Mem. Differential Equations Math. Phys. 13(1997), 132–135

G. Giorgadze

ON OSCILLATORY PROPERTIES OF AN N-TH ORDER SYSTEM OF

LINEAR DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

(Reported on October 13, 20, 1997)

Consider the system

\begin{cases}x′_{i}(t) = p_{i}(t)x_{i+1}(τ_{i+1}(t)) (i = 1, …, n − 1), \\ x′_{n}(t) = p_{n}(t)x_{1}(τ_{1}(t)), \end{cases}  (1)

where n ≥ 2, τ_{i} ∈ C^{1}(R_{+}; R) are nondecreasing functions, lim_{t→+∞} τ_{i}(t) = +∞(i = 1, …, n), p_{i} ∈ L_{loc}(R_{+}; R_{+}) (i = 1, …, n − 1), p_{n} ∈ L_{loc}(R_{+}; R) and

∫ ^{+∞} p_{i}(t)dt = +∞ (i = 1, …, n − 1).  (2)

In the present paper, sufficient conditions for the oscillation of all proper solutions of (1) are established. Analogous questions for deviating and general functional differential equations were studied in a great deal of papers, for example, in [1,2], and for ordinary differential equations, in [3,4,5].

Let t_{0} ∈ R_{+}, t_{∗} = min{t_{0}, τ_{1}(t_{0}), …, τ_{n}(t_{0})}. A continuous vector function x = (x_{i})_{i=1}^{n} : [t_{∗}, +∞[→ R^{n} is said to be a proper solution of the system (1), if it is locally absolutely continuous on [t_{0}, +∞[, a.e. on this interval satifies (1), and

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

A proper solution of (1) is said to be oscillatory, if each of its components has a sequence of zeroes tending to +∞. Otherwise the solution is called nonoscillatory.

We say that the system (1) has the property A provided any of its solutions is oscillatory if n is even, and either is oscillatory or satisfies

|x_{i}(t)| ↓ 0, for t ↑ +∞ (i = 1, …, n)  (3)

if n is odd.

We say that the system (1) has the property B provided any of its solutions either is oscillatory or satisfies (3) if n is even, and either is oscillatory or satisfies (3) or

|x_{i}(t)| ↑ +∞, for t ↑ +∞ (i = 1, …, n)  (4)

if n is odd.

Introduce the notation

τ_{ji}^{∗}(t) = \begin{cases}τ_{j}(τ_{j−1}(⋯ (τ_{i+1}(t)) ⋯)) & for 1 ≤ i < j ≤ n + 1, \\ t & for i = j (i = 1, …, n). \end{cases}

1991 Mathematics Subject Classification. 34K15.

Key words and phrases. System of linear differential equations with deviating arguments, oscillatory solution, property A, property B.

Here we mean that τ_{n+1}(t) = τ_{1}(t).

γ_{ji}(t) = inf {s : s ∈ R_{+}, s ≥ t, 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}^{∗′}(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).

Besides, everywhere below we set

t_{∗i} =γ_{n−1,i}(0) (i = 1, …, n − 1);
α_{e}(t) = \frac{I^{n−e}(t, t_{∗e}; p_{n−1}, …, p_{e})}{I^{n−e−1}(τ_{e+1}(t), τ_{e+1}(t_{∗e}); p_{n−1}, …, p_{e+1})} ;
z_{e}(t) = \frac{J^{e}(t, γ_{e1}(0); p_{1}, …, p_{e})}{J^{1}(τ_{e1}^{∗}(t), 0; p_{e})} .

