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A multi-point boundary value problem with two critical conditions. (English) Zbl 1121.34023
A coincidence degree theorem due to Mawhin is applied to show the existence of a solution to the multipoint boundary value problem for the nonlinear second-order equation $u''=f(t,u,u') , \qquad t \in (0,1)\,,$ $u'(0)= u'(\eta)\,, \qquad u(1)=\sum^{n}_{i=1}\alpha_{i} u(\eta_{i})\,,$ where $$\,0< \eta \leq 1,\, \sum^{n}_{i=1}\alpha_{i}=\sum^{n}_{i=1}\alpha_{i} \eta_{i} =1\,$$ and $$f$$ satisfies the Caratheodory conditions. Sufficient conditions are given in order to prove the existence of at least one solution. Both boundary conditions are responsible for resonance.

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations
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##### References:
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