Fundamental operator-function of an integro-differential operator with derivatives of functionals in Banach spaces

In this paper, we consider a generalized integro-differential operator with derivatives of functionals, which has in its construction an invertible operator in the linear part free of derivatives. The research uses previously obtained results in the field of fundamental operator functions in Banach spaces, as well as the properties of generalized functions, operators, and functionals. For an integro-differential operator with derivatives of functionals in Banach spaces, a fundamental operator-function in the terminology of Jordan sets is obtained and the conditions for the existence of this fundamental operator-function are revealed.