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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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