GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 6, 1996, 571-582

ON OSCILLATION OF SOLUTIONS OF SECOND-ORDER SYSTEMS OF DEVIATED DIFFERENTIAL EQUATIONS

N. PARTSVANIA

Abstract. Sufficient conditions are found for the oscillation of proper solutions of the system of differential equations

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))),

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

Introduction

The problem of oscillation of solutions of second-order ordinary differential equations has been studied well enough. A number of papers were devoted to an analogous problem for deviated differential equations and systems of ordinary differential equations (see [2]–[6]). As to systems of deviated differential equations, for them the problem of oscillation of solutions has not been studied enough. In the present paper, we give the results for systems of differential equations, which generalize some well-known statements for second-order differential equations.

Consider the system of differential equations

\begin{aligned}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{aligned} (0.1)

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

571 that

1991 Mathematics Subject Classification. 34K15.

Key words and phrases. System of deviated differential equations, proper solution, oscillatory solution.

1072-947X/96/1100-0571$09.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1996 Plenum Publishing Corporation

σ_{i}(t) ≤ t for t ∈ R_{+}, lim_{t→+∞} σ_{i}(t) = +∞,

lim_{t→+∞} τ_{i}(t) = +∞(i = 1, …, m). (0.2)

Definition 0.1. Let t_{0} ∈ R_{+}. A continuous vector-function (u_{1}, u_{2}) defined on [a_{0}, +∞[ (where a_{0} = min [ min { inf_{t≥t_{0}} τ_{i}(t) : i = 1, …, m},

min{ inf_{t≥t_{0}} σ_{i}(t) : i = 1, …, m}]) is said to be a proper solution of system

(0.1) in [t_{0}, +∞[ if it is absolutely continuous on each finite segment contained in [t_{0}, +∞[ , satisfies (0.1) almost everywhere in [t_{0}, +∞[ , and

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

Definition 0.2. A proper solution (u_{1}, u_{2}) of system (0.1) 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_{0} ∈ R_{+} such that u_{1}(t)u_{2}(t) ≠ 0 for t ∈ [t_{0}, +∞[ , then (u_{1}, u_{2}) is said to be nonoscillatory.

Throughout the paper the following notation will be used:

τ_{∗}(t) = min {t, τ_{i}(t) : i = 1, …, m}, σ_{∗}(t) = min {σ_{i}(t) : i = 1, …, },

σ̃(t) = inf {σ_{∗}(s) : s ≥ t}, σ^{∗}(t) = max {σ_{i}(t) : i = 1, …, m}.

§ 1. Auxiliary Statements

In this section, we consider the system of differential inequalities

u′_{1}(t) sign u_{2}(t) ≥ ∑_{i=1}^{m} p_{i}(t)|u_{2}(σ_{i}(t))|,

u′_{2}(t) sign u_{1}(t) ≤ − ∑_{i=1}^{m} q_{i}(t)|u_{1}(τ_{i}(t))|,

(1.1)

where p_{i}, q_{i} ∈ L_{loc}(R_{+}; R_{+}) (i = 1, …, m) and σ_{i}, τ_{i} : R_{+} → R (i = 1, …, m) are continuous functions satisfying (0.2).

Lemma 1.1. Let (u_{1}, u_{2}) be a nonoscillatory solution of (1.1) and

h(+∞) = +∞,  (1.2)

where

h(t) = ∫_{0}^{t} p̃(s)ds, p̃(t) = ∑_{i=1}^{m} p_{i}(t).  (1.3)

Then there exists t^{∗} ∈ R_{+} such that

u_{1}(t)u_{2}(t) > 0 for t ∈ [t^{∗}, +∞[ .  (1.4)

If, moreover,

∫_{0}^{+∞}q̃(t)h(t)dt = +∞,  (1.5)

where H is defined by (1.3) and

q̃(t) = ∑_{i=1}^{m} q_{i}(t),  (1.6)

then

lim_{t→+∞} |u_{1}(t)| = +∞.  (1.7)

Proof. Suppose that (u_{1}, u_{2}) does not satisfy (1.4). Then there exists t_{0} ∈ R_{+} such that

u_{1}(t)u_{2}(t) < 0 for t ∈ [t_{0}, +∞[ .  (1.8)

