GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 4, 1999, 335-346

OSCILLATORY BEHAVIOUR OF SOLUTIONS OF TWO-DIMENSIONAL DIFFERENTIAL SYSTEMS WITH

DEVIATED ARGUMENTS

R. KOPLATADZE AND N. PARTSVANIA

Abstract. Sufficient conditions are established for the oscillation of proper solutions of the system

\begin{matrix}u′_{1}(t) = f_{1}(t, u_{1}(τ_{1}(t)), …, u_{1}(τ_{m}(t)), u_{2}(σ_{1}(t)), …, u_{2}(σ_{m}(t))), \\ u′_{2}(t) = f_{2}(t, u_{1}(τ_{1}(t)), …, u_{1}(τ_{m}(t)), u_{2}(σ_{1}(t)), …, u_{2}(σ_{m}(t))), \end{matrix}

where f_{i} : ℝ_{+} × ℝ^{2m} → ℝ (i = 1, 2) satisfy the local Carathéodory conditions and τ_{i}, σ_{i} : ℝ_{+} → ℝ_{+} (i = 1, …, m) are continuous functions such that σ_{i}(t) ≤ t for t ∈ ℝ_{+}, lim_{t→+∞} τ_{i}(t) = +∞, lim_{t→+∞} σ_{i}(t) = +∞ (i = 1, …, m).

§ 1. Statement of the Problem and Formulation of the Main Results

Consider the system

\begin{matrix}u′_{1}(t) = f_{1}(t, u_{1}(τ_{1}(t)), …, u_{1}(τ_{m}(t)), u_{2}(σ_{1}(t)), …, u_{2}(σ_{m}(t))), \\ u′_{2}(t) = f_{2}(t, u_{1}(τ_{1}(t)), …, u_{1}(τ_{m}(t)), u_{2}(σ_{1}(t)), …, u_{2}(σ_{m}(t))), \end{matrix} (1.1)

where f_{i} : ℝ_{+}×ℝ^{2m} → ℝ (i = 1, 2) satisfy the local Carathéodory conditions and τ_{i}, σ_{i} : ℝ_{+} → ℝ_{+} (i = 1, …, m) are continuous functions such that

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

Definition 1.1. Let t_{0} ∈ ℝ_{+} and a_{0} = inf [min {τ_{i}(t), σ_{i}(t) : i = 1, …, m} : t ≥ t_{0}]. A continuous vector-function (u_{1}, u_{2}) defined on [a_{0}, +∞[ is said to be a proper solution of system (1.1) in [t_{0}, +∞[ if it

335 is absolutely continuous on each finite segment contained in [t_{0}, +∞[ , satisfies (1.1) almost everywhere on [t_{0}, +∞[ , and

1991 Mathematics Subject Classification. 34K15.

Key words and phrases. Two-dimensional differential system with deviated arguments, proper solution, oscillatory solution.

1072-947X/99/0700-0335$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1999 Plenum Publishing Corporation

sup {|u_{1}(s)| + |u_{2}(s)| : s ≥ t} > 0 for t ≥ t_{0}.

Definition 1.2. A proper solution (u_{1}, u_{2}) of system (1.1) is said to be weakly oscillatory if either u_{1} or u_{2} has a sequence of zeros tending to infinity. This solution is said to be oscillatory if both u_{1} and u_{2} have sequences of zeros tending to infinity. If there exists t_{∗} ∈ ℝ_{+} such that u_{1}(t)u_{2}(t) ≠ 0 for t ≥ t_{∗}, then (u_{1}, u_{2}) is said to be nonoscillatory.

In this paper, sufficient conditions are obtained for the oscillation of proper solutions of system (1.1) which make the results contained in [1, 2] more complete.

