# Existence Results for a Class of Fourth Order Periodic Boundary Value Problems of Difference Equations

## Abstract

It is well known that difference equations occur in numerous settings and forms, both in mathematics and in its applications to statistics, computing, electrical circuit analysis, dynamical systems, economics, biology, and other fields. In recent years, many researchers have paid a lot of attention to boundary value problems (BVPs) of difference equations with various boundary conditions. In this talk, we focus on the discrete fourth order periodic BVP

\begin{align} \Delta^4 u(t - 2) - \Delta(p(t - 1) \Delta u(t - 1)) + q(t) u(t) & = f(t, u(t)), \quad t \in [1, N]_Z \\ \Delta^i u(-1) & = \Delta^i u(N - 1), \quad i = 0, 1, 2, 3 \end{align}

where $$N \ge 1$$ is an integer, $$[1, N]_z = [1, \ldots, N]$$, $$p \in C([0, N]_Z, R)$$, $$q \in C([1, N]_Z, R)$$, $$f \in C([1, N]_Z \times R, R)$$. Sufficient conditions are obtained for the existence of one and multiple solutions of the problem. Our analysis is mainly based on the variational method and critical point theory. Examples are included to illustrate the results.

This talk is based on some recent joint work with Professors J. R. Graef and M. Wang.