MMN-2391

# Oscillatory properties of certain system of non-linear ordinary differential equations

*Zdenek Oplustil*;

## Abstract

We consider certain two-dimensional system of non-linear differential
equations
u'=g(t)|v|^(1/A)sgn v
v'=-p(t)|u|^(A) sgn u,
where A is a positive number, g,p are locally integrable functions (g is non-negative).
In the case when coefficient g is not inegrable on the half-line, the considered system has been widely studied in particular cases such linear systems as well as
second order linear and half-linear differential equations. However, the case when function g is integrable on the hlaf-line has not been studied in detail in the existing literature.
Moreover, we allow that the coefficient g can have zero points in any neigh-
bourhood of infinity and consequently, considered system can not be rewritten
as the second order linear or half-linear differential equation in this case. In the
paper, new oscillation criteria are established in the case
when function g is integrable on the hlaf-line and without restricted assumption function p preserves its sign (which is usually considered).

Vol. 19 (2018), No. 1, pp. 439-459

DOI: 10.18514/MMN.2018.2391