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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.

34B18Positive solutions of nonlinear boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE
Full Text: DOI
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