On the existence and uniqueness of solution of the Cauchy problem for a class of linear integro-differential equations

summary:Some problems in the theory of viscoelasticity may be described by means of integro-differential equations. In the paper a class of initial-value problems is considered which includes these physical examples, covering also their analogues - equations of the second order in time coordinate. The theory is restricted to the equations only, possessing in the same term both the highest spatial and the highest derivatives. The weak solution is defined on the base of variational principles, introduced in a previous article, and its existence, uniqueness and continuous dependence on the given data is proved, using the theory of integral Volterra's equations in Banach spaces