Positive solutions for singular m-point boundary value problems with sign changing nonlinearities depending on $x'$

Ya Ma, Shandong Normal University, Jinan, P. R. China
Baoqiang Yan, Shandong Normal University, Jinan, P. R. China

E. J. Qualitative Theory of Diff. Equ., No. 43. (2009), pp. 1-14.

Abstract: Using the theory of fixed point theorem in cone, this paper presents the existence of positive solutions for the singular $m$-point boundary value problem
$$
\left\{\begin{array}{ll}
x''(t)+a(t)f(t,x(t),x'(t))=0,0<t<1,\\
x'(0)=0,\ \ x(1)=\dis \sum_{i=1}^{m-2}\alpha_{i}x(\xi_{i}),
\end{array}\right.
$$
where $0<\xi_{1}<\xi_{2}<\cdots<\xi_{m-2}<1, \alpha_{i}\in [0,1)$, $i = 1, 2, \cdots m-2$ , with $0<\sum_{i=1}^{m-2}\alpha_{i}<1$ and $f$ may change sign and may be singular at $x=0$ and $x'=0$.

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