Using critical point theory, some sufficient conditions are obtained for the existence of nonconstant $T$-periodic solutions of a class of second order self-adjoint difference equations.

Explicit quadratic Liapunov functions that provide necessary and sufficient conditions for the asymptotic stability of the system of linear difference equations $x (t + 1) = A x(t)$ are constructed by transforming the original systems to $y (t + 1) = G y(t)$ , where $G$ is a companion matrix associated with the characteristic polynomial of $A$. A necessary and sufficient condition for all roots of the characteristic polynomial to lie in the unit circle $|z| < 1$ on the complex plane is also derived.

By critical point theory, a new approach is provided to study the existence of periodic and subharmonic solutions of the second order difference equation $$\Delta^2 x_{n-1} +f(n, x_n)\,{=}\,0,$$ where $f\in C({\bf R}\times {\bf R}^m, {\bf R}^m), f(t+M,z)=f(t,z)$ for any $(t, z)\in {\bf R}\times{\bf R}^m$ and $M$ is a positive integer. This is probably the first time critical point theory has been applied to deal with the existence of periodic solutions of difference systems.This project is supported by TRATOYT of China and by the State Education Commission Trans-Century Training Program Foundation for the Talents.

By means of generalized Riccati transformation techniques and generalized exponential functions, some oscillation criteria are given for the nonlinear dynamic equation \[ (p(t)x^{\Delta} (t))^{\Delta}+q(t)(f\circ x^{\sigma})=0 \] on time scales. The results are also applied to linear and nonlinear dynamic equations with damping, and some sufficient conditions are obtained for the oscillation of all solutions.

For h € ℝ and ϕ : ℝ2 → ℝ define Lhϕ: ℝ2 → ℝ by (Lhϕ)(x, y) = ϕ (x + h, y) + ϕ (x — h,y) — ϕ (x,y + h) — ϕ (x,y — h) for all (x,y) € ℝ2. The aim of the paper is to establish the following "stability" theorem concerning the functional equation

if δ > 0, f : ℝ2 → ℝ and

then there exists ɛ > 0 ami ϕ : ℝ2 → ℝ such that (Lhϕ)(x, y ) = 0 for all x, y,h ∊ ℝ and