Throughout the paper we will assume that the inequalities

\begin{aligned}f_{1}(t, x_{1}, …, x_{m}, y_{1}, …, y_{m}) sgn y_{1} & ≥ ∑_{i=1}^{m} p_{i}(t)|y_{i}|, \\ f_{2}(t, x_{1}, …, x_{m}, y_{1}, …, y_{m}) sgn x_{1} & ≤ − ∑_{i=1}^{m} q_{i}(t)|x_{i}| \end{aligned}  (1.2)

hold for t ∈ ℝ_{+}, x_{1}x_{i} > 0, y_{1}y_{i} > 0 (i = 1, …, m), where p_{i}, q_{i} ∈ L_{loc}(ℝ_{+}; ℝ_{+}) (i = 1, …, m), and we will use the notation

p(t) = ∑_{i=1}^{m} p_{i}(t), q(t) = ∑_{i=1}^{m} q_{i}(t), h(t) = ∫_{0}^{t} p(s) ds.

Theorem 1.1. Let

h(+∞) = +∞,  (1.3)
∫^{+∞}h_{0}(t)q(t) dt = +∞,  (1.4)

where h_{0}(t) = min{h(t), h(τ_{i}(t)) : i = 1, …, m}, and there exist a nondecreasing function σ ∈ C(ℝ_{+}; ℝ_{+}) such that σ_{i}(t) ≤ σ(t) ≤ t for t ∈ ℝ_{+} (i = 1, …, m),

lim sup_{t→+∞} \frac{h(τ(σ(t)))}{h(t)} < +∞.  (1.5)

If, moreover, there exists ε_{0} > 0 such that for any λ ∈]0, 1],

lim inf_{t→+∞} h^{ε_{0}} (t)h^{1−λ}(τ(σ(t))) ∫_{τ(σ(t))}^{+∞} p(s)h^{−2−ε_{0}} (s)g(s, λ) ds > 1, (1.6)

TWO-DIMENSIONAL DIFFERENTIAL SYSTEMS 337

where

τ(t) = max [ max {τ_{i}(s), η(s) : i = 1, …, m} : 0 ≤ s ≤ t],
η(t) = sup {s : σ(s) < t},
g(t, λ) = ∫_{0}^{σ(t)}h(s) ∑_{i=1}^{m} q_{i}(s)h^{λ}(τ_{i}(s)) ds,  (1.7)

then every proper solution of system (1.1) is oscillatory.

Theorem 1.2. Let conditions (1.3)–(1.5) be fulfilled, where the function σ ∈ C(ℝ_{+}; ℝ_{+}) is nondecreasing, σ_{i}(t) ≤ σ(t) ≤ t for t ∈ ℝ_{+} (i = 1, …, m). If, moreover, there exists ε > 0 such that for any λ ∈]0, 1],

lim inf_{t→+∞} h^{−λ}(t) ∫_{0}^{σ(t)}h(s) ∑_{i=1}^{m} q_{i}(s)h^{λ}(τ_{i}(s)) ds ≥ 1 − λ + ε,  (1.8)

then every proper solution of system (1.1) is oscillatory.

Theorem 1.3. Let conditions (1.3), (1.4) be fulfilled,

lim sup_{t→+∞} \frac{h(τ_{i}(t))}{h(t)} < +∞ (i = 1, …, m),  (1.9)

and there exist ε > 0 such that for any λ ∈]0, 1],

lim inf_{t→+∞} h^{−1}(t) ∫_{0}^{t} h^{2}(s) ∑_{i=1}^{m} q_{i}(s)[\frac{h(τ_{i}(s))}{h(s)}]^{λ} ds ≥ λ(1 − λ) + ε. (1.10)

Then every proper solution of system (1.1) is oscillatory.

Corollary 1.1. Let conditions (1.3), (1.4), (1.9) be fulfilled and α_{i} ∈ ]0, +∞[ (i = 1, …, m), where

α_{i} = lim inf_{t→+∞} \frac{h(τ_{i}(t))}{h(t)} (i = 1, …, m).  (1.11)

If, moreover, there exists ε > 0 such that for any λ ∈]0, 1],

lim inf_{t→+∞} h^{−1}(t) ∫_{0}^{t} h^{2}(s) ∑_{i=1}^{m} α_{i}^{λ}q_{i}(s) ds ≥ λ(1 − λ) + ε,

then every proper solution of system (1.1) is oscillatory.

