On solution of integral equation of Abel-Volterra type

The linear integral equation of Abel-Volterra type $$ \varphi(x)={\frac{a(x)}{\Gamma(\alpha)}}\int^{x}_{0} {\frac{\varphi(t)}{(x-t)^{1-\alpha}}}\ dt+f(x) \quad (0<x<\infty,\ 0<\alpha <1)\tag{*} $$ is investigated. The asymptotic behavior of the solution $\varphi(x)$ as $x\to 0$ is studied, provided that the functions $a(x)$ and $f(x)$ have special power asymptotic expansions near zero. It is shown that in certain cases such an asymptotic solution $\varphi(x)$ of the equation $(*)$ coincides with its explicit solution. Examples are also given.