Mem. Differential Equations Math. Phys. 10(1997), 113–114

N. Partsvania

TO THE QUESTION OF OSCILLATION OF SOLUTIONS OF TWO-DIMENSIONAL DIFFERENTIAL SYSTEMS WITH DEVIATED ARGUMENTS

(Reported on February 26 and March 4, 1996)

Consider the system of differential inequalities

u′_{1}(t) sign u_{2}(σ(t)) ≥ p(t)|u_{2}(σ(t))|,
u′_{2}(t) sign u_{1}(τ(t)) ≤ −q(t)|u_{1}(τ(t))|, (1)

where p, q : ℝ_{+} → ℝ_{+} are locally summable functions, τ, σ : ℝ_{+} → ℝ are nondecreasing continuous functions and σ(t) ≤ t, σ(τ(t)) ≤ t for t ∈ ℝ_{+}, lim_{t→+∞} τ(t) = +∞, lim_{t→+∞} σ(t) = +∞.

Definition. A proper solution (u_{1}, u_{2}) of the system (1), i.e. a nontrivial solution defined in some neighbourhood of +∞, 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.

In this paper, we are especially interested in the question whether every proper solution of (1) is oscillatory.

In the sequel, we assume that the condition

∫_{0}^{+∞}p(s)ds = +∞

is fulfilled.

Theorem 1. Let for some k ∈ ℕ and for any t_{0} ∈ ℝ_{+} the inequality

lim sup_{t→+∞}( ∫_{σ(τ(t))}^{t} q(s)ω_{k}(σ(τ(s)), σ(τ(t)); t_{0}) ∫_{η_{k}(t_{0})}^{τ(s)} p(ξ)dξds +

+g_{k}(σ(τ(t)); t_{0}) ∫_{t}^{+∞}q(s)ds) > 1,

be fulfilled, where

g_{k}(t; t_{0}) = ∫_{η_{k}(t_{0})}^{t} p(s)ω_{k}(σ(s), s; t_{0})ds +

1991 Mathematics Subject Classification. 34K15.

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

+ ∫_{η_{k}(t_{0})}^{t} q(s)ψ_{k}(s; t_{0}) ∫_{η_{k}(t_{0})}^{s} p(ξ)ω_{k}(σ(ξ), ξ; t_{0})dξds for t ≥ η_{k}(t_{0}),

ω_{j}(t, s; t_{0}) = exp { ∫_{t}^{s} q(ξ)φ_{j}(ξ; t_{0}) ∫_{η_{j} (t_{0})}^{τ(ξ)} p(ξ_{1})dξ_{1}dξ}

for s ≥ t ≥ η_{j}(t_{0}) (j = 1, …, k),

φ_{1}(t; t_{0}) = 1, φ_{j}(t; t_{0}) = ω_{j−1}(σ(τ(t)), t; t_{0}) for t ≥ η_{j}(t_{0}) (j = 2, …, k),

ψ_{1}(t; t_{0}) = 0, ψ_{j}(t; t_{0}) = ∫_{η_{j−1}(t_{0})}^{σ(τ(t))} p(s) exp { ∫_{σ(s)}^{t} q(ξ)ψ_{j−1}(ξ; t_{0})dξ}ds  (2)

for t ≥ η_{j}(t_{0}) (j = 2, …, k),
η_{1}(t) = max {s : min (σ(τ(σ(s))), σ(s)) ≤ t}, η_{j}(t) = η_{1}(η_{j−1}(t))
(j = 2, …, k).

Then every proper solution of (1) is oscillatory.

(In the case of second order differential inequalities, an analogous construction of the functions φ_{k}, ψ_{k}, η_{k} can be found in [1]).

Theorem 2. Let for some k ∈ ℕ and for any t_{0} ∈ ℝ_{+} the inequality

lim sup_{t→+∞} ψ_{k}(t; t_{0}) ∫_{t}^{+∞}q(s)ds > 1,

be fulfilled, where the function ψ_{k} is defined by (2).

Then every proper solution of (1) is oscillatory.

References

1. R. Koplatadze, Oscillation criteria of solutions of second order differential inequalities and equations with a delayed argument. (Russian) Trudy Inst. Prikl. Mat. I. N. Vekua 17(1986), 104–120.

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