Solutions of a nonlinear fourth order periodic boundary value problem for difference equations. (English) Zbl 1264.39002
Summary: We consider the fourth order periodic boundary value problem
\[
\begin{aligned}&\Delta^4u(t-2)-\Delta (p(t-1)\Delta u(t-1))+q(t)u(t)=f(t,u(t)),\quad t\in [1,N]_\mathbb Z, \\ &\Delta^iu(-1)= \Delta^iu(N-1),\quad i=0,1,2,3,\end{aligned}
\]
where \(N\geq 1\) is an integer, \(p\in C([0,N]_\mathbb Z,\mathbb R)\), \(q\in C([1,N]_\mathbb Z,\mathbb R)\), and \(f\in C([1,N]_\mathbb Z\times\mathbb R,\mathbb R)\). We obtain sufficient conditions for the existence of one and two solutions of the problem. The analysis is based mainly on the variational method and critical point theory.
MSC:
39A10 | Additive difference equations |
39A12 | Discrete version of topics in analysis |
34B15 | Nonlinear boundary value problems for ordinary differential equations |