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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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