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A note on multi-point boundary value problems. (English) Zbl 1127.34006

This work is concerned with solvability of the multipoint boundary value problem

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

where $0<{\eta }_{j},{\xi }_{j}<1$, ${\eta }_{j},{\xi }_{j}\in ℝ$, $j=1,2,\cdots ,n$, $n\ge 2$, $e\in {L}^{1}\left[0,1\right]$, and $f:\left[0,1\right]×{ℝ}^{2}\to ℝ$ is a Carathéodory function. The additional assumptions

$\sum _{j=1}^{n}{\alpha }_{j}=1=\sum _{j=1}^{n}{\beta }_{j},\phantom{\rule{1.em}{0ex}}\sum _{j=1}^{n}{\alpha }_{j}{\xi }_{j}=0=\sum _{j=1}^{n}{\beta }_{j}{\eta }_{j}$

are “critical”, that is, responsible for resonance.

This note complements the result by N. Kosmatov [Nonlinear Anal., Theory Methods Appl. 65, 622–633 (2006; Zbl 1121.34023)] and extends the result by the first author [Appl. Math. Comput. 143, 275–299 (2003; Zbl 1071.34014)]. The existence result follows from the celebrated coincidence degree theorem due to J. Mawhin [Topological degree methods in nonlinear boundary value problems (Regional Conference Series in Mathematics, No. 40, Providence, R.I., The American Mathematical Society) (1979; Zbl 0414.34025)].

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ODE 34B15 Nonlinear boundary value problems for ODE 47H11 Degree theory (nonlinear operators)