Modified fractional integrals and derivatives in the half-axis and differential equations of fractional order in the space of integrable functions

Fractional integrals and derivatives of positive order, being modifications of the Liouville fractional integrals and derivatives on the half-axis, are introduced. The properties of such modified constructions are investigated in the space of integrable functions. The results obtained are applied to prove the equivalence of the Cauchy-type problem for the nonlinear differential equation of fractional order and of the nonlinear Volterra integral equations. The Banach fixed point theorem is used to prove the existence and uniqueness of a solution of the above Cauchy-type problem. Particular cases are considered and examples are given.