Kolisnyk R. The Cauchy problem for evolution equations with differential operator of infinite order

The spaces of entire functions are constructed; in such spaces the topological structure is investigated and the properties of basic operations are studied. The necessary and sufficient conditionsare found, under which the differential operator of infinite order which acts in the spaces of basic functions is bounded. The estimations and properties of fundamental solution of the Cauchy problem as an abstract function of the time parameter wich values in spaces of basic functions are investigated. The theorems on a correct solution of the Cauchy problem in the spaces of generalized functions of infinite order of ultra decomposition is proved.