Corollary 1.2. Let conditions (1.3), (1.4), (1.9) be fulfilled, α_{i} ∈]0, +∞[ (i = 1, …, m), q_{i}(t) ≥ c_{i}q_{0}(t) for t ∈ ℝ_{+} (i = 1, …, m), where α_{i} (i = 1, …, m) are defined by (1.11), c_{i} > 0 (i = 1, …, m), and q_{0} ∈L_{loc}(ℝ_{+}; ℝ_{+}). Then the condition

lim inf_{t→+∞} h^{−1}(t) ∫_{0}^{t} h^{2}(s)q_{0}(s) ds > max {λ(1 − λ)( ∑_{i=1}^{m} α_{i}^{λ}c_{i})^{−1} : λ ∈ [0, 1]}

is sufficient for the oscillation of every proper solution of system (1.1).

Corollary 1.3. Let q_{0} ∈ L_{loc}(ℝ_{+}; ℝ_{+}), α ∈]0, 1[ , and

lim inf_{t→+∞} t^{−1} ∫_{0}^{t} s^{1+α}q_{0}(s) ds > 0.

Then every proper solution of the equation

u′′(t) + q_{0}(t)u(t^{α}) = 0

is oscillatory.

§ 2. Some Auxiliary Statements

Lemma 2.1. Let condition (1.3) be fulfilled and (u_{1}, u_{2}) be a nonoscillatory solution of system (1.1). Then there exists t_{0} ∈ ℝ_{+} such that

u_{1}(t)u_{2}(t) > 0 for t ≥ t_{0}.  (2.1)

If, moreover,

∫^{+∞}h(t)q(t) dt = +∞,

then

lim_{t→+∞} |u_{1}(t)| = +∞.

Lemma 2.2. Let

∫_{t}^{+∞}p(s) ds > 0, ∫_{t}^{+∞}q(s) ds > 0 for t ∈ ℝ_{+}.  (2.2)

Then every weakly oscillatory solution of system (1.1) is oscillatory.^{1}

1For the proofs of Lemma 2.1 and Lemma 2.2 see [2].

Lemma 2.3. Let condition (1.3) be fulfilled and (u_{1}, u_{2}) be a nonoscillatory solution of system (1.1). Then either the inequality

|u_{1}(t)| < h(t)|u_{2}(t)|  (2.3)

is fulfilled for sufficiently large t or there exists t_{0} ∈ ℝ_{+} such that

|u_{1}(t)| ≥ h(t) ∫_{t}^{+∞}\frac{p(s)}{h^{2}(s)} ∫_{t_{0}}^{σ(s)}h(ξ) ∑_{i=1}^{m} q_{i}(ξ)|u_{1}(τ_{i}(ξ))| dξ ds for t ≥ t_{0}, (2.4)

where σ ∈ C(ℝ_{+}; ℝ_{+}) is a nondecreasing function and σ_{i}(t) ≤ σ(t) ≤ t for t ∈ ℝ_{+} (i = 1, …, m).

Proof. Since (u_{1}, u_{2}) is a nonoscillatory solution of system (1.1) and condition (1.3) is fulfilled, by Lemma 2.1 there exists t_{∗} ∈ ℝ_{+} such that condition (2.1) holds for t ≥ t_{∗}. By (1.2) and (2.1), from (1.1) we have

\begin{aligned}|u_{1}(t)|′ & ≥ ∑_{i=1}^{m} p_{i}(t)|u_{2}(σ_{i}(t))| \\ & & for t ≥ t_{∗}. \\ |u_{2}(t)|′ & ≤ − ∑_{i=1}^{m} q_{i}(t)|u_{1}(τ_{i}(t))| \end{aligned}  (2.5)