Theorem 1. Let (2) be satisfied, p_{n} ∈ L_{loc}(R_{+}; R_{−}) and for any l ∈ {1, …, n − 1} with l + n odd,

τ_{e1}^{∗}(τ_{n+1,e}^{∗}(t)) ≥ t,  (5)

lim_{t→+∞} sup α_{e}(t) ∫_{t}^{+∞} I^{n−e−1}(τ_{e+1}(s), τ_{e+1}(t_{∗e}); p_{n−1}, ⋯, p_{e+1}) ×

×z_{e}(τ_{n+1,e}(s))|p_{n}(τ_{ne}^{∗}(s))|τ_{ne}^{∗′}(s)ds > 1.  (6)

If, moreover, the condition

∫ ^{+∞} I^{n−1}(ξ, t_{∗1}; p_{n−1}, …, p_{1})|p_{n}(τ_{n1}^{∗}(ξ))|τ_{n1}^{∗′}(ξ)dξ = +∞,  (7)

is fulfilled for odd n, then the system (1) has the property A.

Theorem 2. Let (2) be satisfied, p_{n} ∈ L_{loc}(R_{+}; R_{−}) and for any l ∈ {1, …, n − 1} with l + n odd

τ_{e1}^{∗}(τ_{n+1,e}^{∗}(t)) ≤ t,  (8)

lim_{t→+∞} sup α_{e}(τ_{e1}^{∗}(τ_{n+1,e}^{∗}(t)) ∫_{t}^{+∞} I^{n−e−1}(τ_{e+1}(s),τ_{e+1}(t_{∗e}); p_{n−1}, …, p_{e+1}) ×

×|p_{n}(τ_{ne}^{∗}(s))|z_{e}(τ_{n+1,e}^{∗}(s))τ_{ne}^{∗′}(s)ds > 1.  (9)

If, moreover, the condition (7) is fulfilled for odd n, then the system (1) has the property A.

Theorem 3. Let (2) be satisfied, p_{n} ∈ L_{loc}(R_{+}; R_{−}) and for any l ∈ {1, …, n − 1} with l + n odd, (5) be fulfilled along with

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

×|p_{n}(τ_{ne}^{∗}(s))|z_{e}(τ_{n+1,e}^{∗}(s))|I^{1}(s, t_{∗e}; p_{e})τ_{ne}^{∗′}(s)ds > 1.  (10)

If, moreover, the condition (7) is fulfilled for odd n, then the system (1) has the property A.

Theorem 4. Let (2) be satisfied, p_{n} ∈ L_{loc}(R_{+}; R_{−}) and for any l ∈ {1, …, n − 1} with l + n odd, (8) be fulfilled along with

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

×|p_{n}(τ_{ne}^{∗}(s))|z_{e}(τ_{n+1,e}^{∗}(s))|I^{1}(τ_{e1}^{∗}(τ_{n+1,e}(s)), t_{∗e}; p_{e})τ_{ne}^{∗′}(s)ds > 1.  (11)

If, moreover, the condition (7) is fulfilled, then the system (1) has the property A.

Theorem 5. Let (2) be satisfied, p_{n} ∈ L_{loc}(R_{+}; R_{+}) and for any l ∈ {1, …, n − 2} with l + n even, (5) and (6) ((8) and (9)) be fulfilled. If, moreover,

∫ ^{+∞} J^{n−1}(τ_{1}(t), t_{∗1}; p_{1}, …, p_{n−1})p_{n}(t)dt = +∞,  (12)

and, for even n, the condition (7) is fulfilled, then the system (1) has the property B.

Theorem 6. Let (2) be satisfied, p_{n} ∈ L_{loc}(R_{+}; R_{+}) and for any l ∈ {1, …, n − 2} with l + n even, the conditions (5) and (10) ((8) and (11)) be fulfilled. If, moreover, (12) is satisfied and, for even n, the condition (7) is fulfilled, then the system (1) has the property B.

Now consider the system

\begin{cases}x′_{i}(t) = x_{i+1}(β_{i+1}t) (i = 1, …, n − 1), \\ x′_{n}(t) = p(t)x_{1}(β_{1}t), \end{cases}  (13)

where p ∈ L_{loc}(R_{+}; R), β_{i} ∈]0; +∞[ (i = 1, …, n), which is a special case of (1). For this system, the above results can be written in a more effective form.

