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Solutions to a three-point boundary value problem. (English) Zbl 1222.34022
Authors’ abstract: By using the fixed-point index theory and the Leggett-Williams fixed-point theorem, we study the existence of multiple solutions to the three-point boundary value problem
$u'''(t)+a(t)f(t,u(t),u'(t))=0,\quad 0<t<1,$
$u(0)=u'(0)=0,\quad u'(1)-\alpha u'(\eta)=\lambda,$
where $$\eta\in(0,\frac12]$$, $$\alpha\in[-\frac{1}{2\eta},\frac{1}{\eta})$$ are constants, $$\lambda\in(0,\infty)$$ is a parameter, and $$a, f$$ are given functions. New existence theorems are obtained, which extend and complement some existing results. Examples are also given to illustrate our results.
##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 47N20 Applications of operator theory to differential and integral equations
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##### References:
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