Convolutional integro-differential equations in Banach spaces with a Noetherian operator in the main part

An initial-value problem for an integro-differential equation of convolution type with a finite index operator for the higher order derivative in Banach spaces is considered. The equations under consideration model the evolution of the processes with "memory" when the current state of the system is influenced not only by the entire history of observations but also by the factors that have formed it and that remain relevant to the current moment of observation. Solutions are constructed in the class of generalized functions with a left bounded support with the use of the theory of fundamental operator functions of degenerate integro-differential operators in Banach spaces. A fundamental operator function that corresponds to the equation under consideration is constructed. Using this function the generalized solution is restored. The relationship between the generalized solution and the classical solution of the original initial-value problem is studied. Two examples of initial-boundary value problems for the integro-differential equations with partial derivatives are considered.