Consider the function ρ(t) = |u_{1}(t)| − h(t)|u_{2}(t)|. Taking into account (2.5) and the fact that |u_{2}(t)| is a nonincreasing function, we get

ρ′(t) = |u_{1}(t)|′ − h(t)|u_{2}(t)|′ − p(t)|u_{2}(t)| ≥

≥ ∑_{i=1}^{m} p_{i}(t)|u_{2}(σ_{i}(t))| − p(t)|u_{2}(t)| − h(t)|u_{2}(t)|′ ≥

≥ −h(t)|u_{2}(t)|′ ≥ 0 for t ≥ t_{∗}.

Thus there exists t_{1} > t_{∗} such that either

|u_{1}(t)| − h(t)|u_{2}(t)| < 0 for t ≥ t_{1}  (2.6)

or

|u_{1}(t)| − h(t)|u_{2}(t)| ≥ 0 for t ≥ t_{1}.  (2.7)

If condition (2.6) holds, then the validity of the lemma is obvious. Thus assume that (2.7) is fulfilled and show that in that case estimate (2.4) is valid.

Multiplying the second inequality of system (2.5) by h(t) and integrating from t_{1} to σ(t), we obtain

∫_{t_{1}}^{σ(t)}h(s) ∑_{i=1}^{m} q_{i}(s)|u_{1}(τ_{i}(s))| ds ≤ − ∫_{t_{1}}^{σ(t)}h(s)|u_{2}(s)|′ ds =

= −h(σ(t))|u_{2}(σ(t))| + ∫_{t_{1}}^{σ(t)}p(s)|u_{2}(s)| ds + h(t_{1})|u_{2}(t_{1})| for t ≥ t_{1}.

Multiplying the latter inequality by \frac{p(t)}{h^{2}(t)} , integrating from t to +∞, and taking into account (2.5), (2.7) and the fact that |u_{2}(t)| is a nonincreasing function, we get

∫_{t}^{+∞}\frac{p(s)}{h^{2}(s)} ∫_{t_{1}}^{σ(s)}h(ξ) ∑_{i=1}^{m} q_{i}(ξ)|u_{1}(τ_{i}(ξ))| dξ ds ≤

≤ ∫_{t}^{+∞}\frac{p(s)}{h^{2}(s)} ∫_{t_{1}}^{σ(s)}p(ξ)|u_{2}(ξ)| dξ ds − ∫_{t}^{+∞}\frac{p(s)}{h^{2}(s)} h(σ(s))|u_{2}(σ(s))| ds +

+h(t_{1})|u_{2}(t_{1})| ∫_{t}^{+∞}\frac{p(s)}{h^{2}(s)} ds = ∫_{t}^{+∞}\frac{p(s)}{h^{2}(s)} ∫_{t_{1}}^{σ(s)}p(ξ)|u_{2}(ξ)| dξ ds −

− ∫_{t}^{+∞}\frac{p(s)}{h^{2}(s)}[ ∫_{0}^{s} p(ξ) dξ − ∫_{σ(s)}^{s} p(ξ) dξ]|u_{2}(σ(s))| ds + \frac{h(t_{1})|u_{2}(t_{1})|}{h(t)} ≤

≤ ∫_{t}^{+∞}\frac{p(s)}{h^{2}(s)} ∫_{t_{1}}^{σ(s)}p(ξ)|u_{2}(σ(ξ))| dξ ds − ∫_{t}^{+∞}\frac{p(s)}{h(s)} |u_{2}(σ(s))| ds +

+ ∫_{t}^{+∞}\frac{p(s)}{h^{2}(s)} ∫_{σ(s)}^{s} p(ξ)|u_{2}(σ(ξ))| dξ ds + \frac{h(t_{1})|u_{2}(t_{1})|}{h(t)} =

