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

A. K. Tripathy, Kakatiya Institute of Technology and Science, Warangal, India

E. J. Qualitative Theory of Diff. Equ., No. 48. (2010), pp. 1-19.

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) $ \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
$ \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).

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