a_{0}.

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→+∞} ∫_{τ_{i}(t)}^{t} p_{i}(s)ds > 0,  (2.2)

lim sup_{t→+∞} ∫_{σ(t)}^{t} \overline{p}(s)ds < +∞  (2.3)

and

inf \{lim inf_{t→+∞} exp (λ ∫_{0}^{t} \overline{p}(s)ds)

× ∑_{i=1}^{m} ∫_{t}^{+∞}p_{i}(s) exp (−λ ∫_{0}^{τ_{i}(s)}\overline{p}(ξ)dξ)ds) : λ ∈ (0, ∞)\} > 1,  (2.4)

where

\begin{matrix}\overline{p}(t) = ∑_{i=1}^{m} p_{i}(t), σ(t) = inf \{τ_{∗}(s) : s ≥ t ≥ 0\} , \\ τ_{∗}(t) = min \{τ_{i}(t) : i = 1, …, m\} . \end{matrix}  (2.5)

Then equation (1.1) is oscillatory.

Remark 2.1. Condition (2.3) is not an essential restriction because if for some i ∈ {1, …, m} ,

lim sup_{t→+∞} ∫_{τ_{i}(t)}^{t} p_{i}(s)ds > 1,

then equation (1.1) is oscillatory (see, e.g., [13]).

Theorem 2.2. Let conditions (2.2), (2.3) be fulfilled, \overline{p}(t) > 0 for sufficiently large t, and

lim inf_{t→+∞} ∫_{τ_{i}(t)}^{t} \overline{p}(s)ds = α_{i} > 0 (i = 1, …, m).  (2.6)

If, moreover, for some t_{0} ∈ R_{+},

inf \{\frac{1}{λ}vrai inf_{t≥t_{0}} (\frac{1}{\overline{p}(t)} ∑_{i=1}^{m} p_{i}(t)e^{α_{i}λ}) : λ ∈ (0, +∞)\} > 1,  (2.7)

then equation (1.1) is oscillatory.

Theorem 2.3. Let conditions (2.2), (2.3), (2.6) be fulfilled, and \overline{p}(t) > 0 for sufficiently large t. Let, moreover, for some t_{0} ∈ R_{+},

vrai inf_{t≥t_{0}} (\frac{1}{\overline{p}(t)} ∑_{i=1}^{m} α_{i}p_{i}(t)) > \frac{1}{e}.  (2.8)

Then equation (1.1) is oscillatory.

Theorem 2.4. If conditions (2.2), (2.3), (2.6) are fulfilled, and

min \{α_{i} : i = 1, …, m\} > \frac{1}{e},  (2.9)

then equation (1.1) is oscillatory.

Theorem 2.5. Let τ_{i}(t) (i = 1, …, m) be nondecreasing,

∫_{0}^{∞} | p_{i}(t) − p_{j}(t) | dt < +∞ (i, j = 1, …, m),  (2.10)

lim inf_{t→+∞} ∫_{τ_{i}(t)}^{t} p_{i}(s)ds = β_{i} > 0 (i = 1, …, m)  (2.11)

and

min \{∑_{i=1}^{m} \frac{e^{β_{i}λ}}{λ} : λ ∈ (0, +∞) \} > 1.  (2.12)

Then equation (1.1) is oscillatory.

Theorem 2.6. Let conditions (2.10), (2.11) hold, and

∑_{i=1}^{m} β_{i} > \frac{1}{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_{t→+∞}τ(t) = +∞. If, moreover,

lim inf_{t→+∞} ∫_{τ(t)}^{t} p(s)ds > 0  (3.1)

and u : [a_{0}, +∞) → (0, +∞) is a solution of the equation

u′(t) + p(t)u(τ(t)) = 0,  (3.2)

then there exists λ > 0 such that

lim_{t→+∞} u(t) {exp (λ ∫_{0}^{t} p(s)ds)} = +∞.  (3.3)

Proof. First we will show that

lim sup_{t→+∞} \frac{u(τ(t))}{u(t)} < +∞.  (3.4)

By virtue of (3.1) there are c > 0 and t_{0} ∈ R_{+} such that

∫_{τ(t)}^{t} p(s)ds ≥ c for t ≥ t_{0}.

