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In this paper, the existence of solutions for a system of nonlinear equations is considered. 2 n nonzero real solutions are obtained by using the critical point theory. Additionally, the Dirichlet boundary value problems of even order difference equations and partial difference equations are investigated.

The dynamical systems considered have scalar state, are multivariate, linear, time-discrete, and time-variable and are described by an initial value problem for a class of evolutionary partial difference equations. The time set is the nonnegative part of the integer lattice in several dimensions. Parts of the asymptotical stability set in the parameter space spanned by the time-variable coefficients are explicitly found. To assess the quality of the sufficient stability criteria, a comparison with the exact stability set is made in an example.

This paper is concerned with a class of neutral partial difference equations. The conditions for the existence of monotone eventually positive solutions are established which improve and extend some of the criteria existing in the literature. Comparison theorems are also derived. Results are obtained on the existence of a monotone eventually positive solution of dual equation.

We develop criteria for the nonexistence of eventually positive (negative) and nondecreasing (nonincreasing) solutions of the partial difference equation ∇ m ∇ n y(m, n) + P(m, n, y(m + k, n + l)) = Q(m, n, y(m + k, n + l)) and ∇ m ∇ n y(m, n) + ∑ i = 1 τ P i (m, n, y(m + k i , n + l i )) = ∑ i = 1 τ Q i (m, n, y(m + k i , n + l i )). Several examples which dwell upon the importance of our results are also included.

This paper is concerned with two partial difference equations of the form A m + 1, n + A m, n + 1 − A m, n + P m, n A m − k, n − l = 0 and A m + 1, n + A m, n + 1 − A m, n + ∑ i = 1 u P i (m, n)f i (A m − k i , n − l i ) = 0 respectively. We obtain sufficient conditions for the oscillation of all solutions of these equations.

Sufficient conditions are derived for the nonexistence of positive solutions of a class of positive nonlinear delay partial difference equations. Results are also obtained for the nonexistence of positive solutions of the dual equation.

In this remark, we shall show a counter example for the main result of the paper [S.T. Liu, Y.Q. Liu, Oscillation theorems for second-order nonlinear partial difference equations, J. Comput. Appl. Math. 132 (2001) 479–482].