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Positive solutions for nonlinear \(n\)th-order singular nonlocal boundary value problems. (English) Zbl 1148.34015

Summary: We study the existence and multiplicity of positive solutions for a class of \(n\)th-order singular nonlocal boundary value problems
\[ u^{(n)}(t)+a(t)f(t,u) = 0,\quad t\in (0,1),\quad u(0) = 0,\quad u'(0) = 0,\dots,u^{(n-2)}(0) =0,\quad \alpha u(\eta) =u(1), \]
where \(0<\eta <1\), \(0<\alpha\eta^{n-1}<1\). The singularity may appear at \(t=0\) and/or \(t = 1\). The Krasnosel’skii-Guo theorem on cone expansion and compression is used in this study. The main results improve and generalize the existing results.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
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References:

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