Thus for any t > t_{0} there exists t^{∗} > t such that

∫_{t}^{t^{∗}} p(s)ds = \frac{c}{2}, ∫_{τ(t^{∗})}^{t} p(s)ds ≥ \frac{c}{2}.  (3.5)

Without loss of generality we can assume that u(τ(t)) > 0 for t ≥ t_{0}. In view of (3.5) from (3.2) we have

u(t) ≥ ∫_{t}^{t^{∗}} p(s)u(τ(s))ds ≥ u(τ(t^{∗})) ∫_{t}^{t^{∗}} p(s)ds = \frac{c}{2}u(τ(t^{∗}))

and

u(τ(t^{∗})) ≥ ∫_{τ(t^{∗})}^{t} p(s)u(τ(s))ds ≥ \frac{c}{2}u(τ(t)).

The last two inequalities result in u(t) ≥ (c^{2}/4)u(τ(t)). This, in view of the arbitrariness of t, means that (3.4) is valid. Thus from (3.2) we get

u(t) = u(t_{0}) exp (− ∫_{t_{0}}^{t} p(s)\frac{u(τ(s))}{u(s)}ds) ≥ u(t_{0}) exp (−\frac{4}{c^{2}} ∫_{t_{0}}^{t} p(s)ds) .  (3.6)

On the other hand, from (3.1) it obviously follows that

∫_{t_{0}}^{+∞}p(s)ds = +∞.

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, τ_{0} : R_{+} → R_{+} be continuous functions,

lim_{t→+∞} τ(t) = lim t → +∞τ_{0}(t) = +∞,

q(t) ≥ p(t), τ_{0}(t) ≤ τ(t) ≤ t for t ≥ t_{0}.

(3.7)

If, moreover, v : [t_{0}, +∞) → (0, +∞) is a solution of the inequality

v′(t) + q(t)v(τ_{0}(t)) ≤ 0,  (3.8)

then equation (3.2) has a solution u : [t_{1}, +∞) → (0, +∞) satisfying the condition

0 < u(t) ≤ v(t) for t ≥ t_{1},  (3.9)

where t_{1} ≥ t_{0} is a sufficiently large number.

Proof. Let v : [t_{0}, +∞) → (0, +∞) be a solution of inequality (3.8). By (3.1) and (3.7) there is t_{1} > t_{0} such that v(τ_{0}(t)) > 0 for t > t_{1} and

∫_{τ(t)}^{t} p(s)ds > 0 for t ≥ t_{1}.  (3.10)

From (3.8) we have

v(t) ≥ ∫_{t}^{+∞}q(s)v(τ(s))ds for t ≥ t_{1}.  (3.11)

Denote t_{1}^{∗} = inf {τ(t) : t ≥ t_{1}} and consider the sequence of functions u_{i} : [t_{1}^{∗}, +∞) → [0, +∞) (i = 1, 2, 3, …) defined by the following equalities:

u_{1}(t) = v(t) for t ≥ t_{1}^{∗},

u_{i}(t) = \begin{cases}∫_{t}^{+∞}p(s)u_{i−1}(τ(s))ds & for t ≥ t_{1} \\ v(t) − v(t_{1}) + u_{i}(t_{1}) & for t_{1}^{∗} ≤ t < t_{1} \end{cases} (i = 2, 3, …).  (3.12)

On account of the last inequality of (3.7) and conditions (3.10), (3.11) it is clear that 0 < u_{i}(t) ≤ u_{i−1}(t) ≤ v(t) (i = 2, 3, …) for t ≥ t_{1}. Thus 0 ≤ u(t) ≤ v(t) for t ≥ t_{1}, where u(t) = lim_{i→+∞} u_{i}(t). Let us show that u(t) > 0 for t ≥ t_{1}. Otherwise there is t_{2} ≥ t_{1}such that u(t) ≡ 0 for t ≥ t_{2} and u(t) > 0 for t ∈ [t_{1}^{∗}, t_{2}). Denote by U the set of points t satisfying τ(t) = t_{2}, and put t_{∗} = min U. Evidently t_{∗} ≥ t_{2}. Therefore, by (3.10) and (3.12), we get

u(t_{2}) = ∫_{t_{2}}^{+∞}p(s)u(τ(s))ds ≥ ∫_{τ(t^{∗})}^{t^{∗}} p(s)u(τ(s))ds > 0.

