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Solvability of a three-point nonlinear boundary-value problem. (English) Zbl 1206.34033
Summary: Using the Leray Schauder nonlinear alternative, we prove the existence of a nontrivial solution for the three-point boundary-value problem
$u''+f(t,u)= 0,\quad 0<t<1$
$u(0)= \alpha u'(0),\quad u(1)=\beta u'(\eta ),$
where $$\eta \in (0,1)$$, $$\alpha ,\beta \in \mathbb{R}$$, $$f\in C([0,1] \times\mathbb{R},\mathbb{R})$$. Some examples are given to illustrate the results obtained.

MSC:
 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 47N20 Applications of operator theory to differential and integral equations
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