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Positive solutions to a singular third order nonlocal boundary value problem. (English) Zbl 1168.34317

Summary: The existence of a positive solution is shown for the third-order nonlocal boundary value problem
\[ \begin{aligned} y'''&= f(x,y), \quad 0<x\leq 1,\\ y(0)&= y'(p)= \int_q^1 w(x)y''(x)\,dx=0, \end{aligned} \]
where \(\frac12<p<q<1\) are fixed, and where \(f(x,y)\) is singular at \(x=0\), \(y=0\), and possibly at \(y=\infty\). The method involves a fixed point theorem for operators that are describing with respect to a cone.

MSC:

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