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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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