Variational formulation of the Cauchy problem for equations with operator coefficients

summary:Several variational principles are suggested, which are equivalent to initialvalue (Cauchy) problems for equations of the first and second order in time coordinate. Their coefficients are linear operators, acting in the space $L_2(l,H)$ of square-integrable mappings of a time interval $l$ into a Hilbert space $H$. In particular, the theory includes some classes of partial differential equations and of integro-differential equations. Some kinds of symmetry in the sense of convolutions are required for the operator coefficients. In the following two papers, the variational principles were employed for the definitions of weak solutions for particular classes of integro-differential equations