Subordination Principle for Equations with the Hilfer Fractional Derivative

The subordination principle for differential equations in Banach spaces means that if a linear operator A generates a strongly continuous resolving family of operators for an equation of a certain order, then it also generates a resolving family of operators for an equation of a smaller order. Earlier, such a principle was proved for equations with the Gerasimov–Caputo derivative, including the distributed and discretely distributed cases, and for equations with the Riemann–Liouville derivative. In this paper, the subordination principle with respect to the order of the derivative is proved for equations with Hilfer fractional derivatives independently of the types of these derivatives. Sufficient conditions for the validity of the reverse subordination principle are obtained. In addition, the subordination principle with respect to the type of Hilfer derivatives is proved for equations whose orders coincide. The abstract results are used in the study of initial value problems in the space of uniformly continuous and bounded functions on the real line for equations with a differential or difference operator A with respect to the spatial variables in order to prove the existence and uniqueness of their solution and to obtain the form of the solution.