Integro-differential equations with degenerating in Banach spaces

In this paper integro-differential equations of the special type with Noetherian operator in main part in Banach spaces is investigated. All investigations conduct in space of generalized functions with value in Banach spaces and with support on positive semiaxis. In this paper instrument of fundamental operator- fonctions is application, this instrument make possible reconstruct of generalized solutions in convolution form and on this basis obtain theorem about solvability being investigated problems in set of fonction of finite smoothness. This method of investigations make possible automatically solve problem of agreement input data for existence of classical (smooth) solutions and also obtain formulas for representation of solutions both in spaces of distributions and in spaces of fonctions of finite smooth. Examined in this paper integro-differential equations make possible in most general form investigated mathematical models of theory of oscillations in visco-elastic medium or of theory of electric chain.