Oscillatory behaviour of a class of nonlinear second order mixed difference equations

Abstract

In this paper oscillatory and asymptotic behaviour of solutions of a class of nonlinear second order neutral difference equations with positive and negative coefficients of the form (E) $\quad \Delta (r(n) \Delta (y(n) + p(n) y(n-m))) + f(n) H_1(y(n-k_1))-g(n) H_2 (y(n-k_2)) = q(n) $ and $\quad \Delta (r(n) \Delta (y(n) + p(n) y(n-m))) + f(n) H_1(y(n-k_1))-g(n) H_2 (y(n-k_2)) = 0 $ are studied under the assumptions \begin{eqnarray}\sum\limits_{n=0}^{\infty} \frac{1}{r(n)} < \infty \nonumber \end{eqnarray} and \begin{eqnarray}\sum\limits_{n=0}^{\infty} \frac{1}{r(n)} = \infty \nonumber \end{eqnarray} for various ranges of $p(n)$. Using discrete Krasnoselskii's fixed point theorem sufficient conditions are obtained for existence of positive bounded solutions of (E).