Mem. Differential Equations Math. Phys. 5(1995), 122–124

A. Lomtatidze and N. Partsvania

ON OSCILLATORY PROPERTIES OF SECOND ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

(Reported on May 8, 1995)

Consider the system

\begin{aligned}u′ & = p(t)v, \\ v′ & = −q(t)u, \end{aligned}  (1)

where p, q : [0, +∞[→ [0, +∞[ are locally summable functions.

Definition. The system (1) is said to be oscillatory if it has at least one oscillatory solution and nonoscillatory otherwise.

It is known (see [1]) that if

∫^{+∞}p(s) ds = +∞ and ∫^{+∞}q(s) ds = +∞,

then the system (1) is oscillatory, but if

∫^{+∞}p(s) ds < +∞ and ∫^{+∞}q(s) ds < +∞,

then it is nonoscillatory.

Therefore we will assume that

∫^{+∞}p(s) ds = +∞ and ∫^{+∞}q(s) ds < +∞.  (2)

(The case where ∫^{+∞}p(s) ds < +∞ and ∫^{+∞}q(s) ds = +∞ can be easily reduced to the case (2)).

As shown in [2], if the condition (2) is fulfilled and for some λ < 1

∫^{+∞}f^{λ}(s)q(s) ds = +∞,

where

f(t) = ∫_{0}^{t} p(s) ds,  (3)

1991 Mathematics Subject Classification. 34C10.

Key words and phrases. Second order system of ordinary differential equations, oscillatory solution. then the system (1) is oscillatory.

Therefore we will consider the case where

∫^{+∞}p(s) ds = +∞ and ∫^{+∞}f^{λ}(s)q(s) ds < +∞ for all λ < 1,

where f is the function defined by (3).

Introduce the notation

g_{∗}(λ) = lim inf_{t→+∞} f^{1−λ}(t) ∫_{t}^{+∞}f^{λ}(s)q(s) ds,
g^{∗}(λ) = lim sup_{t→+∞} f^{1−λ}(t) ∫_{t}^{+∞}f^{λ}(s)q(s) ds

for λ < 1 and

g_{∗}(λ) = lim inf_{t→+∞} f^{1−λ}(t) ∫_{0}^{t} f^{λ}(s)q(s) ds,
g^{∗}(λ) = lim sup_{t→+∞} f^{1−λ}(t) ∫_{0}^{t} f^{λ}(s)q(s) ds

for λ > 1.

Theorem 1. Let g_{∗}(0) ≤ \frac{1}{4} , g_{∗}(2) ≤ \frac{1}{4} and

g^{∗}(0) > g_{∗}(0) + \frac{1}{2}(\sqrt{1 − 4g_{∗}(0)} + \sqrt{1 − 4g_{∗}(2)} ).

Then the system (1) is oscillatory.

Theorem 2. Let g_{∗}(0) ≤ \frac{1}{4} and g_{∗}(2) ≤ \frac{1}{4} . Moreover, let either

g^{∗}(λ) > \frac{λ^{2}}{4(1 − λ)} + \frac{1}{2}(1 + \sqrt{1 − 4g_{∗}(2)} )

for some λ < 1 or

g^{∗}(λ) > \frac{λ^{2}}{4(λ − 1)} − \frac{1}{2}(1 − \sqrt{1 − 4g_{∗}(0)} )

for some λ > 1. Then the system (1) is oscillatory.

Corollary 1. Let

lim_{λ→1} |1 − λ|g^{∗}(λ) > \frac{1}{4} .

Then the system (1) is oscillatory.

Corollary 2. Let for some λ ≠ 1

|1 − λ|g_{∗}(λ) > \frac{1}{4} .

Then the system (1) is oscillatory.

Theorem 3. If for some λ ≠ 1

|1 − λ|g_{∗}(λ) > \frac{(2λ − 1)(3 − 2λ)}{4} , |1 − λ|g^{∗}(λ) < \frac{1}{4} ,

then the system (1) is nonoscillatory.

References

1. J. D. Mirzov, Asymptotic behaviour of solutions of systems of nonlinear nonautonomous ordinary differential equations. (Russian) Maĭkop, 1993.

Authors’ addresses:

A. Lomtatidze N. Muskhelishvili Institute of Computational Mathematics Georgian Academy of Sciences 8, Akuri St., Tbilisi 380093 Georgia

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