## 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

### Keywords:

coincidence degree
Full Text:

### References:

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