Mem. Differential Equations Math. Phys. 6(1995), 127–129

G. Giorgadze

ON OSCILLATORY PROPERTIES OF THE n-TH ORDER SYSTEM OF DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

(Reported on November 27, 1995)

Consider the system

x′_{i}(t) = f_{i}(t, x_{1}(δ_{i1}(t)), …, x_{n}(δ_{in}(t))), (i = 1, …, n),  (1)

where n ≥ 2, the vector function (f_{i})_{i=1}^{n} : R_{+}×R^{n} → R^{n} satisfies the local Caratheodory conditions, δ_{ij} : R_{+} → R are nondecreasing and

lim_{t→+∞} δ_{ij}(t) = +∞ (i, j = 1, …, n), δ_{i,i+1} ∈ C′(R_{+}, R) (i = 1, …, n − 1).

Define σ : R_{+} → R_{+} by

σ(t) = inf {s : s ∈ R_{+}, s ≥ t, δ_{ij}(ξ) ≥ t, for ξ ∈ [s, +∞[ (i, j = 1, …, n)}.

Definition 1. A continuous vector function X = (X_{i})_{i=1}^{n} : [t_{0}, +∞[→ R^{n} with t_{0} ∈ R_{+} is said to be a proper solution of the system (1) if it is locally absolutely continuous on [σ(t_{0}), +∞[, almost everywhere on this interval the equality (1) is fulfilled, and

sup {∥x(s)∥ : s ∈ [t, ∞[} > 0, for t ∈ [t_{0}, +∞[

Definition 2. A proper solution of the system (1) is said to be oscillatory if every component of this solution has a sequence of zeroes tending to +∞. Otherwise the solution is said to be nonoscillatory.

Definition 3. We say that the system (1) has the property A provided its every proper solution is oscillatory if n is even, and either is oscillatory or satisfies

|x_{i}(t)| ↓ 0, for t ↑ +∞, (i = 1, …, n),  (2)

if n is odd.

Definition 4. We say that the system (1) has the property B provided its every proper solution either is oscillatory or satisfies either (2) or

|x_{i}(t)| ↑ +∞, for t ↑ +∞, (i = 1, …, n)  (3)

if n is even, and either is oscillatory or satisfies (3) if n is odd.

1991 Mathematics Subject Classification. 34K15.

Key words and phrases. Oscillatory solution, nonoscillatory solution, property A, property B.

We will assume that there exist ν_{i} ∈ {0; 1} such that

(−1)^{ν_{i}}f_{i}(t, x_{1}, …, x_{n}) sign x_{i+1} ≥ p_{i}(t)|x_{i+1}|,

(−1)^{ν_{n}}f_{n}(t, x_{1}, …, x_{n}) sign x_{1} ≥ g(t, |x_{1}|), for t ∈ R_{+}; x_{1}, …, x_{n} ∈ R, (4)

where the function g ∈ K_{loc}(R_{+} × R_{+}; R_{+}) is nondecreasing in the second argument, p_{i} ∈ L_{loc}(R_{+}, R_{+}) and

∫_{0}^{+∞}p_{i}(t)dt = +∞ (i = 1, …, n − 1).  (5)

Besides, introduce the notation

ν = ∑_{i=1}^{n} ν_{i}.
τ_{i}(t) = δ_{i−1,i}(t) (i = 2, …, n), τ_{1}(t) = τ_{n+1}(t) = δ_{n1}(t).
τ_{ji}^{∗}(t) = \begin{cases}τ_{j}(τ_{j−1}(… (τ_{i+1}(t)) …)), & if 1 ≤ i < j ≤ n + 1, \\ t, & if i = j (i = 1, …, n), \end{cases}
γ_{ji}^{∗}(t) = inf {s : s ∈ R_{+}, τ_{ki}^{∗}(s) ≥ t (k = i, …, j)} (1 ≤ i ≤ j ≤ n),
I^{0} = 1, I^{j}(s, t; p_{i+j−1}, …, p_{i}) =

= ∫_{t}^{s} p_{i+j−1}(τ_{i+j−1,i}(ξ))(τ_{i+j−1,i}^{∗′}(ξ)I^{j−1}(ξ, t; p_{i+j−2}, …, p_{i})dξ,

J_{0} = 1, J^{j}(t, s; p_{i}, …, p_{i+j−1}) = ∫_{s}^{t} p_{i}(ξ)J^{j−1}(τ_{i+1}(ξ), τ_{i+1}(s); p_{i+1}, …, p_{i+j−1}dξ),

(i = 1, …, n − 1; j = 1, …, n − i).