= ∫_{t}^{+∞}\frac{p(s)}{h^{2}(s)} ∫_{t_{1}}^{s} p(ξ)|u_{2}(σ(ξ))| dξ ds −

− ∫_{t}^{+∞}\frac{1}{h(s)} d ∫_{t_{1}}^{s} p(ξ)|u_{2}(σ(ξ))| dξ + \frac{h(t_{1})|u_{2}(t_{1})|}{h(t)} =

= \frac{1}{h(t)} ∫_{t_{1}}^{t} p(s)|u_{2}(σ(s))| ds + \frac{h(t_{1})|u_{2}(t_{1})|}{h(t)} ≤

≤ \frac{1}{h(t)} [|u_{1}(t_{1})| + ∫_{t_{1}}^{t} ∑_{i=1}^{m} p_{i}(s)|u_{2}(σ_{i}(s))| ds] ≤ \frac{|u_{1}(t)|}{h(t)} for t ≥ t_{0},

where t_{0} > t_{1} is a sufficiently large number. Therefore (2.4) is fulfilled. Thus the lemma is proved.

Lemma 2.4. Let t_{0} ∈ ℝ_{+}, φ, ψ ∈ C([t_{0}, +∞[ ; ]0, +∞[ ),

lim inf_{t→+∞} φ(t) = 0, ψ(t) ↑ +∞ for t ↑ +∞,  (2.8)

and

lim_{t→+∞} φ̃(σ(t))ψ(t) = +∞,  (2.9)

where

φ̃(t) = min {φ(s) : t_{0} ≤ s ≤ t}  (2.10)

and σ : ℝ_{+} → ℝ_{+} is a nondecreasing continuous function such that σ(t) ≤ t for t ∈ ℝ_{+}, lim_{t→+∞} σ(t) = +∞. Then there exists a sequence of points

{t_{k}}_{k=1}^{+∞} such that t_{k} ↑ +∞ for k ↑ +∞ and

φ̃(σ(t_{k}))ψ(t_{k}) ≤ φ̃(σ(s))ψ(s) for s ≥ t_{k},
φ̃(σ(t_{k})) = φ(σ(t_{k})) (k = 1, 2, …).

Proof. Define the sets E_{i} (i = 1, 2) in the following manner:

t ∈ E_{1} ⟺ φ̃(σ(t))ψ(t) ≤ φ̃(σ(s))ψ(s) for s ≥ t,
t ∈ E_{2} ⟺ φ̃(σ(t)) = φ(σ(t)).

In view of (2.8)–(2.10) it is clear that

sup E_{i} = +∞ (i = 1, 2).  (2.11)

Show that E_{1} ∩E_{2} is a nonempty set. Let m ∈ ℕ. According to (2.11) there exist t_{m}^{(i)} ∈ E_{i} (i = 1, 2) such that m ≤ t_{m}^{(2)} ≤ t_{m}^{(1)}. Suppose that t_{m}^{(1)} ∉ E_{2}. Then we can find t_{m}^{∗} ∈ [t_{m}^{(2)}, t_{m}^{(1)}[ such that

φ̃(σ(t)) = φ̃(σ(t_{m}^{(1)})) for t ∈ [t_{m}^{∗}, t_{m}^{(1)}]  (2.12)

and

φ̃(σ(t_{m}^{∗})) = φ(σ(t_{m}^{∗})).  (2.13)

On the other hand, since t_{m}^{(1)} ∈ E_{1}, on account of (2.8), (2.12), we have

φ̃(σ(t_{m}^{∗}))ψ(t_{m}^{∗}) ≤ φ̃(σ(s))ψ(s) for s ≥ t_{m}^{∗}.  (2.14)

By virtue of (2.13) and (2.14), t_{m}^{∗} ∈ E_{1} ∩ E_{2}. Taking into account the arbitrariness of m, by the above reasoning we can easily conclude that sup E_{1} ∩ E_{2} = +∞. This implies that the lemma is valid.

