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Zbl 1166.39008
Ma, Ruyun; Ma, Huili
Unbounded perturbations of nonlinear discrete periodic problem at resonance.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 70, No. 7, A, 2602-2613 (2009). ISSN 0362-546X

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.
[Fozi Dannan (Damascus)]
MSC 2000:
*39A12 Discrete version of topics in analysis
39A10 Difference equations
34L30 Nonlinear ordinary differential operators

Keywords: difference equations; eigenvalue; resonance; Leray-Schauder continuation method; nonlinear discrete boundary value problems

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