Subordination Principle for Equations with Proportional Distributed Gerasimov–Caputo Derivatives

The Cauchy problem for linear equations in Banach spaces with a distributed order derivative of Gerasimov–Caputo is considered. The equation contains a linear closed densely defined operator $$A$$ at the unknown function. It is proved that in the case of certain proportionality of two distributed order equations, the existence of resolving family of operators for a lower order equation follows from the existence of such family for higher order one. The subordination formula and the analyticity of surordinated family are proved. The inversion for the subordination formula in the case of analytic resolving family of a lower order equation and under some additional restrictions are obtained. Abstract results are used in the study of initial boundary value problem to a class of partial differential equations with a discretely distributed Gerasimov–Caputo derivative in time.