A class of distributed order semilinear equations in Banach spaces

The Cauchy problem is studied for a class of semilinear equations which are resolved with respect to the distributed Gerasimov — Caputo derivative in Banach spaces with a linear part generating a resolving family of operators. Using previously obtained results on the solvability of the Cauchy problem for the corresponding linear inhomogeneous equation, the found operator form of its solution, and the contraction mapping theorem, under the improved smoothness condition with respect to spatial variables for the nonlinear operator in the equation the local unique solvability of the Cauchy problem for the considered semilinear equation is proved. The obtained result is applied to the study of a class of initial-boundary value problems for semilinear partial differential equations.