By (1.8) we have from (1.1)

|u_{1}(t)|′ ≤ − ∑_{i=1}^{m} p_{i}(t)|u_{2}(σ_{i}(t))|,  (1.9)

|u_{2}(t)|′ ≥ ∑_{i=1}^{m} q_{i}(t)|u_{1}(τ_{i}(t))| for t ∈ [t_{0}, +∞[ .  (1.10)

Thus by (1.10) from (1.9) we have

|u_{1}(t)| ≤ |u_{1}(t_{1})| − c ∫_{t_{1}}^{t} p̃(s)ds for t ∈ [t_{1}, +∞[ ,

where t_{1} ∈]t_{0}, +∞[ is a sufficiently large number and

c = min { inf_{t∈[t_{1},+∞[} |u_{2}(σ_{i}(t))| : i = 1, …, m} > 0.

By (1.2) the latter inequality implies |u_{1}(t)| → −∞ as t → +∞. The contradiction obtained proves (1.4).

Now prove that (1.5) implies (1.7). Since (1.4) is satisfied, from (1.1) we find

\begin{aligned}|u_{1}(t)|′ & ≥ ∑_{i=1}^{m} p_{i}(t)|u_{2}(σ_{i}(t))|, \\ |u_{2}(t)|′ & ≤ − ∑_{i=1}^{m} q_{i}(t)|u_{1}(τ_{i}(t))| for t ∈ [t^{∗}, +∞[ . \end{aligned}  (1.11)

Since |u_{1}(t)| is a nondecreasing function, by virtue of (0.2) there exist t_{2} ∈ [t^{∗}, +∞[ and a ∈]0, +∞[ such that

|u_{1}(τ_{i}(t))| ≥ a for t ∈ [t_{2}, +∞[ (i = 1, …, m),

whence by (1.11) we get

|u_{2}(t)|′h(t) ≤ −aq̃(t)h(t) for t ∈ [t_{2}, +∞[ .

Integrating from t_{2} to t, we obtain

|u_{2}(t)|h(t) − |u_{2}(t_{2})|h(t_{2}) + a ∫_{t_{2}}^{t} q̃(s)h(s)ds ≤

≤ ∫_{t_{2}}^{t} p̃(s)|u_{2}(s)|ds.

Taking into account (0.2) and the fact that |u_{2}(t)| is a decreasing function, from the latter inequality we have

∫_{t_{2}}^{t} ∑_{i=1}^{m} p_{i}(s)|u_{2}(σ_{i}(s))|ds ≥ ∫_{t_{2}}^{t} p̃(s)|u_{2}(s)|ds ≥

≥ a ∫_{t_{2}}^{t} q̃(s)h(s)ds − |u_{2}(t_{2})|h(t_{2}).  (1.12)

From (1.11) we find

|u_{1}(t)| ≥ |u_{1}(t_{2})| + ∫_{t_{2}}^{t} ∑_{i=1}^{m} p_{i}(s)|u_{2}(σ_{i}(s))|ds for t ∈ [t_{2}, +∞[ ,

which by virtue of (1.5), (1.12) implies that (1.7) is valid.

Lemma 1.2. If

∫_{t}^{+∞}p̃(s)ds > 0, ∫_{t}^{+∞}q̃(s)ds > 0 for t ∈ R_{+},  (1.13)

where p̃ and q̃ are defined by (1.3) and (1.6), respectively, then every weakly oscillatory solution of (1.1) is oscillatory.

Proof. Let (u_{1}, u_{2}) be a weakly oscillatory solution of (1.1), and suppose that this solution is not oscillatory. Without loss of generality it can be assumed that there exists an increasing sequence of points {t_{k}} tending to +∞ such that

\begin{matrix}u_{1}(t_{k}) = 0 (k = 1, 2, …), \\ u_{2}(σ_{i}(t)) > 0 for t ∈ [t_{1}, +∞[ (i = 1, …, m). \end{matrix}  (1.14)

On account of (1.13), (1.14), there is k ∈ N such that

∫_{t_{1}}^{t_{k}} ∑_{i=1}^{m} p_{i}(s)u_{2}(σ_{i}(s))ds > 0.

On the other hand, by (1.14) from (1.1) we have

0 ≥ ∫_{t_{1}}^{t_{k}} ∑_{i=1}^{m} p_{i}(s)u_{2}(σ_{i}(s))ds > 0.

The contradiction obtained proves the validity of the lemma.

