Mem. Differential Equations Math. Phys. 6(1995), 119–120

R. Koplatadze and N. Partsvania

CRITERIA FOR OSCILLATION OF SOLUTIONS OF TWO–DIMENSIONAL DIFFERENTIAL SYSTEMS WITH DEVIATING ARGUMENTS

(Reported on October 16 and 23, 1995)

Consider the system of differential equations

\begin{aligned}u′_{1}(t) & = f_{1}(t, u_{1}(τ(t)), u_{2}(σ(t))), \\ u′_{2}(t) & = f_{2}(t, u_{1}(τ(t)), u_{2}(σ(t))), \end{aligned}  (1)

where f_{i} : R_{+} × R^{2} → R (i = 1, 2) satisfy the local Carathéodory conditions, 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) = +∞.

In this paper, we establish sufficient conditions for oscillation of so-called proper solutions of (1), i.e., of nontrivial solutions defined in some neighbourhood of +∞. Analogous problems for higher order functional differential equations have been considered in [1].

In the sequel, we assume that the inequalities

f_{1}(t, x, y) sign y ≥ p(t)|y|,
f_{2}(t, x, y) sign x ≤ −q(t)|x| for t ∈ R_{+}, (x, y) ∈ R^{2}

are fulfilled, where p, q : R_{+} → R_{+} are locally summable functions.

Theorem 1. Let the function

h(t) = ∫_{0}^{t} p(s) ds  (2)

be such that

h(+∞) = +∞,  (3)

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

Then every proper solution of (1) is oscillatory.

Remark. Note that the inequality (4) cannot be replaced by

\underline{lim}_{t→+∞} ∫_{σ(τ(t))}^{t} q(s)h(τ(s)) ds > \frac{1}{e} − ε  (5)

1991 Mathematics Subject Classification. 34K15.

Key words and phrases. Two-dimensional differential system with deviating arguments, oscillatory solution. for no ε ∈]0, \frac{1}{e} [ .

Indeed, let ε ∈]0, \frac{1}{e} [ and μ ∈ [0, 1[ . Let us choose λ ∈]0, 1 − μ[ such that λ > (1 − eε)(1 − μ). Obviously the system

u′_{1}(t) = \frac{c_{1}}{t^{μ}} u_{2}(αt), u′_{2}(t) = \frac{c_{2}}{t^{2−μ}} u_{1}(βt) for t ≥ 1,

where α ∈]0, 1], β ∈]0, +∞[ , αβ = e^{\frac{1}{λ+μ−1}} , c_{1} = λα^{1−λ−μ} and c_{2} = (λ + μ − 1)β^{−λ}, has the nonoscillatory solution (t^{λ}, t^{λ+μ−1}) despite the fact that (5) is fulfilled.

Theorem 2. Let (3) be fulfilled. Let, moreover, there exist α ∈]0, +∞[ , β ∈ [1, +∞[ such that

\underline{lim}_{t→+∞} \frac{h(σ(τ(t)))}{h(t)} ≥ α, \underline{lim}_{t→+∞} \frac{h(t)}{h(σ(t))} ≥ β,

and for some λ ∈]0, 1],

\underline{lim}_{t→+∞}h^{λ}(t) ∫_{t}^{+∞}h^{1−λ}(s)q(s) ds > \frac{1}{λ^{2}} max \{α^{x−1}β^{−x}(1 − x)x : x ∈ [0, 1]\},

where the function h is defined by (2). Then every proper solution of (1) is oscillatory.

Theorem 3. Let condition (3) be fulfilled. If, moreover,

\underline{lim}_{t→+∞} \frac{h(σ(τ(t)))}{h^{μ}(t)} > 0,

lim_{λ→0} ( \underline{lim}_{t→+∞}h^{λ}(t) ∫_{t}^{+∞}h^{μ−λ}(s)q(s) ds) > 0,

where μ ∈]0, 1[ and the function h is defined by (2), then every proper solution of (1) is oscillatory.

References

1. R. Koplatadze, On oscillatory properties of solutions of functional differential equations. Mem. Differential Equations Math. Phys. 3(1994), 1–179.

Authors’ addresses:

R. Koplatadze I. Vekua Institute of Applied Mathematics Tbilisi State University 2, University St., Tbilisi 380043 Ceorgia

N. Partsvania A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia