## Solvability of a multi-point boundary value problem at resonance.(English)Zbl 1043.34016

The existence of a solution of the following multipoint boundary value problem is studied: $x^{''}(t)=f(t,x(t),x'(t)),\quad 0<t<1,$
$x(0)=0, x(1)=\sum_{i=1}^{k} \xi_i x(\eta_i),$ with $$k \geq 1$$ and $$0<\eta_1<... <\eta_k <1$$. The authors study this problem under the resonance condition $$\sum_{i=1}^{k} \xi_i \eta_i =1.$$ The function $$f(t,x,y)$$ satisfies $$\| f(t,x,y)\| \leq c(x) + d(x) y^2, \, t\in [0,1], \, x,y \in \mathbb{R}$$ and some sign condition. The multidimensional case is also considered.

### MSC:

 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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### References:

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