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Existence and multiplicity of positive solutions of a nonlinear discrete fourth-order boundary value problem. (English) Zbl 1243.39007

Summary: We show the existence and multiplicity of positive solutions of the nonlinear discrete fourth-order boundary value problem \(\Delta^4 u(t - 2) = \lambda h(t)f(u(t)), ~t \in \mathbb T_2, ~u(1) = u(T + 1) = \Delta^2 u(0) = \Delta^2 u(T) = 0\), where \(\lambda > 0, ~h : \mathbb T_2 \rightarrow (0, \infty)\) is continuous, and \(f : \mathbb R \rightarrow [0, \infty)\) is continuous, \(T > 4, ~T_2 = \{2, 3, \dots, T\}\). The main tool is the Dancer’s global bifurcation theorem.

MSC:

39A12 Discrete version of topics in analysis
39A22 Growth, boundedness, comparison of solutions to difference equations
34B15 Nonlinear boundary value problems for ordinary differential equations
39A28 Bifurcation theory for difference equations
34C23 Bifurcation theory for ordinary differential equations
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References:

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