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Positive solutions of a nonlinear \(n\)th order boundary value problem with nonlocal conditions. (English) Zbl 1074.34022
Summary: We discuss the existence of positive solutions of a nonlinear \(n\)th-order boundary value problem \[ u^{(n)}+ a(t) f(u)= 0,\quad t\in (0,1), \]
\[ u(0)= 0,\quad u'(0)= 0,\;u'(0)= 0,\dots, u^{(n-2)}(0)= 0,\quad\alpha u(\eta)= u(1), \] with \(0< \eta< 1\), \(0< \alpha\eta^{n-1}< 1\). In particular, we establish the existence of at least one positive solution if \(f\) is either superlinear or sublinear by applying the fixed-point theorem in cones due to Krasnoselskij and Guo.

MSC:
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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