§ 3. Proof of the Main Results

Proof of Theorem 1.1. Let (u_{1}, u_{2}) be a proper solution of system (1.1). Suppose that this solution is not oscillatory. From (1.3), (1.4) follow inequalities (2.2). Thus by Lemma 2.2 (u_{1}, u_{2}) is nonoscillatory. Therefore, due to Lemma 2.1, one can find t_{0} ∈ ℝ_{+} so that condition (2.1) will be fulfilled for t ≥ t_{0} and

lim_{t→+∞} |u_{1}(t)| = +∞.  (3.1)

By (1.2) and (2.1), from (1.1) we have

\begin{aligned}|u_{1}(t)|′ & ≥ ∑_{i=1}^{m} p_{i}(t)|u_{2}(σ_{i}(t))| \\ & & for t ≥ t_{0}. \\ |u_{2}(t)|′ & ≤ − ∑_{i=1}^{m} q_{i}(t)|u_{1}(τ_{i}(t))| \end{aligned}  (3.2)

Since (u_{1}, u_{2}) is a nonoscillatory solution of system (1.1) and condition (1.3) holds, by Lemma 2.3 either (2.3) or (2.4) is fulfilled.

Suppose that (2.3) is fulfilled. Then taking into account (3.2) and the fact that |u_{2}(t)| is a nonincreasing function, we obtain

(\frac{|u_{1}(t)|}{h(t)})′ = \frac{|u_{1}(t)|′h(t) − p(t)|u_{1}(t)|}{h^{2}(t)} ≥

≥ \frac{h(t) ∑_{i=1}^{m} p_{i}(t)|u_{2}(σ_{i}(t))| − p(t)|u_{1}(t)|}{h^{2}(t)} ≥

≥ \frac{p(t)[h(t)|u_{2}(t)| − |u_{1}(t)|]}{h^{2}(t)} ≥ 0 for t ≥ t_{1},

where t_{1} > t_{0} is a sufficiently large number. Thus there exist c > 0 and t^{∗} ≥ t_{1} such that

|u_{1}(τ_{i}(t))| ≥ ch(τ_{i}(t)) for t ≥ t^{∗} (i = 1, …, m).

In view of the latter inequalities, from the second inequality of system (3.2) we get

|u_{2}(t^{∗})| ≥ c ∫_{t^{∗}}^{+∞}∑_{i=1}^{m} q_{i}(s)h(τ_{i}(s)) ds ≥ c ∫_{t^{∗}}^{+∞}h_{0}(s)q(s) ds.

But the latter inequality contradicts (1.4). Therefore below (2.4) will be assumed to be fulfilled.

Denote by Δ the set of all λ ∈]0, 1] satisfying

lim inf_{t→+∞} \frac{|u_{1}(t)|}{h^{λ}(t)} = 0.

By (3.1) it is obvious that 0 ∉ Δ, and by using (1.4) we can easily show that 1 ∈ Δ.

Let λ_{0} = inf Δ. Then by (1.6) there exist λ^{∗} ∈]0, 1] ∩ [λ_{0}, 1] and ε_{1} ∈ ]0, ε_{0}] such that λ^{∗} − ε_{1} ∈ [0, 1],

lim inf_{t→+∞} \frac{|u_{1}(t)|}{h^{λ^{∗}}(t)} = 0, lim_{t→+∞} \frac{|u_{1}(t)|}{h^{λ^{∗}−ε_{1}} (t)} = +∞,  (3.3)

and

lim inf_{t→+∞} h^{ε_{1}} (t)h^{1−λ^{∗}}(τ(σ(t))) ∫_{τ(σ(t))}^{+∞} p(s)h^{−2−ε_{1}} (s)g(s, λ^{∗}) ds > 1, (3.4)

where g(t, λ) is defined by (1.7).

Introduce the notation

φ̃(t) = min \{\frac{|u_{1}(τ(s))|}{h^{λ^{∗}}(τ(s))} : t_{0} ≤ s ≤ t\}.

