The existence problem for a nonlinear Abel equation on the half-line (Q988851)

A nonlinear Volterra integral equation of the form \[ u(x)= \int^x_{-\infty} (x-s)^{\alpha- 1}r(s) g(u(s))\,ds\qquad (-\infty< x,\;0<\alpha) \] is considered. It is shown that if the following conditions hold: {\parindent8mm \begin{itemize}\item[(i)] \(g: [0,+\infty)\to [0,+\infty)\) is continuous, nondecreasing and \(g(0)= 0\), \(g(x)> 0\) for \(x> 0\), \item[(ii)] \(r: (-\infty,+\infty)\to (0,+\infty)\) is continuous, nondecreasing with \(r(-\infty)= 0\) and \(r(x)> 0\) for \(x >-\infty\),\item[(iii)] \(\int^x_{-\infty} r(s)\,ds< +\infty\) if \(0<\alpha< 1\) and \(\int^x_{-\infty} r(s)^{1/\alpha} ds<+\infty\) if \(1\leq\alpha\), \end{itemize}} then the condition: (iv) \(I(\delta)= \int^\delta_0 [{s\over g(s)}]^{1/\alpha}\,{ds\over s}<+\infty\) is necessary and sufficient condition for the existence of a nontrivial solution that is everywhere positive.