Existence of positive solutions for singular four-point boundary value problem with a p-Laplacian

In this paper we deal with the four-point singular boundary value problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left\{ \begin{gathered} (\phi _p (u'(t)))' + q(t)f(t,u(t),u'(t),u'(t)) = 0,t \in (0,1), \hfill \\ u'(0) - \alpha u(\xi ) = 0,u'(1) + \beta u(\eta ) = 0, \hfill \\ \end{gathered} \right. $$\end{document} where φp(s) = |s|p−2s, p > 1, 0 < ξ < η < 1, α, β > 0, q ∈ C[0, 1], q(t) > 0, t ∈ (0, 1), and f ∈ C([0,1] × (0, +∞) × ℔, (0, + ∞)) may be singular at u = 0. By using the well-known theory of the Leray-Schauder degree, sufficient conditions are given for the existence of positive solutions.