Linear equations with discretely distributed fractional derivative in Banach spaces

We study the unique solvability of linear equations in Banach spaces with discretely distributed Gerasimov–Caputo fractional derivative in terms of analytic resolving families of operators. Necessary and sufficient conditions for the existence of such a family of operators are obtained in terms of the resolvent of a closed operator from the right-hand side of the equation, and the properties of this family are studied. These results are used to prove the existence of a unique solution to the Cauchy problem for a linear equation of the corresponding class with inhomogeneity which is either continuous in the norm of the graph of the operator from the right-hand side of the equation or Hölderian. Based on the abstract results obtained, we investigate the unique solvability of initial–boundary value problems for a class of equations with discretely distributed fractional time derivative and with polynomials in an elliptic self-adjoint differential operator with respect to spatial variables.