Fundamental operator functions of integro-differential operators under spectral or polynomial boundedness

We study a Cauchy problem for a degenerate higher order integro-differential equation in Banach spaces. The operator kernel of the integral part of the equation is a linear combination of the operator coefficients of its differential part, which corresponds to the physical meaning of some technological processes. The solution is constructed in the space of generalized functions (distributions) in Banach spaces using the methods of the theory of fundamental operands. The convolutional representation of the original equation leads to a further active use of the convolutional technique and its properties. For the considered equations, the corresponding fundamental operator functions are constructed. By means of this operator, a unique generalized solution to the original Cauchy problem in the class of distributions with a left-bounded support is recovered. The analysis of the resulting generalized solution allows us to study the solvability problem in the classical sense. The fundamental operator function is constructed in terms of the theory of semigroups of operators with kernels. Abstract results are illustrated by examples of initial-boundary value problems from visco-elasticity theory.