Multiple positive solutions for a second-order boundary value problem with integral boundary conditions

In view of the Avery-Peterson fixed point theorem, this paper investigates the existence of three positive solutions for the second-order boundary value problem with integral boundary conditions {u″(t)+h(t)f(t,u(t),u′(t))=0,0<t<1,u(0)−αu′(0)=∫01g1(s)u(s)ds,u(1)+βu′(1)=∫01g2(s)u(s)ds.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left \{ \textstyle\begin{array}{@{}l} u''(t)+h(t)f(t,u(t),u'(t))=0,\quad 0< t< 1, \\ u(0)-\alpha u'(0)=\int_{0}^{1}g_{1}(s)u(s)\,ds, \\ u(1)+\beta u'(1)=\int_{0}^{1}g_{2}(s)u(s)\,ds. \end{array}\displaystyle \right . $$\end{document} The interesting point is that the nonlinear term involves the first-order derivative explicitly.