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Zbl 1193.39002
Xu, Youji; Gao, Chenghua; Ma, Ruyun
Solvability of a nonlinear fourth-order discrete problem at resonance.
(English)
[J] Appl. Math. Comput. 216, No. 2, 662-670 (2010). ISSN 0096-3003

The authors consider the nonlinear discrete boundary value problem: $$\Delta^4 u(t- 2)= \lambda_1 u(t0= f(t,u(t))+ \tau\varphi(t)+\overline h(t),\quad t\in\Pi_2,$$ $$u(1)= u(T+ 1)= \Delta^2 u(0)= \Delta^2 u(T)= 0,$$ where $T$ is an integer with $T\ge 5$; $\Pi_2= \{2,3,\dots, T\}$; $\lambda_1$ is the first eigenvalue of the associated linear eigenvalue problem; $\varphi(0)$ is the corresponding eigenfunction; $f:\Pi_2\times\bbfR\to\bbfR$ is continuous and $|f(t,s)\le A|s|^\alpha_B$, $t\in\Pi_2$, $s\in\bbfR$ for some $0\le\alpha< 1$ and $A,B\in[0,+\infty)$; $\overline h: \Pi_2\to \bbfR$ with $\sum^T_{s= 2}\overline h(t)\varphi(t)= 0$. The existence of the solution of the above problem is shown.
[Stefan Balint (Timişoara)]
MSC 2000:
*39A12 Discrete version of topics in analysis

Keywords: Krein-Rutman theorem; connectivity of solution set; continuum; fourth-order difference equations; eigenvalue

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