Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 63, pp. 1-9.

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

Author: Yun Chen (Shandong Normal Univ., Jinan, China)
        Baoqiang Yan (Shandong Normal Univ., Jinan, China)
        Lili Zhang (Shandong Normal Univ., Jinan, China)

Abstract:
  Using a fixed point theorem in cones, this paper shows 
 the existence of positive solutions for the singular
 three-point boundary-value problem
 \begin{gather*}
 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{gather*}
 where $0<\alpha <1$, $0<\eta<1 $, and $f$ may change sign and may
 be singular at $x=0$ and $x'=0$.
 
Submitted September 15, 2006. Published April 25, 2007.
Math Subject Classifications: 34B15, 34B10.
Key Words: Asymptotic formulas; $L^q$-norm; simple pendulum.