Volume 6, no. 2Pages 40 - 48
Strongly Continuous Operator Semigroups. An Alternative ApproachA.A. Zamyshlyaeva
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. Full text
- Sobolev type equation, strongly continuous semigroups of operators with kernals, approximations of semigroups.
- 1. Hille E., Phillips R.S. Functional Analysis and Semi-Groups. American Mathematical Society, Providence, Rhode Island, 1957.
2. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, Koln, Tokyo, VSP, 2003.
3. Demidenko G.V., Uspenskii S.V. Partial Differential Equations and Systems Not Solvable with Respect to the Highest Order Derivative. N.Y., Basel, Hong Kong, Marcel Dekker, Inc., 2003.
4. Favini A., Yagi A. Degenerate Differential Equations in Banach Spaces. N.Y., Basel, Hong Kong, Marcel Dekker, Inc., 1999.
5. Sidorov N., Loginov B., Sinithyn A., Falaleev M. Lyapunov-Shmidt Methods in Nonlinear Analysis and Applications. Dordrecht, Boston, London, Kluwer Academic Publishers, 2002.
6. Al'shin A. B., Korpusov M.O., Sveshnikov A.G. Blow-up in Nonlinear Sobolev Type Equations. Series in Nonlinear Analisys and Applications, 15, De Gruyter, 2011.
7. Sviridyuk G.A. Linear Sobolev Type Equations and Strongly Continuous Semigroups of the Resolving Operators with Kernels [Lineynye uravneniya tipa Soboleva i sil'no nepreryvnye polugruppy razreshayushchikh operatorov s yadrami]. Doklady akademii nauk, 1994, vol. 337, no. 5, pp. 581-584.
8. Sviridyuk G.A., Zamyshlyaeva A.A. The Phase Spaces of a Class of Linear Higher-order Sobolev Type Equations. Differential Equations, 2006, vol. 42, no. 2. pp. 269-278.