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Zbl 1188.39008
Ma, Ruyun; Li, Jiemei; Gao, Chenghua
Existence of positive solutions of a discrete elastic beam equation.
(English)
[J] Discrete Dyn. Nat. Soc. 2010, Article ID 582919, 15 p. (2010). ISSN 1026-0226; ISSN 1607-887X/e

Let $T$ be an integer with $T\ge 5$ and let $\Bbb T_2=\{2,3,\dots,T\}$. We consider the existence of positive solutions of the nonlinear boundary value problems of fourth-order difference equations $$\Delta^4u(t-2)-ra(t)f(u(t))=0,\quad t\in \Bbb T_2,\ u(1)=u(T+1)=\Delta^2u(0)=\Delta^2u(T)=0,$$ where $r$ is a constant, $a:\Bbb T_2\to (0,\infty)$, and $f:[0,\infty)\to [0,\infty)$ is continuous. Our approaches are based on the Krein-Rutman theorem and the global bifurcation theorem.
MSC 2000:
*39A12 Discrete version of topics in analysis
34B15 Nonlinear boundary value problems of ODE
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
39A10 Difference equations

Keywords: positive solutions; discrete elastic beam equation; nonlinear boundary value problems; fourth-order difference equations; Krein-Rutman theorem; global bifurcation theorem

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