A class of singular nonlinear boundary value problems (Q5915453)

The authors study boundary value problems of the form \[ y''= g (t,y) f(y'),\;0<t<1,\;y'(0)=m\leq 0,\;y(1)=0, \] where \(g(t,y)\in C^1([0,1] \times (0,\infty))\) is positive and nonincreasing in \(y\) for each fixed \(t\); \(\int^1_0 g(t,y)(1-t)dt <\infty\) for each fixed \(y>0\); \(f\in C^1[m_0,m_1]\), with \(m_0<0\), \(m\in(m_0,m_1)\) and \(f(m_0)=0\), \(f(z)<0\) for \(m_0<z<m_1\). The authors provide conditions under which the boundary value problem has a unique positive solution, which then satisfies \(m_0<y'(t)\leq m\). They also obtain information on the qualitative behavior of this solution as \(t\to 1^-\).