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Positive solutions for a singular second-order three-point boundary value problem. (English) Zbl 1138.34015
Summary: This paper is concerned with the existence of positive solutions for the singular three-point boundary value problem $$u''(t)+a(t)u'(t)+b(t)u(t)+h(t)f(t,u)=0,\quad 0<t<1\quad u(0)=0,\quad u(1)=\alpha u(\eta),$$ where $h(t)$ is allowed to be singular at $t=0,1$ and $f$ may be singular at $u = 0$. Existence criteria for positive solutions are established by applying the fixed point index theorem under some weaker conditions concerning the first eigenvalue corresponding to the relevant linear operator.

MSC:
 34B18 Positive solutions of nonlinear boundary value problems for ODE 34B10 Nonlocal and multipoint boundary value problems for ODE 34B16 Singular nonlinear boundary value problems for ODE 47H11 Degree theory (nonlinear operators)
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References:
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