On generation of an analytic in a sector resolving operators family for a distributed order equation

We investigate the unique solvability of the Cauchy problem for a class of differential equations of a distributed order not greater than one with an unbounded operator in a Banach space. The necessary and sufficient conditions for the existence of an analytic in the sector resolving family of operators for the homogeneous equation are obtained. Two versions of the theorem on the unique solvability of the Cauchy problem for the corresponding inhomogeneous equation are proved: with the condition of extra smoothness in spatial variables (the condition of continuity in the graph norm of the unbounded operator) functions in the right-hand side of the equation and its increased smoothness in the time variable (condition of Hölder continuity with respect to time). The results are obtained using the Laplace transform theory and represent the extension to the case of distributed order equations of some results of the analytical theory of operator semigroups and its generalizations to the case of integral equations, fractional differential equations. Abstract results are used in the study of a class of initial boundary value problems for equations with polynomials of an elliptic differential operator with respect to spatial variables.