Note that the functions γ_{ji}^{∗} : R → R_{+} are increasing,

γ_{ki}^{∗}(t) ≥ γ_{ji}^{∗}(t) (1 ≤ i ≤ j ≤ k ≤ n),
γ_{ji}^{∗}(t) ≥ t (1 ≤ i ≤ j ≤ n), for t ∈ R,

and the expressions I^{j}(s, t; p_{i+j−1}, …, p_{i}) and J^{j}(t, s; p_{i}, …, p_{i+j−1}) have the meaning iff t, s ≥ γ_{i+j−1,i}^{∗}(0) (i = 1, …, n − 1; j = 1, …, n − i).

Theorem -1. Suppose that the conditions (4) and (5) are fulfilled, ν is odd and for every l ∈ {1, …, n − 1} such that l + n is odd, the equation

v′(t) = I^{n−l}(τ_{l1}^{∗}(t), t_{∗l}; p_{n−1}, …, p_{l})g(τ_{n1}^{∗}(t), z_{l}(τ_{n+1,1}^{∗}(t)))τ_{n1}^{∗′}(t),  (6)

with z_{l}(t) = \frac{J^{l}(t, γ_{l1}^{∗}(0); p_{1}, …, p_{l})}{J′(τ_{l1}^{∗}(t), 0; p_{l})} , t_{∗l} = γ_{n−1,l}^{∗}(0), has no positive proper solution. In

the case where n is odd, let, moreover,

∫_{γ_{n1}^{∗}(0)}^{+∞} I^{n−1}(ξ, γ_{n−1,1}^{∗}(0); p_{n−1,...,p_{1}})g(τ_{n1}^{∗}(ξ), c)τ_{n1}^{∗′}(ξ)dξ = +∞, for c > 0. (7)

Then the system (1) has the property A.

Theorem 0. Suppose that the conditions (4) and (5) are fulfilled, ν is even and for every l ∈ {1, …, n − 2} such that l + n is even, the equation (6) has no positive proper solution. Let, moreover,

∫_{γ(0)}^{+∞}g(t, cJ^{n−1}(τ_{1}(t), γ_{n1}^{∗}(0); p_{1}, …, p_{n−1}))dt = +∞.  (8)

for any c > 0, and, in the case where n is even, the condition (7) be fulfilled. Then the system (1) has the property B.

Consider now the case where the inequalities

(−1)^{ν_{i}}f_{i}(t, x_{1}, …, x_{n}) sign x_{i+1} ≥ p_{i}(t)|x_{i+1}| (i = 1, …, n; x_{n+1=x_{1}})  (9)

are fulfilled, where p_{i} ∈ L_{loc}(R_{+}, R_{+}) (i = 1, …, n) and (5) holds.

Theorem 1. Suppose that (12) is fulfilled, ν is odd and for every i ∈ {1, …, n − 1} such that i + n is odd, the inequalites

lim_{t→+∞} sup \frac{I^{n−i}(t, t_{∗i}, p_{n−1}, …, p_{i})}{I^{n−i−1}(τ_{i+1}(t), τ_{i+1}(t_{∗i}); p_{n−1}, …, p_{i+1})} ×

× ∫_{t}^{+∞}I^{n−i−1}(τ_{i+1}(s), τ_{i+1}(t_{∗i}); p_{n−1}, …, p_{i+1}) ×

× \frac{J^{i}(τ_{n+1}^{∗}(s), γ_{i1}^{∗}(0); p_{1}, …, p_{i})}{J′(τ_{i1}^{∗}(τ_{n+1}^{∗}(s)), 0; p_{i})} p_{n}(τ_{ni}^{∗}(s))τ_{ni}^{∗′}(s)ds > 1  (10)

and

τ_{i1}^{∗}(τ_{n+1,i}^{∗}(t)) ≥ t, textfor t ≥ γ(0)  (11)

hold, where t_{∗i} = γ_{n−1,i}^{∗}(0). In the case where n is odd, let, moreover,

∫_{γ_{n1}^{∗}(0)}^{+∞} I^{n−1}(ξ, γ_{n−1,1}^{∗}(0); p_{n−1}, …, p_{1})p_{n}(τ_{n1}^{∗}(ξ))τ_{n1}^{∗′}(ξ)dξ = +∞  (12)

Then the system (1) has the property A.

Theorem 2. Suppose that (12) is fulfilled and for every i ∈ {1, …, n − 1} such that i + n is even, the inequalities (13) and (14) are fulfilled. Let. moreover,

∫_{γ(0)}^{+∞}J^{n−1}(τ_{1}(t), γ_{n−1,1}^{∗}(0); p_{1}, …, p_{n−1})p_{n}(t)dt = +∞,

and, in the case where n is odd, (15) hold. Then the system (1) has the property B.

Author’s address:

Georgian Technical University

72, Kostava St., Tbilisi 380093

Georgia