The obtained contradiction proves that u(t) > 0 for t ≥ t_{1}. Consequently we have 0 < u(t) ≤ v(t) for t ≥ t_{1}. □

Lemma 3.3. Let condition (2.1) hold, for some i ∈ {1, …, m},

lim inf_{t→+∞} ∫_{τ_{i}(t)}^{t} p_{i}(s)ds > 0,  (3.13)

and u : [t_{0}, +∞) → (0, +∞) be a positive solution of equation (1.1). Then there exists λ > 0 such that

lim_{t→+∞} u(t) exp (λ ∫_{0}^{t} p_{i}(s)ds) = +∞.  (3.14)

Proof. It is obvious that u is a solution of the differential inequality

u′(t) + p_{i}(t)u(τ_{i}(t)) ≤ 0 for t ≥ t_{1},

where t_{1} > t_{0} 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 t_{0} ∈ R_{+}, φ, ψ ∈ C([t_{0}, +∞); (0, +∞)), ψ(t) be non-in-creasing and

lim_{t→+∞} φ(t) = +∞, lim inf_{t→+∞}ψ(t)φ̃(t) = 0,

where φ̃(t) = inf {φ(s) : s ≥ t ≥ t_{0}.} Then there exists an increasing sequence of points {t_{k}}_{k=1}^{+∞} such that t_{k} ↑ +∞ as k ↑ +∞ and

{ \atop φ(t_{k})} = φ(t_{k}), ψ(t)φ̃(t) ≥ ψ(t_{k})φ̃(t_{k}) for t_{0} ≤ t ≤ t_{k}.

For the proof of Lemma 3.4 see [11, Lemma 7.1].

4. Proof of the Main results

Proof of Theorem 2.1. Assume the contrary. Let equation (1.1) have a nonoscillatory proper solution u : [t_{0}, +∞) → (0, +∞). According to condition (2.2) and Lemma 3.3, there exists λ > 0 such that

lim_{t→+∞} u(t) exp (λ ∫_{0}^{t} \overline{p}(s)ds) = +∞,  (4.1)

where the function \overline{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 λ_{0} = inf Λ. Since u(t) is non-increasing, in view of (4.1) it is obvious that λ_{0} ≥ 0. By the definition of λ_{0} and condition (2.4), there exist ε > 0 and λ^{∗} > λ_{0} such that

lim inf_{t→+∞} \{exp (λ^{∗} ∫_{0}^{t} \overline{p}(s)ds) ∑_{i=1}^{m} ∫_{t}^{+∞}p_{i}(ξ) exp (−λ ∫_{0}^{τ_{i}(ξ)}\overline{p}(s)ds) dξ\}
> (1 + ε)e^{(1+M)ε},  (4.2)
lim_{t→+∞} u(t) exp (λ^{∗} ∫_{0}^{t} \overline{p}(ξ)dξ) = +∞,  (4.3)

lim inf_{t→+∞} exp ((λ^{∗} − ε) ∫_{0}^{t} \overline{p}(ξ)dξ) = 0,  (4.4)

where

M = lim sup_{t→+∞} ∫_{σ(t)}^{t} \overline{p}(s)ds.  (4.5)

Due to (4.3) and (4.4) it is clear that the functions φ and ψ satisfy the conditions of Lemma 3.4 where

φ(t) = u(σ(t)) exp (λ^{∗} ∫_{0}^{σ(t)}\overline{p}(s)ds) , ψ(t) = exp (−ε ∫_{0}^{t} \overline{p}(s)ds)

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 {t_{k}}_{k=1}^{+∞} such that