§ 2. Oscillatory Solutions

Theorem 2.1. Let

f_{1}(t, x_{1}, …, x_{m}, y_{1}, …, y_{m}) sign y_{1} ≥ ∑_{i=1}^{m} p_{i}(t)|y_{i}|,

f_{2}(t, x_{1}, …, x_{m}, y_{1}, …, y_{m}) sign x_{1} ≤ − ∑_{i=1}^{m} q_{i}(t)|x_{i}|

for t ∈ R_{+}, x_{1}x_{i} > 0, y_{1}y_{i} > 0 (i = 1, …, m),

(2.1)

where p_{i}, q_{i} ∈ L_{loc}(R_{+}; R_{+}) (i = 1, …, m), and conditions (1.2), (1.5), and

∫_{0}^{+∞}∑_{i=1}^{m} q_{i}(t)h(τ_{i}(t))dt = +∞  (2.2)

be fulfilled. Let, moreover, there exist ε_{0} > 0 and a nondecreasing function δ : R_{+} → R such that δ(t) ≥ σ_{∗}(t), and for any λ ∈ [0, 1[

∫_{0}^{τ_{∗}(_{σ}(t))}h^{ε_{0}}(δ(s)) ∑_{i=1}^{m} p_{i}(s) ∫_{σ_{i}(s)}^{+∞} ∑_{i=1}^{m} q_{i}(ξ)h^{λ}(τ_{i}(ξ))dξds >

> h^{λ}(τ_{∗}(σ̃(t)))h^{ε_{0}}(δ(t)) for t ∈ [t_{0}, +∞[ ,  (2.3)

where t_{0} ∈ R_{+} and h is defined by (1.3). Then every proper solution of (0.1) is oscillatory.

Proof. Let (u_{1}, u_{2}) be a proper solution of (0.1). By virtue of (2.1) it will be a solution of (1.1) as well. Suppose that this solution is not oscillatory. By (1.2) and (2.2) it is obvious that all the conditions of Lemma 1.2 are fulfilled. Thus (u_{1}, u_{2}) is nonoscillatory. Therefore by virtue of condition (1.5) and Lemma 1.1 one can find t_{0} ∈ R_{+} such that

u_{1}(t)u_{2}(t) > 0 for t ∈ [t_{0}, +∞[ ,  (2.4)

and

lim_{t→+∞} |u_{1}(t)| = +∞.  (2.5)

By (2.4) we have from (1.1)

\begin{aligned}|u_{1}(t)|′ & ≥ ∑_{i=1}^{m} p_{i}(t)|u_{2}(σ_{i}(t))|, \\ |u_{2}(t)|′ & ≤ − ∑_{i=1}^{m} q_{i}(t)|u_{1}(τ_{i}(t))| for t ∈ [t_{0}, +∞[ . \end{aligned}  (2.6)

From these inequalities we get

|u_{1}(t)| ≥ ∫_{t_{1}}^{t} ∑_{i=1}^{m} p_{i}(s) ∫_{σ_{i}(s)}^{+∞} ∑_{i=1}^{m} q_{i}(ξ)|u_{1}(τ_{i}(ξ))|dξds  (2.7)
for t ∈ [t_{1}, +∞[ ,

where t_{1} ∈ [t_{0}, +∞[ is a sufficiently large number.

Denote by S the set of all λ ∈ R_{+} satisfying

\frac{|u_{1}(t)|}{h^{λ}(t)} → +∞ as t → +∞.  (2.8)

By (2.5), it is obvious that 0 ∈ S. Let λ_{0} = sup S. Suppose that λ_{0} > 1. By virtue of (2.8) there exists t̃∈ [t_{0}, +∞[ such that

|u_{1}(τ_{i}(t))| ≥ h(τ_{i}(t)) for t ∈ [t̃, +∞[ (i = 1, …, m).

Thus on account of (2.2) we find from (2.6)

|u_{2}(t)| ≤ |u_{2}(t̃)| − t ; ∫ ; _{t} ∑_{i=1}^{m} q_{i}(s)h(τ_{i}(s))ds → −∞ as t → +∞.

