This paper is concerned with the delay partial difference equation (1) $A_{m+1,n}+A_{m,n+1}-A_{m,n}+{\Sigma }_{i=1}^u{p}_i{A}_{m-k_i,n-l_i}=0$ where $p_i$ are real numbers, $k_i$ and $l_i$ are nonnegative integers, u is a positive integer. Sufficient and necessary conditions for all solutions of (1) to be oscillatory are obtained.

In this paper we study two classes of delay partial difference equations with constant coefficients. Explicit necessary and sufficient conditions for the oscillation of the solutions of these equations are obtained.

In this paper the authors give necessary and sufficient conditions for the oscillation of solutions of nonlinear delay difference equations of Emden– Fowler type in the form ${\Delta }^2{y}_{n-1}+q_n{y}_{\sigma \left(n\right)}^{\gamma }=g_n$, where $\gamma$ is a quotient of odd positive integers, in the superlinear case $(\gamma >1)$ and in the sublinear case $(\gamma <1)$.

We present necessary conditions for linear noncooperative N-player delta dynamic games on an arbitrary time scale. Necessary conditions for an open-loop Nash-equilibrium and for a memoryless perfect state Nash-equilibrium are proved.

In the paper, conditions are obtained, in terms of coefficient functions, which are necessary as well as sufficient for non-oscillation/oscillation of all solutions of self-adjoint linear homogeneous equations of the form $$\Delta \left(p_{n-1}\Delta y_{n-1}\right)+q{y}_n=0,n\ge 1,$$ where $q$ is a constant. Sufficient conditions, in terms of coefficient functions, are obtained for non-oscillation of all solutions of nonlinear non-homogeneous equations of the type $$\Delta \left(p_{n-1}\Delta y_{n-1}\right)+q_{n}g\left(y_n\right)=f_{n-1},n\ge 1,$$ where, unlike earlier works, $f_n\ge 0$ or $\le 0$ (but $\neg \equiv 0)$ for large $n$. Further, these results are used to obtain...

This paper aims at discussing asymptotic behaviour of nonoscillatory solutions of the forced fractional difference equations of the form $$\begin{array}{cc}\hfill {\Delta }^{\gamma }u\left(\kappa \right)& +\Theta [\kappa +\gamma ,w(\kappa +\gamma \left)\right]\hfill \\ \hfill =& \Phi (\kappa +\gamma )+{\rm Y}(\kappa +\gamma )w^{\nu }(\kappa +\gamma )+\Psi [\kappa +\gamma ,w(\kappa +\gamma )],\kappa \in {\mathbb{N}}_{1-\gamma },\hfill \\ \hfill u_0=& c_0,\hfill \end{array}$$ where ${\mathbb{N}}_{1-\gamma }=\{1-\gamma ,2-\gamma ,3-\gamma ,\cdots \}$, $0<\gamma \le 1$, ${\Delta }^{\gamma }$ is a Caputo-like fractional difference operator. Three cases are investigated by using some salient features of discrete fractional calculus and mathematical inequalities. Examples are presented to illustrate the validity of the theoretical results.