Existence of solutions to a discrete fourth order boundary value problem. (English) Zbl 1391.39006
Summary: Criteria are established for the existence of at least two nontrivial solutions to the discrete fourth order boundary value problem
\[
\begin{cases} \Delta^4 u(t-2)-\alpha\Delta^2 u(t-1) +\beta u(t)=f(t,u(t)),\quad t\in [1,N]_{\mathbb Z}, \\ u(-1)=\Delta u(-1)=0,\quad u(N+1)=\Delta^2 u(N)=0,\end{cases}
\]
where \(N\geq 1\) is an integer, \(\alpha\), \(\beta\geq 0\) and \(f:[1,N]_{\mathbb Z}\times \mathbb R\to \mathbb R\) is continuous in the second argument. Applications of the results to a related eigenvalue problem are also presented. The proofs are mainly based on the variational method and the classic mountain pass lemma of A. Ambrosetti and P. H. Rabinowitz [J. Funct. Anal. 14, 349–381 (1973; Zbl 0273.49063)]. Examples are included to illustrate the applicability of the results.
MSC:
| 39A10 | Additive difference equations |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 39A12 | Discrete version of topics in analysis |
Citations:
Zbl 0273.49063References:
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