The contradiction obtained proves that λ_{0} ∈ [0, 1]. Then by (2.3) there exist λ_{∗} ∈ [0, λ_{0}] ∩ [0, 1[ and t_{0} ∈ R_{+} such that

∫_{0}^{τ_{∗}(_{σ}(t))}h^{ε_{0}}(δ(s)) ∑_{i=1}^{m} p_{i}(s) ∫_{σ_{i}(s)}^{+∞} ∑_{i=1}^{m} q_{i}(ξ)h^{λ_{∗}}(τ_{i}(ξ))dξds >

> h^{λ_{∗}}(τ_{∗}(σ̃(t)))h^{ε_{0}}(δ(t)) for t ∈ [t_{0}, +∞[ ,  (2.9)

and

lim_{t→+∞} \frac{|u_{1}(t)|}{h^{λ_{∗}}(t)} = +∞, \underline{lim}_{t→+∞} \frac{|u_{1}(t)|}{h^{λ_{∗}+ε_{0}}(t)} = 0.  (2.10)

Introduce the notation

φ(t) = inf {h^{−λ_{∗}}(τ_{∗}(s))|u_{1}(τ_{∗}(s))| : s ≥ t}.  (2.11)

By (0.2) and (2.10) it is clear that

φ(t) ↑ +∞ as t ↑ +∞,  (2.12)
\underline{lim}_{t→+∞} φ(σ̃(t))h^{−ε_{0}}(δ(t)) = 0.  (2.13)

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

t ∈ S_{1} ⟺ φ(σ̃(t))h^{−ε_{0}}(δ(t)) ≤ φ(σ̃(s))h^{−ε_{0}}(δ(s))
for s ∈ [t_{0}, t],

t ∈ S_{2} ⟺ φ(σ̃(t)) = \frac{|u_{1}(τ_{∗}(σ̃(t)))|}{h^{λ_{∗}}(τ_{∗}(σ̃(t)))} .

It is clear that by (2.12) and (2.13), sup S_{i} = +∞ (i = 1, 2). Show that

sup S_{1} ∩ S_{2} = +∞^{1}.  (2.14)

1Analogous discussions for nth order equations are given in [4], [5].

Indeed, if we assume that t_{2} ∈ S_{1} and t_{2} ∉ S_{2}, by (2.11) and (2.12) there exists t_{3} > t_{2} such that

\begin{aligned}φ(σ̃(t_{3})) & = \frac{|u_{1}(τ_{∗}(σ̃(t_{3})))|}{h^{λ_{∗}}(τ_{∗}(σ̃(t_{3})))} , \\ φ(σ̃(t)) & = φ(σ̃(t_{2})) for t ∈ [t_{2}, t_{3}]. \end{aligned}  (2.15)

On the other hand, since t_{2} ∈ S_{1} and δ is nondecreasing, we have

φ(σ̃(t_{3}))h^{−ε_{0}}(δ(t_{3})) ≤ φ(σ̃(s))h^{−ε_{0}}(δ(s)) for s ∈ [t_{0}, t_{3}].  (2.16)

Therefore from (2.15), (2.16) it follows that t_{3} ∈ S_{1} ∩ S_{2}. By the above reasoning we easily ascertain that (2.14) is fulfilled. Thus there exists an increasing sequence of points {t_{k}} such that

\begin{matrix}lim_{k→+∞} t_{k} = +∞, \\ φ(σ̃(t_{k}))h^{−ε_{0}}(δ(t_{k})) ≤ φ(σ̃(s))h^{−ε_{0}}(δ(s)) for s ∈ [t_{0}, t_{k}], \\ φ(σ̃(t_{k})) = \frac{|u_{1}(τ_{∗}(σ̃(t_{k})))|}{h^{λ_{∗}}(τ_{∗}(σ̃(t_{k})))} (k = 1, 2, …). \end{matrix}  (2.17)

On the other hand, it is obvious that

φ(t) ≤ \frac{|u_{1}(τ_{i}(t))|}{h^{λ_{∗}}(τ_{i}(t))} for t ∈ [t_{0}, +∞[ (i = 1, …, m).

