A nonlinear Volterra integral equation with a weak singularity (Q2719421)

The nonlinear Volterra equation NEWLINE\[NEWLINE\varphi(x) = \frac{a}{\varGamma(\alpha)}\int_0^x \frac{\varphi^2(t) dt}{(x-t)^{1-\alpha}} + f(x),\quad 0<x<d\leq\infty,\quad \alpha>0,\quad a\neq 0 NEWLINE\]NEWLINE is studied. The author proves the uniqueness of a locally bounded and continuous solution \(\varphi(x)\). The asymptotics of the solution \(\varphi(x)\) is established and the solution \(\varphi(x)\) is represented in the form of a series under the condition of a special power behaviour of \(f(x)\) near zero. Some examples are presented.