On a generalization of the Osgood condition (Q1595131)

The authors prove as main result a generalization of the Osgood uniqueness condition for a nonlinear Volterra integral equation of the second kind. Namely, it concerns a nonlinear equation of the Abel type. The main theorem says in which cases a nonlinear Volterra equation of this kind has trivial and nontrivial solutions for different values of \(\alpha\geq 1\) in the Abel kernel \((x-s)^{\alpha-1}\). In particular, for \(\alpha=1\) we obtain the classical Osgood condition. The existence of trivial or nontrivial solutions depends upon the behavior of the function, which gives the nonlinearity of the equation. As it is mentioned the case \(\alpha\in(1,2)\) is still open to conclude that the Osgood type condition is necessary for the existence of nontrivial solutions of a nonlinear Volterra equation of this type.