Positive solutions of differential equations with nonlinear boundary conditions (Q1422542)

The author is interested in the following differential equation \[ u''(t)+f(t,u(t))=0\quad \text{a.e.\;on}\;[0,1],\tag{1} \] with nonlinear boundary conditions \[ u'(0)=0, \qquad u(1)+B(u'(\eta))=0,\tag{2} \] with \(\eta\in [0,1]\). By using the fixed-point index theory, he shows that under weaker restrictions on \(B\) and the growth of \(f\), the BVP (1)-(2) has at least one or multiple positive solutions.