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Existence of solutions for a three-point boundary value problem at resonance. (English) Zbl 1104.34007
The author studies the existence of solutions to the second-order three-point boundary value problem $$u''(t)=f(t, u(t)),\quad u(0)=\varepsilon u'(0), \quad u(1)=\alpha u'(\eta),$$ where $f:[0,1]\times \bbfR\rightarrow \bbfR$ is continuous, $\varepsilon\in [0,\alpha)$, $\alpha\in (0,\infty)$ and $\eta\in (0,1)$ are given constants such that $\alpha(\eta+\varepsilon)=1+\varepsilon$. The proof of the main result is based upon the connectivity properties of parameterized families of compact vector fields. For related work, see {\it R. Ma} [Nonlinear Anal., Theory Methods Appl. 53 A, No. 6, 777--789 (2003; Zbl 1037.34011)].

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ODE 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces 47H11 Degree theory (nonlinear operators)
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##### References:
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