φ̃(t_{k}) exp (−ε ∫_{0}^{t_{k}} \overline{p}(s)ds)) ≤ φ̃(t) exp (−ε ∫_{0}^{t} \overline{p}(s)ds) for t_{0} ≤ t ≤ t_{k}, (4.6)

φ̃(t_{k}) = u(σ(t_{k})) exp (λ^{∗} ∫_{0}^{σ(t_{k})}\overline{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

ρ̃_{i}(t) = inf \{u(τ_{i}(s)) exp (λ^{∗} ∫_{0}^{τ_{i}(s)}\overline{p}(ξ)dξ) : s ≥ t\}

≥ inf \{u(σ(s)) exp (λ^{∗} ∫_{0}^{σ(s)}\overline{p}(ξ)dξ) : s ≥ t\} = φ̃(t) (i = 1, …, m).

Thus from (1.1) we get

u(σ(t_{k})) ≥ ∑_{i=1}^{m} ( ∫_{σ(t_{k})}^{t_{k}} p_{i}(s)u(τ_{i}(s))ds + ∫_{t_{k}}^{+∞}p_{i}(s)u(τ_{i}(s))ds)

≥ ∑_{i=1}^{m} ∫_{σ(t_{k})}^{t_{k}} p_{i}(s) exp (−λ^{∗} ∫_{0}^{τ_{i}(s)}\overline{p}(ξ)dξ) ρ̃_{i}(s)ds

+ ∑_{i=1}^{m} ∫_{t_{k}}^{+∞}p_{i}(s) exp (−λ^{∗} ∫_{0}^{τ_{i}(s)}\overline{p}(ξ)dξ) ρ̃_{i}(s)ds

≥ ∑_{i=1}^{m} ∫_{σ(t_{k})}^{t_{k}} p_{i}(s) exp (−λ^{∗} ∫_{0}^{τ_{i}(s)}\overline{p}(ξ)dξ) φ̃(s)ds

+ ∑_{i=1}^{m} ∫_{t_{k}}^{+∞}p_{i}(s) exp (−λ^{∗} ∫_{0}^{τ_{i}(s)}\overline{p}(ξ)dξ) φ̃(s)ds

whence, in view of (4.6), we find

u(σ(t_{k})) ≥ ∑_{i=1}^{m} φ̃(t_{k}) exp (−ε ∫_{0}^{t_{k}} \overline{p}(ξ)dξ)

× ∫_{σ(t_{k})}^{t_{k}} exp (ε ∫_{0}^{s} \overline{p}(ξ)dξ) p_{i}(s) exp (−λ^{∗} ∫_{0}^{τ_{i}(s)}\overline{p}(ξ)dξ) ds

+ ∑_{i=1}^{m} φ̃(t_{k}) ∫_{t_{k}}^{+∞}p_{i}(s) exp (−λ^{∗} ∫_{0}^{τ_{i}(s)}\overline{p}(ξ)dξ) ds

= φ̃(t_{k}) ∑_{i=1}^{m} ∫_{t_{k}}^{+∞}p_{i}(s) exp (−λ^{∗} ∫_{0}^{τ_{i}(s)}\overline{p}(ξ)dξ) ds − φ̃(t_{k}) exp (−ε ∫_{0}^{t_{k}} \overline{p}(s)ds)

RETARDED DE 71

× ∑_{i=1}^{m} ∫_{σ(t_{k})}^{t_{k}} exp (ε ∫_{0}^{s} \overline{p}(ξ)dξ) d ∫_{s}^{+∞}p_{i}(ξ) exp (−λ^{∗} ∫_{0}^{τ_{i}(ξ)}\overline{p}(ξ_{1}) dξ_{1})dξ

= φ̃(t_{k}) ∑_{i=1}^{m} exp (−ε ∫_{σ(t_{k})}^{t_{k}} \overline{p}(s)ds) ∫_{σ(t_{k})}^{+∞}p_{i}(s) exp (−λ^{∗} ∫_{0}^{τ_{i}(s)}\overline{p}(ξ)dξ) ds.

