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Fractional integrals and derivatives on the half-axis and differential equations of fractional order in weighted spaces of continuous functions. (Russian) Zbl 1260.34146
Summary: The properties of fractional integrals and derivatives of any positive order on the half-axis are investigated in weighted spaces of continuous 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 existence and uniqueness of the solution of the above Cauchy-type problem is proved using the Banach fixed point theorem. The corresponding assertions for the Cauchy problem for the ordinary differential equation are presented.
MSC:
34K37Functional-differential equations with fractional derivatives
26A33Fractional derivatives and integrals (real functions)