Cauchy Problem for an abstract Evolution Equation of fractional order

TL;DR

This paper develops a framework for solving fractional order evolution equations in abstract Hilbert spaces by defining operator functions via series expansions, extending previous Laurent series approaches to entire functions with growth conditions.

Contribution

It introduces a new method for defining operator functions using Taylor series, broadening the class of functions applicable to fractional evolution equations in Hilbert spaces.

Findings

Extended the class of functions for operator calculus to include entire functions with growth regularity.

Provided conditions under which the evolution equation's solutions exist in an abstract Hilbert space.

Broadened the applicability of fractional evolution equations by relaxing constraints on the second term.

Abstract

In this paper, we define an operator function as a series of operators corresponding to the Taylor series representing the function of the complex variable. In previous papers, we considered the case when a function has a decomposition in the Laurent series with the infinite principal part and finite regular part. Our central challenge is to improve this result having considered as a regular part an entire function satisfying the special condition of the growth regularity. As an application we consider an opportunity to broaden the conditions imposed upon the second term not containing the time variable of the evolution equation in the abstract Hilbert space.