Riccati Techniques, Discrete Oscillation and Conjugacy Criteria for Fourth Order Nonlinear Difference Equations

We discuss the discrete Oscillatory properties of Fourth order non linear difference equation. whereα >1. In particular we establish the discrete oscillation using Riccati Techniques and Conjugacy criteria. The Proofs of all the results in this paper are based on the Riccati technique.

Oscillation of impulsive conformable fractional differential equations

In this paper, we investigate oscillation results for the solutions of impulsive conformable fractional differential equations of the form$$\left\{ \begin{array}{l} {t_k}{D^\alpha }\left( {p\left( t \right)\left[ {{t_k}{D^\alpha }x\left( t \right) + r\left( t \right)x\left( t \right)} \right]} \right) + q\left( t \right)x\left( t \right) = 0,\quad t \ge {t_0},\;t \ne {t_k},\\ x\left( {t_k^ + } \right) = {a_k}x(t_k^ - ),\quad {t_k}{D^\alpha }x\left( {t_k^ + } \right) = {b_{k\;{t_{k - 1}}}}{D^\alpha }x(t_k^ - ),\quad \;k = 1,2, \ldots. \end{array} \right.$$Some new oscillation results are obtained by using the equivalence transformation and the associated Riccati techniques.

OSCILLATION AND COMPARISON THEOREMS FOR NEUTRAL DIFFERENCE EQUATIONS

In this paper we study qualitative properties of solutions of the neutral difference equation $$ \Delta(y_n-py_{n-k})+\sum_{i=1}^m q_n^i y_{n-k_i} =0 $$ $$ y_n=A_n \quad \text{ for } n=-M, \cdots, -1, 0$$ where $p \ge 1$, $M =\max\{k, k_1, \cdots, k_m\}$, and $k$, $k_i$, $i =1, \cdots, m$, are nonnegative integers. Riccati techniques are used.