On a hyper-singular equation

Abstract

The equation $v=v_0+{\int }_0^t(t-s{)}^{\lambda -1}v(s)ds$ is considered, $\lambda \ne 0,-1,-\mathrm{2...}$ and $v_0$ is a smooth function rapidly decaying with all its derivatives. It is proved that the solution to this equation does exist, is unique and is smoother than the singular function $t^{-\frac{5}{4}}$.