It is obvious that φ̃(t) ↓ 0 for t ↑ +∞ and

\frac{|u_{1}(τ_{i}(t))|}{h^{λ^{∗}}(τ_{i}(t))} ≥ φ̃(t) for t ≥ t_{0} (i = 1, …, m).  (3.5)

By virtue of (1.3), (1.5) and (3.3) all the conditions of Lemma 2.4 are fulfilled. Thus there exists a sequence of points {t_{k}}_{k=1}^{+∞} such that t_{k} ↑ +∞ for k ↑ +∞,

φ̃(σ(t_{k}))h^{ε_{1}} (t_{k}) ≤ φ̃(σ(s))h^{ε_{1}} (s) for s ≥ t_{k},  (3.6)
φ̃(σ(t_{k})) = \frac{|u_{1}(τ(σ(t_{k})))|}{h^{λ^{∗}}(τ(σ(t_{k})))} (k = 1, 2, …).  (3.7)

Taking into account conditions (3.5)–(3.7), for sufficiently large k from (2.4) we get

|u_{1}(τ(σ(t_{k})))| ≥

≥ h(τ(σ(t_{k}))) ∫_{τ(σ(t_{k}))}^{+∞} \frac{p(s)}{h^{2}(s)} ∫_{t_{0}}^{σ(s)}h(ξ) ∑_{i=1}^{m} q_{i}(ξ) \frac{|u_{1}(τ_{i}(ξ))|}{h^{λ^{∗}}(τ_{i}(ξ))} h^{λ^{∗}}(τ_{i}(ξ)) dξ ds ≥

≥ h(τ(σ(t_{k}))) ∫_{τ(σ(t_{k}))}^{+∞} \frac{p(s)}{h^{2}(s)} ∫_{t_{0}}^{σ(s)}φ̃(ξ)h(ξ) ∑_{i=1}^{m} q_{i}(ξ)h^{λ^{∗}}(τ_{i}(ξ)) dξ ds ≥

≥ h(τ(σ(t_{k}))) ∫_{τ(σ(t_{k}))}^{+∞} p(s)h^{−2}(s)φ̃(σ(s)) ∫_{t_{0}}^{σ(s)}h(ξ) ∑_{i=1}^{m} q_{i}(ξ)h^{λ^{∗}}(τ_{i}(ξ)) dξ ds ≥

≥ φ̃(σ(t_{k}))h^{ε_{1}} (t_{k})h(τ(σ(t_{k}))) ∫_{τ(σ(t_{k}))}^{+∞} p(s)h^{−2−ε_{1}} (s) ×

× ∫_{t_{0}}^{σ(s)}h(ξ) ∑_{i=1}^{m} q_{i}(ξ)h^{λ^{∗}}(τ_{i}(ξ)) dξ ds =

= |u_{1}(τ(σ(t_{k})))|h^{ε_{1}} (t_{k})h^{1−λ^{∗}}(τ(σ(t_{k}))) ×

× ∫_{τ(σ(t_{k}))}^{+∞} p(s)h^{−2−ε_{1}} (s) ∫_{t_{0}}^{σ(s)}h(ξ) ∑_{i=1}^{m} q_{i}(ξ)h^{λ^{∗}}(τ_{i}(ξ)) dξ ds.

Therefore

h^{ε_{1}} (t_{k})h^{1−λ^{∗}}(τ(σ(t_{k}))) ∫_{τ(σ(t_{k}))}^{+∞} p(s)h^{−2−ε_{1}} (s) ×

× ∫_{t_{0}}^{σ(s)}h(ξ) ∑_{i=1}^{m} q_{i}(ξ)h^{λ^{∗}}(τ_{i}(ξ)) dξ ds ≤ 1.

But the latter inequality contradicts (3.4). The contradiction obtained proves that the theorem is valid.

