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Positive solutions for singular three-point boundary-value problems with sign changing nonlinearities depending on \(x'\). (English) Zbl 1141.34309

Summary: Using a fixed point theorem in cones, this paper shows the existence of positive solutions for the singular three-point boundary-value problem
\[ \begin{aligned} & x''(t)+a(t)f(t,x(t),x'(t))=0,\quad 0<t<1,\\ & x'(0)=0,\quad x(1)=\alpha x(\eta),\end{aligned} \]
where \(0<\alpha <1\), \(0<\eta<1 \), and \(f\) may change sign and may be singular at \(x=0\) and \(x'=0\).

MSC:

34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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