On Generation of Family of Resolving Operators for a Distributed Order Equation Analytic in Sector

The questions of the existence and uniqueness solution to Cauchy problem for an equation in a Banach space distributed order at most one are investigated. Necessary sufficient conditions resolving family operators this analytic sector obtained. An explicit form these is found. Two versions theorem on unique solvability corresponding inhomogeneous obtained: with condition increased smoothness spatial variables (the continuity norm graph generator family) time variable Hölder condition). Abstract results obtained using theory Laplace transform generalization some from operator semigroups its extensions case integral equations fractional differential equations. used study class initial boundary value problems polynomials elliptic respect variables.