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Necessary and sufficient condition for the existence of solutions to a discrete second-order boundary value problem. (English) Zbl 1242.34035

Summary: This paper is concerned with the existence of solutions for the discrete second-order boundary value problem \(\Delta^2 u(t - 1) + \lambda_1 u(t) + g(\Delta u(t)) = f(t), t \in \{1, 2, \dots, T\}, u(0) = u(T + 1) = 0\), where \(T > 1\) is an integer, \(f : \{1, \dots, T\} \rightarrow \mathbb R, g : \mathbb R \rightarrow \mathbb R\) is bounded and continuous, and \(\lambda_1\) is the first eigenvalue of the eigenvalue problem \(\Delta^2 u(t - 1) + \lambda u(t) = 0, t \in \mathbb T, u(0) = u(T + 1) = 0\).

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
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