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Existence of positive solutions of a discrete elastic beam equation. (English) Zbl 1188.39008

Let \(T\) be an integer with \(T\geq 5\) and let \(\mathbb T_2=\{2,3,\dots,T\}\). We consider the existence of positive solutions of the nonlinear boundary value problems of fourth-order difference equations
\[ \Delta^4u(t-2)-ra(t)f(u(t))=0,\quad t\in \mathbb T_2,\;u(1)=u(T+1)=\Delta^2u(0)=\Delta^2u(T)=0, \]
where \(r\) is a constant, \(a:\mathbb T_2\to (0,\infty)\), and \(f:[0,\infty)\to [0,\infty)\) is continuous. Our approaches are based on the Krein-Rutman theorem and the global bifurcation theorem.

MSC:

39A12 Discrete version of topics in analysis
34B15 Nonlinear boundary value problems for ordinary differential equations
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
39A10 Additive difference equations
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