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Solvability of a second-order multi-point boundary value problem at resonance. (English) Zbl 1168.34310
Summary: Based on the coincidence degree theory of Mawhin, we get a general existence result for the following second-order multi-point boundary value problem at resonance
$x''(t)= f(t,x(t),x'(t))+e(t), \quad t\in(0,1),$
$x(0)= \sum_{i=1}^m \alpha_ix(\xi_i), \qquad x'(1)= \sum_{j=1}^n \beta_jx'(\eta_j),$
where $$f:[0,1]\times\mathbb R^2\to\mathbb R$$ is a Carathéodory function, $$e\in L^1[0,1]$$, $$0<\xi_1<\xi_2<\cdots< \xi_m<1$$, $$\alpha_i\in\mathbb R$$, $$i=1,2,\dots,m$$, $$m\geq 2$$ and $$0<\eta_1<\cdots<\eta_n<1$$, $$\beta_j\in\mathbb R$$, $$j=1,\dots,n$$, $$n\geq 1$$. In this paper, both of the boundary value conditions are responsible for resonance.

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 47N20 Applications of operator theory to differential and integral equations
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##### References:
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