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Hlaváček, Ivan
On the existence and uniqueness of solution of the Cauchy problem for linear integro-differential equations with operator coefficients. (English). Aplikace matematiky, vol. 16 (1971), issue 1, pp. 64-80
MSC: 45J05, 47G05 | MR 0300158 | Zbl 0217.15801 | DOI: 10.21136/AM.1971.103326
Summary:
In the theory of neutron fields some problems arise, which may be described by means of integro-differential equations with initial conditions. The aim of the paper is to state a class of problems covering this physical example and to prove the existence, uniqueness and continuous dependence of their solutions on the given data. A variational approach, presented in a previous author's article, is used to establish the definition of generalized (weak) solutions of the Cauchy problem, which is an extension of the concept of generalized solutions of differential equations.
References:
[1] Hlaváček I.: Variational formulation of the Cauchy problem for equations with operator coefficients. Aplikace matematiky 16 (1971) 1, 46 - 63. MR 0283652
[2] Lions J. L.: Equations differentielles operationnelles et problèmes aux limites. Grundlehren Math. Wiss., Bd. 111, Springer 1961. MR 0153974 | Zbl 0098.31101
[3] Ладыженская О. А.: Смешанная загача для гиперболических уравнений. Москва 1953. Zbl 1151.94459
[4] Люстерник Л. А., Соболев В. И.: Элементы функционального анализа. Москва 1951. Zbl 1165.94313
[5] Achieser N. I., Glasmann I. M.: Theorie der linearen Operatoren im Hilbert-Raum. Akademie-Verlag Berlin 1958. Zbl 0082.32702
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