The Cauchy problem for a semilinear equation of the distributed order

A semilinear equation of distributed order (with the Gerasimov — Caputo derivative) in a Banach space with a bounded operator at the unknown function is considered. Using previously obtained results on the solvability of the Cauchy problem for the corresponding linear inhomogeneous equation of distributed order, the found operator form of its solution, and the contraction mapping theorem, the local unique solvability of the Cauchy problem for the considered semilinear equation is proved. An example of applying the obtained abstract results is given.