Thus on account of (0.2), (2.12), (2.17), for sufficiently large k we have from (2.7)

|u_{1}(τ_{∗}(σ̃(t_{k})))| ≥

≥ ∫_{t_{1}}^{τ_{∗}(_{σ}(t_{k}))}∑_{i=1}^{m} p_{i}(s) ∫_{σ_{i}(s)}^{+∞} ∑_{i=1}^{m} q_{i}(ξ)\frac{|u_{1}(τ_{i}(ξ))|}{h^{λ_{∗}}(τ_{i}(ξ))}h^{λ_{∗}}(τ_{i}(ξ))dξds ≥

≥ ∫_{t_{1}}^{τ_{∗}(_{σ}(t_{k}))}∑_{i=1}^{m} p_{i}(s)φ(σ̃(s)) ∫_{σ_{i}(s)}^{+∞} ∑_{i=1}^{m} q_{i}(ξ)h^{λ_{∗}}(τ_{i}(ξ))dξds ≥

≥ φ(σ̃(t_{k}))h^{−ε_{0}}(δ(t_{k})) ×

× ∫_{t_{1}}^{τ_{∗}(_{σ}(t_{k}))}h^{ε_{0}}(δ(s)) ∑_{i=1}^{m} p_{i}(s) ∫_{σ_{i}(s)}^{+∞} ∑_{i=1}^{m} q_{i}(ξ)h^{λ_{∗}}(τ_{i}(ξ))dξds,

whence we obtain

∫_{t_{1}}^{τ_{∗}(_{σ}(t_{k}))}h^{ε_{0}}(δ(s)) ∑_{i=1}^{m} p_{i}(s) ∫_{σ_{i}(s)}^{+∞} ∑_{i=1}^{m} q_{i}(ξ)h^{λ_{∗}}(τ_{i}(ξ))dξds ≤

≤ h^{λ_{∗}}(τ_{∗}(σ̃(t_{k})))h^{ε_{0}}(δ(t_{k})) for k ≥ k_{0},

where k_{0} is a sufficiently large number. But the latter inequality contradicts (2.9). The contradiction obtained proves the validity of the theorem.

Theorem 2.2. Let conditions (1.2), (1.5), (2.1), and (2.2) be fulfilled. Let, moreover,

\underline{lim}_{t→+∞} \frac{h(τ_{∗}(σ̃(t)))}{h(t)} > 0  (2.18)

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

h^{1−λ}(t) ∫_{σ^{∗}(t)}^{+∞} ∑_{i=1}^{m} q_{i}(ξ)h^{λ}(τ_{i}(ξ))dξ ≥ λ + ε  (2.19)
for t ∈ [t_{0}, +∞[ ,

where t_{0} ∈ R_{+} and h is defined by (1.3). Then every proper solution of (0.1) is oscillatory.

Proof. By virtue of Theorem 2.1, to prove the theorem it is sufficient to show that (2.19) implies (2.3) with δ(t) ≡ t.

Indeed, choose ε_{0} > 0 such that

\frac{λ + ε}{λ + ε_{0}} γ^{ε_{0}} > 1 for λ ∈ [0, 1[ ,  (2.20)

where γ = \underline{lim}_{t→+∞} \frac{h(τ_{∗}(_{σ}(t)))}{h(t)}.

On account of (1.2) and (2.18)–(2.20) we obtain

h^{−λ}(τ_{∗}(σ̃(t)))h^{−ε_{0}}(t) ×

× ∫_{0}^{τ_{∗}(_{σ}(t))}h^{ε_{0}}(s) ∑_{i=1}^{m} p_{i}(s) ∫_{σ_{i}(s)}^{+∞} ∑_{i=1}^{m} q_{i}(ξ)h^{λ}(τ_{i}(ξ))dξds ≥

≥ (λ + ε)h^{−λ}(τ_{∗}(σ̃(t)))h^{−ε_{0}}(t) ∫_{0}^{τ_{∗}(_{σ}(t))}p̃(s)h^{ε_{0}+λ−1}(s)ds =

= \frac{λ + ε}{λ + ε_{0}} h^{−λ}(τ_{∗}(σ̃(t)))h^{−ε_{0}}(t)h^{λ+ε}(τ_{∗}(σ̃(t))) ≥

≥ \frac{λ + ε}{λ + ε_{0}} γ^{ε_{0}} > 1 for t ∈ [t^{∗}, +∞[ ,

where t^{∗} ∈]t_{0}, +∞[ is a sufficiently large number. Therefore (2.3) is fulfilled.

Remark. In a certain sense, condition (2.19) is optimal. If we assume ε = 0, then, in general, Theorem 2.2 is not valid.