Proof of Theorem 1.2. By Theorem 1.1 it is sufficient to show that we can find ε_{0} ∈]0, ε] such that condition (1.6) will hold. Indeed, choose ε_{0} ∈]0, ε] so that

\frac{1 + ε}{1 + ε_{0}} (\frac{1}{2γ} )^{ε_{0}} > 1,  (3.8)

where

γ = lim sup_{t→+∞} \frac{h(τ(σ(t)))}{h(t)} .  (3.9)

In view of (1.8), (3.8) and (3.9), we obtain

h^{ε_{0}} (t)h^{1−λ}(τ(σ(t))) ∫_{τ(σ(t))}^{+∞} p(s)h^{−2−ε_{0}} (s)g(s, λ) ds ≥

≥ (1 − λ + ε)h^{ε_{0}} (t)h^{1−λ}(τ(σ(t))) ∫_{τ(σ(t))}^{+∞} h^{λ−2−ε_{0}} (s) dh(s) =

= \frac{1 − λ + ε}{1 − λ + ε_{0}} h^{ε_{0}} (t)h^{−ε_{0}} (τ(σ(t))) ≥

≥ \frac{1 − λ + ε}{1 − λ + ε_{0}} (\frac{1}{2γ} )^{ε_{0}} ≥ \frac{1 + ε}{1 + ε_{0}} (\frac{1}{2γ} )^{ε_{0}} > 1 for t ≥ t_{0},

where t_{0} ∈ ℝ_{+} is a sufficiently large number. Therefore condition (1.6) is fulfilled. Thus the theorem is proved.

Proof of Theorem 1.3. By virtue of Theorem 1.2 it is sufficient to show that condition (1.8) is fulfilled with σ(t) ≡ t. Indeed, on account of (1.10) we get^{2}

h^{−λ}(t) ∫_{0}^{t} h(s) ∑_{i=1}^{m} q_{i}(s)h^{λ}(τ_{i}(s)) ds =

= h^{−λ}(t) ∫_{0}^{t} h^{1+λ}(s) ∑_{i=1}^{m} q_{i}(s)[\frac{h(τ_{i}(s))}{h(s)}]^{λ} ds =

= h^{−λ}(t) ∫_{0}^{t} h^{λ−1}(s) d ∫_{0}^{s} h^{2}(ξ) ∑_{i=1}^{m} q_{i}(ξ)[\frac{h(τ_{i}(ξ))}{h(ξ)}]^{λ} dξ =

= h^{−1}(t) ∫_{0}^{t} h^{2}(s) ∑_{i=1}^{m} q_{i}(s)[\frac{h(τ_{i}(s))}{h(s)}]^{λ} ds +

2Here we mean that λ < 1. In the case where λ = 1 the validity of (1.8) is obvious.

+(1 − λ)h^{−λ}(t) ∫_{0}^{t} p(s)h^{λ−2}(s) ∫_{0}^{s} h^{2}(ξ) ∑_{i=1}^{m} q_{i}(ξ)[\frac{h(τ_{i}(ξ))}{h(ξ)}]^{λ} dξ ds ≥

≥ (λ(1 − λ) + ε) + (1 − λ)(λ(1 − λ) + ε)h^{−λ}(t) ∫_{0}^{t} p(s)h^{λ−1}(s) ds =

= (λ(1 − λ) + ε)(1 + \frac{1 − λ}{λ}) = \frac{λ(1 − λ) + ε}{λ} ≥ 1 − λ + ε for t ≥ t_{0},

where t_{0} ∈ ℝ_{+} is a sufficiently large number. Therefore condition (1.8) holds. Thus the theorem is proved.

Acknowledgement

This investigation was supported by Grant No. 1.6/1998 of the Georgian Academy of Sciences.

References

1. R. Koplatadze and N. Partsvania, Oscillatory properties of solutions of two-dimensional differential systems with deviated arguments. (Russian) Differentsial’nye Uravneniya 33(1997), No. 10, 1312–1320.

2. N. Partsvania, On oscillation of solutions of second order systems of deviated differential equations. Georgian Math. J. 3(1996), No. 6, 571–582.

(Received 21.04.1998)

Authors’ address: A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia