On a hyper-singular equation

he equation v = v 0 + ∫ t 0 ( t − s ) λ − 1 v ( s ) d s is considered, λ ≠ 0 , − 1 , − 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 − 5 4 .