Oscillatory properties of solutions of second order nonlinear neutral differential inequalities with oscillating coefficients

Abstract

summary:This paper deals with the second order nonlinear neutral differential inequalities $(A_\nu )$: $(-1)^\nu x(t)\,\lbrace \,z^{\prime \prime }(t)+(-1)^\nu q(t)\,f(x(h(t))) \rbrace \le 0,\$ $t\ge t_0\ge 0,$ where $\ \nu =0\$ or $\ \nu =1,\$ $\ z(t)\,=\,x(t)\,+\,p(t)\,x(t-\tau ),\$ $\ 0<\tau =\$ const, $\ p,q,h:[t_0,\infty )\rightarrow R\$ $\ f:R\rightarrow R\$ are continuous functions. There are proved sufficient conditions under which every bounded solution of $(A_\nu )$ is either oscillatory or $\ \liminf \limits _{t\rightarrow \infty }|x(t)|=0.