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

Zdenek Oplustil;


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 in finity 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

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