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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.
References:
[1] M. M. Вайнберг: Вариационные методы исследования нелинейных операторов. Москва 1956. Zbl 0995.90522
[2] N. Dunford J. T. Schwartz: Linear operators, Part I. General theory. Interscience Publishers, 1958. MR 1009162
[3] I. Hlaváček: On the existence and uniqueness of solution of the Cauchy problem for a class of linear integro-differential equations. Aplikace Matematiky 16 (1971), 2. MR 0296633
[4] I. Hlaváček: Variational principles for parabolic equations. Aplikace matematiky 14 (1969), 4, 278-297. MR 0255988
[5] J. L. Lions: Equations diflferentielles operationelles et problèmes aux limites. Grundlehren Math. Wiss., Bd. 111, Springer 1961. MR 0153974
[6] I. Hlaváček: On the existence and uniqueness of solution of the Cauchy problem for linear integro-differential equations with operator coefficients. Aplikace matematiky 16 (1971), 1, 64-80 MR 0300158
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