Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 106, pp. 1-18.

Title: Positive solutions of three-point boundary-value problems 
       for p-Laplacian singular differential equations

Authors: George N. Galanis (Naval Academy, Greece)
         Alex P. Palamides (Univ. of Peloponnese, Greece)

Abstract: 
 In this paper we prove the existence of positive solutions for the
 three-point singular boundary-value problem
 $$
 -[\phi _{p}(u')]'=q(t)f(t,u(t)),\quad 0<t<1
 $$
 subject to
 $$
 u(0)-g(u'(0))=0,\quad u(1)-\beta u(\eta )=0
 $$
 or to
 $$
 u(0)-\alpha u(\eta )=0,\quad u(1)+g(u'(1))=0,
 $$
 where $\phi _{p}$ is the $p$-Laplacian operator, $0<\eta <1$; 
 $0<\alpha ,\beta <1$ are fixed points and $g$ is a monotone 
 continuous function defined on the real line $\mathbb{R}$ with 
 $g(0)=0$ and $ug(u)\geq 0$. Our approach is a combination of 
 Nonlinear Alternative of Leray-Schauder with the properties 
 of the associated vector field at the $(u,u')$ plane.
 More precisely, we show that the solutions of the above 
 boundary-value  problem remains away from the origin for 
 the case where  the nonlinearity is sublinear and so we avoid 
 its singularity at $u=0$.
  
Submitted May 13, 2005. Published October 7, 2005.
Math Subject Classifications: 34B15, 34B18.
Key Words: Three-point singular boundary-value problem; p-Laplacian;
           positive and negative solutions; vector field;
           Nonlinear alternative of Leray-Schauder.