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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

${x}^{\text{'}\text{'}}\left(t\right)=f\left(t,x\left(t\right),{x}^{\text{'}}\left(t\right)\right)+e\left(t\right),\phantom{\rule{1.em}{0ex}}t\in \left(0,1\right),$
$x\left(0\right)=\sum _{i=1}^{m}{\alpha }_{i}x\left({\xi }_{i}\right),\phantom{\rule{2.em}{0ex}}{x}^{\text{'}}\left(1\right)=\sum _{j=1}^{n}{\beta }_{j}{x}^{\text{'}}\left({\eta }_{j}\right),$

where $f:\left[0,1\right]×{ℝ}^{2}\to ℝ$ is a Carathéodory function, $e\in {L}^{1}\left[0,1\right]$, $0<{\xi }_{1}<{\xi }_{2}<\cdots <{\xi }_{m}<1$, ${\alpha }_{i}\in ℝ$, $i=1,2,\cdots ,m$, $m\ge 2$ and $0<{\eta }_{1}<\cdots <{\eta }_{n}<1$, ${\beta }_{j}\in ℝ$, $j=1,\cdots ,n$, $n\ge 1$. In this paper, both of the boundary value conditions are responsible for resonance.

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