By (4.7), for sufficiently large k we obtain

e^{−(1+M)ε} ∑_{i=1}^{m} exp (λ^{∗} ∫_{0}^{σ(t_{k})}\overline{p}(ξ)dξ) ∫_{σ(t_{k})}^{+∞}p_{i}(s) exp (−λ^{∗} ∫_{0}^{τ_{i}(s)}\overline{p}(ξ)dξ) ds ≤ 1

Consequently,

lim inf_{t→+∞} exp (λ^{∗} ∫_{0}^{t} \overline{p}(ξ)dξ) ∑_{i=1}^{m} ∫_{t}^{+∞}p_{i}(s) exp (−λ^{∗} ∫_{0}^{τ_{i}(s)}\overline{p}(ξ)dξ) ds ≤ e^{(1+M)ε}.

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 t_{1} > t_{0} such that

∫_{τ_{i}(t)}^{t} \overline{p}(s)ds > α_{i} − ε for t ≥ t_{1} (i = 1, …, m)  (4.8)

and for any λ ∈ (0, +∞),

\frac{1}{\overline{p}(t)} ∑_{i=1}^{m} p_{i}(t)e^{λ(α_{i}−ε)} ≥ (1 + ε)λ for t ≥ t_{1}.  (4.9)

According to (4.8) and (4.9), for any λ ∈ (0, +∞) we find

exp (λ ∫_{0}^{t} \overline{p}(s)ds) ∑_{i=1}^{m} ∫_{t}^{+∞}p_{i}(s) exp (−λ ∫_{0}^{τ_{i}(s)}\overline{p}(ξ)dξ) ds

≥ exp (λ ∫_{0}^{t} \overline{p}(s)ds) ∫_{t}^{+∞}∑_{i=1}^{m} p_{i}(s)e^{λ(α_{i}−ε)} exp (−λ ∫_{0}^{s} \overline{p}(ξ)dξ) ds

≥ λ(1 + ε) exp (λ ∫_{0}^{t} \overline{p}(s)ds) ∫_{t}^{+∞}exp (−λ ∫_{0}^{s} p^{−}(ξ)dξ) \overline{p}(s)ds

= 1 + ε for t ≥ t_{1}. Therefore condition (2.4) holds and the proof of the theorem is complete. □

Proof of Theorem 2.3. From (2.8), using the inequality e^{x} ≥ 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 t_{1} ∈ R_{+} and ε > 0 such that

∫_{τ_{i}(t)}^{t} \overline{p}(s)ds ≥ \frac{1 + ε}{e} for t ≥ t_{1} (i = 1, …, m).

Thus for any λ ∈ (0, +∞) we have

exp (λ ∫_{0}^{t} \overline{p}(s)ds) ∑_{i=1}^{m} ∫_{t}^{+∞}p_{i}(s) exp (−λ ∫_{0}^{τ_{i}(s)}\overline{p}(ξ)dξ) ds

≥ e^{\frac{(1+ε)λ}{e}} exp (λ ∫_{0}^{t} \overline{p}(s)ds) ∫_{t}^{+∞}\overline{p}(s) exp (−λ ∫_{0}^{τ_{i}(s)}\overline{p}(ξ)dξ) ds

≥ \frac{e(1 + ε)λ}{λe} = 1 + ε.

Consequently, (2.4) is satisfied. □

Proof of Theorem 2.5. Below we will assume that

lim sup_{t→+∞} ∫_{τ_{i}(t)}^{t} p_{i}(s)ds ≤ 1 (i = 1, …, m).

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 t_{1} ∈ R_{+} and ε ∈ (0, β_{i}) such that

∫_{τ_{i}(t)}^{t} p_{i}(s)ds > β_{i} − ε fort ≥ t_{1} (i = 1, …, m)  (4.10)

and for any λ ∈ (0, +∞),

∑_{i=1}^{m} \frac{e^{(β_{i}−ε)λ}}{λ} > 1 + ε.  (4.11)

Put

p_{i}(t) − p_{1}(t) = q_{i}(t), η(t, s) = exp (−λ ∑_{i=1}^{m} | ∫_{t}^{τ_{i}(s)}q_{i}(ξ)dξ| ) for s ≥ t ≥ t_{1}

(i = 1, …, m) and

ψ(t, λ) = exp (λ ∫_{0}^{t} \overline{p}(s)ds) ∑_{i=1}^{m} ∫_{t}^{+∞}p_{i}(s) exp (−λ ∫_{0}^{τ_{i}(s)}\overline{p}(ξ)dξ) ds.