Theorem 7. Let p ∈ L_{loc}(R_{+}; R_{−}), ∏_{i=1}^{n} β_{i} ≥ 1 ( ∏_{i=1}^{n} β_{i} ≤ 1) and

\overline{lim}_{t→+∞}t ∫_{t}^{+∞} s^{n−2}|p(s)|ds > (n − 1)! ∏_{i=2}^{n} β_{i}^{i−1} ((n − 1)! ∏_{i=1}^{n−1} β_{i}^{i−n}).

Then the system (13) has the property A.

Theorem 8. Let p ∈ L_{loc}(R_{+}; R_{−}), ∏_{i=1}^{n} β_{i} ≥ 1 ( ∏_{i=1}^{n} β_{i} ≤ 1) and

\overline{lim}_{t→+∞} \frac{1}{t} ∫_{0}^{+∞} s^{n}|p(s)|ds > (n − 1)! ∏_{i=2}^{n} β_{i}^{i−1} ((n − 1)! ∏_{i=1}^{n−1} β_{i}^{i−n}).

Then the system (13) has the property A.

Theorem 9. Let p ∈ L_{loc}(R_{+}; R_{+}) and ∏_{i=1}^{n} β_{i} ≥ 1 ( ∏_{i=1}^{n} β_{i} ≤ 1). Moreover,

let

\overline{lim}_{t→+∞}t ∫_{t}^{+∞} s^{n−2}p(s)ds > 2(n − 2)! ∏_{i=1}^{n} β_{i}^{i−2} (2(n − 2)! ∏_{i=1}^{n} β_{i}^{i+1−n}),

if n is even, and

\overline{lim}_{t→+∞}t ∫_{t}^{+∞} s^{n−2}p(s)ds > (n − 1)! ∏_{i=2}^{n} β_{i}^{i−1} ((n − 1)! ∏_{i=1}^{n} β_{i}^{i+1−n}).

if n is odd. Then the system (13) has the property B.

Theorem 10. Let p ∈ L_{loc}(R_{+}; R_{+}) and ∏_{i=1}^{n} β_{i} ≥ 1 ( ∏_{i=1}^{n} β_{i} ≤ 1). Moreover,

let

\overline{lim}_{t→+∞} \frac{1}{t} ∫_{0}^{t} s^{n}p(s)ds > 2(n − 2)! ∏_{i=1}^{n} β_{i}^{i−2} (2(n − 2)! ∏_{i=1}^{n} β_{i}^{i+1−n})

if n is even, and

\overline{lim}_{t→+∞} \frac{1}{t} ∫_{0}^{t} s^{n}p(s)ds > (n − 1)! ∏_{i=2}^{n} β_{i}^{i−1} ((n − 1)! ∏_{i=1}^{n} β_{i}^{i+1−n}),

if n is odd. Then the system (13) has the property B.

References

1. R. G. Koplatadze and T. A. Chanturia, On oscillatory properties of differential equations with deviating arguments. (Russian) Tbilisi University Press, Tbilisi, 1977.

2. R. G. Koplatadze, On oscillatory properties of sulutions of functional differential equations. Mem. Differential Equations Math. Phys. 3(1994), 1–177.

3. T. A. Chanturia, On oscillatory properties of systems of nonlinear ordinary differential equations. (Russian) Trudy Inst. Prikl. Mat. I. N. Vekua 14(1983), 163– 202.

4. R. G. Koplatadze and N. L. Partsvania, Oscillatory properties of solutions of systems of second order differential equations with deviating arguments. (Russian) Differentsial’nye Uravneniya 33(1997), No. 10.

5. G. Giorgadze, In oscillatory properties of the n-th order system of differential equations with deviating arguments. Mem. Differential Equations Math. Phys. 6(1995), 127–129.

Author’s address:

Georgian Technical University

72, Kostava St., Tbilisi 380093

Georgia