Strongly Continuous Operator Semigroups. An Alternative Approach

Inheriting and continuing the tradition, dating back to the Hill-Iosida-Feller-Phillips-Miyadera theorem, the new way of construction of the approximations for strongly continuous operator semigroups with kernels is suggested in this paper in the framework of the Sobolev type equations theory, which experiences an epoch of blossoming. We introduce the concept of relatively radial operator, containing condition in the form of estimates for the derivatives of the relative resolvent, the existence of $C_0$-semigroup on some subspace of the original space is shown, the sufficient conditions of its coincidence with the whole space are given. The results are very useful in numerical study of different nonclassical mathematical models considered in the framework of the theory of the first order Sobolev type equations, and also to spread the ideas and methods to the higher order Sobolev type equations.

Keywords

Sobolev type equation, strongly continuous semigroups of operators with kernals, approximations of semigroups.

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