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Unbounded perturbations of nonlinear discrete periodic problem at resonance. (English) Zbl 1166.39008
The authors study the existence of solutions of nonlinear discrete boundary value problems $$\cases\Delta^2u(t-1)+\lambda_k u(t)+g(t,u(t))=h(t),\\ u(0)=u(T),\ u(1)=u (T+1),\endcases\tag$*$ $$ where $\Bbb T:=[1,\dots,T]$, $h:\Bbb T\to\Bbb R$, $\lambda_k$ is the $k$-th eigenvalue of the linear problem $$ \cases \Delta^2u(t-1)+\lambda u(t)=0,\\ u(0)=u(T),\ u(1)=u (T+1),\endcases\tag$**$ $$ $g:\Bbb N\times\Bbb R\to\Bbb R$ satisfies some asymptotic nonuniform resonance conditions, and $g(t,u)u\geq 0$ for $u\in \Bbb R$. The eigenvalues of the linear problem ($**$) are studied in detail. Some examples are considered.

39A12Discrete version of topics in analysis
39A10Additive difference equations
34L30Nonlinear ordinary differential operators
Full Text: DOI
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