Oscillation properties of a class of second order equations


Zornitza Petrova




We establish sufficient conditions for oscillation of the following equations: \[ z''(t)+um_{i=0}^nheta_i(t)z'(t-au_i)+um_{k=0}^{ilde n}\beta_k(t)z(t-igma_k)+G(z''(t),z'(t),z(t))=F(t), \] We suppose that $n$, $\tilde n\in\mathbf N$, $\tau_i\geq0$, $\forall i=\overline{0,n}$ and $\sigma_k\geq0$, $\forall k=\overline{0,n}$ are given constants as well as $T\geq0$ is a large enough constant such that all the functions $\{\theta_i(t)\}^n_{i=0}$, $\{\beta_k(t)\}^{\tilde n}_{k=0}$ and $F(t)$ are of the class $C([T,\infty)^3;\mathbf R)$. Also, $G(z''(t),z'(t),z(t))\in C([T,\infty)^3;\mathbf R)$. We obtain two types of results: the first is concerned with the monotonicity of the solutions, and the second one is a sufficient condition for the distributions of their zeros.