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Multi-point boundary value problems on time scales at resonance. (English) Zbl 1107.34008

Summary: We study the nonlinear second-order differential equation on a time scale \[ u^{\Delta\nabla}(t)=f\bigl(t,u(t),u^\Delta(t)\bigr)+e(t),\quad t \in(0,1)\cap \mathbb{T}, \] subject to the multipoint boundary conditions \[ u^\Delta(0)=0,\quad u(1)=\sum^m_{i=1}\alpha_iu(\eta_i)\quad \text{and}\quad u(0)=0,\quad u(1)= \sum^m_{i=1}\alpha_iu(\eta_i), \] where \(\mathbb{T}\) is a time scale such that \(0\), \(1\in\mathbb{T}\), \(\eta_i\in(0,1) \cap\mathbb{T}\), \(i=1,\dots,m\), and \(f\) is a continuous function satisfying a Carathéodory-type growth condition and \(e\) is a Lebesgue integrable function. Our existence results are obtained by applying a coincidence degree theorem due to Mawhin.

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
39A12 Discrete version of topics in analysis
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