On solvability of some classes of equations with Hilfer derivative in Banach spaces

The issues of unique solvability of a Cauchy-type problem for an evolutionary equation in a Banach space with a Hilfer fractional derivative and with a bounded operator are investigated. The existence of a unique solution to the problem is proved, the solution is presented using Mittag-Leffler operator functions. The abstract result is used when considering a class of initial boundary value problems for partial differential equations