According to (4.10), (4.11), for any λ ∈ (0, +∞) we have

ψ(t, λ) = exp (λ ∑_{i=1}^{m} ∫_{0}^{t} q_{i}(s))ds) exp (λm ∫_{0}^{t} p_{1}(s)ds)

× ∑_{i=1}^{m} ∫_{t}^{+∞}(q_{i}(s) + p_{1}(s)) exp (−λ ∑_{i=1}^{m} ∫_{0}^{τ_{i}(s)}q_{i}(ξ)dξ) exp (−λm ∫_{0}^{τ_{i}(s)}p_{1}(ξ)dξ) ds

≥ exp (λm ∫_{0}^{t} p_{1}(s)ds) ∑_{i=1}^{m} ∫_{t}^{+∞}mp_{1}(s) exp (−λm ∫_{0}^{τ_{i}(s)}p_{1}(ξ)dξ) η(t, s)ds

− exp (λm ∫_{0}^{t} p_{1}(s)ds) ∫_{t}^{+∞}∑_{i=1}^{m} |q_{i}(s)|η(t, s)

× exp (−λm ∫_{0}^{s} p_{1}(ξ)dξ) exp (λm ∫_{τ_{i}(s)}^{s} p_{1}(ξ)dξ) ds.

Therefore, if we take into account the condition lim_{t→+∞}η(t, s) = 1, then by (2.3) and (2.10) we obtain for any λ ∈ (0, +∞),

lim inf_{t→+∞}ψ(t, λ) ≥ lim inf_{t→+∞} exp (λm ∫_{0}^{t} p_{1}(s)ds)

× ∑_{i=1}^{m} ∫_{t}^{+∞}mp_{1}(s) exp (−λm ∫_{0}^{s} p_{1}(ξ)dξ) e^{(β_{i}−ε)λ}ds

−lim sup_{t→+∞} e^{λm(1+M)} ∫_{t}^{+∞}∑_{i=1}^{m} |q_{i}(ξ)|η(t, ξ)dξ

= lim inf_{t→+∞} exp (λm ∫_{0}^{t} p_{1}(s)ds) ∫_{t}^{+∞}mp_{1}(s) exp (−λm ∫_{0}^{s} p_{1}(ξ)dξ)

× ∑_{i=1}^{m} e^{(β_{i}−ε)λ} = ∑_{i=1}^{m} \frac{e^{(β_{i}−ε)λ}}{λ} .

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 e^{x} ≥ ex.

Remark 4.1. As it is noted in the Introduction, several sufficient conditions for the oscillation of equation (1.1) for τ_{i}(t) = t − τ_{i} (i = 1, …, m), where τ_{i} (i = 1, …, m) are positive constants, are established in [1,15,17], while a nonintegral condition is given in [9] for τ_{i}(t) = t − T_{i}(t), where T_{i} 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, \frac{1}{e}), τ ≠ 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

min \{(\frac{e^{τλ}}{λ} + \frac{e^{( \frac{1}{e} −τ)λ}}{λ} ) : λ ∈ (0, ∞)\} > 1.  (4.13)

Since

min \{\frac{e^{τλ}}{λ} : λ ∈ (0, ∞)\} = τe, min \{\frac{e^{( \frac{1}{e} −τ)λ}}{λ} : λ ∈ (0, ∞)\} = 1 − τe

and the functions \frac{e^{τλ}}{λ} , \frac{e^{( \frac{1}{e} −τ)λ}}{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.

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(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: mgarmmat@cc.uoi.gr

ipstav@cc.uoi.gr R. Koplatadze A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia E-mail: roman@rmi.acnet.ge