The solvability of the boundary value problem \[ -(ru')'+ qu=\lambda f(t,u),\quad 0< t< 1,\quad \lambda u(0)- \beta u'(0)=\gamma u(1)+\delta u'(1)= 0, \] is studied for \(\lambda\geq 0\). Here, \(\alpha\), \(\beta\), \(\gamma\), \(\delta\geq 0\), \(\alpha\delta+ \alpha\gamma+ \beta\gamma> 0\), \(r\in C([0, 1],(0,\infty))\), \(q\in C([0,1], [0,\infty))\), and \(f: (0,1)\times \mathbb{R}\to\mathbb{R}\) is continuous and may be singular at \(t= 0\) and \(t=1\).
To prove the existence of positive solutions to this problem, the authors look for positive fixed-points for the mapping \(T_\lambda\) defined on \(([0,1], [0,\infty))\) by \[ (T_\lambda u)(t)= \lambda \int^1_0 G(t,s) f(s, u(s)) ds, \] where \(G(t,s)\) is Green’s function for the problem considered.