Initial value problems for some classes of linear evolution equations with several fractional derivatives

The problems of unique solvability of initial problems for linear inhomogeneous equations of a general form with several Gerasimov–Caputo fractional derivatives in Banach spaces are investigated. The Cauchy problem is considered for an equation solved with respect to the highest fractional derivative containing bounded operators at the lowest derivatives. The solution is presented with the use of Dunford–Taylor type integrals. The obtained result allowed us to study an initial problem for a linear inhomogeneous equation with a degenerate operator at the highest fractional derivative, provided that, with respect to this operator, the operator at the second largest derivative is 0-bounded. Abstract results are applied to the study of a class of initial-boundary value problems for equations with several Gerasimov–Caputo time derivatives and with polynomials with respect to a self-adjoint elliptic differential operator in space variables.