Theorem 2.3. Let conditions (1.2), (1.5), (2.1), (2.2), and (2.18) be fulfilled. Let, moreover,

\underline{lim}_{t→+∞} \frac{h(t)}{h(σ^{∗}(t))} = α > 0  (2.21)

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

h(t) ∫_{t}^{+∞}∑_{i=1}^{m} q_{i}(ξ)[\frac{h(τ_{i}(ξ))}{h(ξ)}]^{λ}dξ ≥ α^{λ−1}(λ(1 − λ) + ε)  (2.22)
for t ∈ [t_{0}, +∞[ ,

where t_{0} ∈ R_{+} and h is defined by (1.3). Then every proper solution of (0.1) is oscillatory.

Proof. By virtue of Theorem 2.2, to prove the theorem it is sufficient to show that (2.21) and (2.22) imply (2.19).

Indeed, on account of (1.2), (2.21), and (2.22) we have

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

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

= h^{1−λ}(t)h^{λ}(σ^{∗}(t)) ∫_{σ^{∗}(t)}^{+∞} ∑_{i=1}^{m} q_{i}(ξ)[\frac{h(τ_{i}(ξ))}{h(ξ)}]^{λ}dξ +

+λh^{1−λ}(t) ∫_{σ^{∗}(t)}^{+∞} p̃(s)h^{λ−1}(s) ∫_{s}^{+∞}∑_{i=1}^{m} q_{i}(ξ)[\frac{h(τ_{i}(ξ))}{h(ξ)}]^{λ}dξds ≥

≥ α^{λ−1}(λ(1 − λ)+ε)([\frac{h(t)}{h(σ^{∗}(t))}]^{1−λ}+ λh^{1−λ}(t) ∫_{σ^{∗}(t)}^{+∞}p̃(s)h^{λ−2}(s)ds) ≥

≥ α^{λ−1}(λ(1 − λ) + ε)(α^{1−λ} + \frac{λ}{1 − λ}[\frac{h(t)}{h(σ^{∗}(t))}]^{1−λ}) ≥

≥ (λ(1 − λ) + ε)(1 + \frac{λ}{1 − λ}) = λ + \frac{1}{1 − λ} ε ≥ λ + ε

for t ∈ [t_{∗}, +∞[ ,

where t_{∗} ∈]t_{0}, +∞[ is a sufficiently large number, and p̃ is defined by (1.3). Therefore (2.19) is fulfilled.

Corollary 2.1. Let conditions (1.2), (2.1), (2.2), (2.18), and (2.21) be fulfilled. Let, moreover,

\underline{lim}_{t→+∞} \frac{h(τ_{i}(t))}{h(t)} = β_{i} > 0,  (2.23)

and

\underline{lim}_{t→+∞} h(t) ∫_{t}^{+∞}q̃(s)ds > α^{−1} max_{λ∈[0,1]} \{(\frac{α}{β})^{λ}λ(1 − λ)\},

where h, q̃, and α are defined by (1.3), (1.6), and (2.21), respectively, and

β = min{β_{i} : i = 1, …, m}.  (2.24)

Then every proper solution of (0.1) is oscillatory.

Corollary 2.2. Let conditions (1.2), (2.1), (2.2), (2.18), (2.21), and (2.23) be fulfilled. Let, moreover, α = β and

\underline{lim}_{t→+∞} h(t) ∫_{t}^{+∞}q̃(s)ds > \frac{1}{4α} ,

where h, q̃, α, and β are defined by (1.3), (1.6), (2.21), and (2.23), (2.24), respectively. Then every proper solution of (0.1) is oscillatory.

Corollary 2.3. Let p, q ∈ L_{loc}(R_{+}; R_{+}),

h(+∞) = +∞,

and

\underline{lim}_{t→+∞} h(t) ∫_{t}^{+∞}q(s)ds > \frac{1}{4} ,

where

h(t) = ∫_{0}^{t} p(s)ds.

Then every proper solution of the system

u′_{1}(t) = p(t)u_{2}(t),
u′_{2}(t) = −q(t)u_{1}(t)

is oscillatory.

Corollary 2.3 immediately implies Hille’s well-known theorem for secondorder ordinary linear differential equations (see [1]).

References

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2. R. Koplatadze, On oscillation of solutions of second-order retarded differential inequalities and equations. (Russian) Mathematica Balkanica 5:29(1975), 163–172.

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(Received 19.06.1995)

Author’s address:

A. Razmadze Mathematical Institute

Georgian Academy of Sciences

1, M. Alexidze St., Tbilisi 380093

Republic of Georgia