Geodesic
Holomorphic degenerate groups of operators in quasi-Banach spaces
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 7 (2015) no. 1, pp. 20-27 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Probably, Sobolev type equations, i.e. unsolved with respect to the highest derivative, first appeared in the late nineteenth century. Due to the fact that the interest to the Sobolev type equations recently significantly increased, the need arose for their consideration in quasi-Banach spaces. Specifically, this study aimed at understanding non-classical models of mathematical physics in quasi-Banach spaces. The theory of holomorphic degenerate groups of operators, developed in Banach spaces and Frechet spaces is transferred to quasi-Banach spaces. Abstract results are illustrated by specific examples. The article besides the introduction and the references contains three parts. The first part provides the necessary information regarding the theory of relatively $p$-bounded operators in quasi-Banach spaces. The second one represents the construction of the holomorphic group of solving operators. The third part contains the sufficient conditions for pair of operators to generate group of solving operators.
Keywords: degenerate groups of operators; quasi-Banach spaces; Sobolev type equations. 
@article{VYURM_2015_7_1_a2,
     author = {A. V. Keller and J. K. Al-Delfi},
     title = {Holomorphic degenerate groups of operators in {quasi-Banach} spaces},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matematika, mehanika, fizika},
     pages = {20--27},
     year = {2015},
     volume = {7},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VYURM_2015_7_1_a2/}
}
TY  - JOUR
AU  - A. V. Keller
AU  - J. K. Al-Delfi
TI  - Holomorphic degenerate groups of operators in quasi-Banach spaces
JO  - Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika
PY  - 2015
SP  - 20
EP  - 27
VL  - 7
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/VYURM_2015_7_1_a2/
LA  - ru
ID  - VYURM_2015_7_1_a2
ER  - 
%0 Journal Article
%A A. V. Keller
%A J. K. Al-Delfi
%T Holomorphic degenerate groups of operators in quasi-Banach spaces
%J Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika
%D 2015
%P 20-27
%V 7
%N 1
%U http://geodesic.mathdoc.fr/item/VYURM_2015_7_1_a2/
%G ru
%F VYURM_2015_7_1_a2
A. V. Keller; J. K. Al-Delfi. Holomorphic degenerate groups of operators in quasi-Banach spaces. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 7 (2015) no. 1, pp. 20-27. http://geodesic.mathdoc.fr/item/VYURM_2015_7_1_a2/

[1] Sviridyuk G. A., “On the general theory of operator semigroups”, Russian Mathematical Surveys, 49:4 (1994), 45–74 | DOI

[2] R. E. Showalter, “The Sobolev type equations. I; II”, Appl. Anal., 5:1 (1975), 15–22 ; 2, 81–99 | DOI

[3] G. A. Sviridyuk, V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht–Boston, 2003, 216 pp.

[4] Shestakov A. L., Keller A. V., Nazarova E. I., “Numerical Solution of the Optimal Measurement Problem”, Automation and Remote Control, 73:1 (2012), 97–104 | DOI

[5] Manakova N. A., Dylkov A. G., “Optimal Control of Solutions of Initial-Finish Problem for the Linear Sobolev Type Equations”, Bulletin of the South Ural State University. Series “Mathematical Modelling, Programming Computer Software”, 17(234):8 (2011), 113–114 (in Russ.)

[6] Sviridyuk G. A., Keller A. V., “Invariant spaces and dichotomies of solutions of a class of linear equations of Sobolev type”, Russian Math. (Iz. VUZ), 41:5 (1997), 57–65

[7] Sagadeeva M. A., Dichotomies of the Solutions for the Linear Sobolev Type Equations, Publishing center of SUSU, Chelyabinsk, 2012, 139 pp. (in Russ.)

[8] Zamyshlyaeva A. A., Linear Sobolev Type Equations of Hihg Order, Publishing center of SUSU, Chelyabinsk, 2012, 88 pp. (in Russ.)

[9] Fedorov V. E., “Holomorphic Solution Semigroups for Sobolev Type Equations in Locally Convex Spaces”, Sbornik: Mathematics, 195:8 (2004), 1205–1234 | DOI | DOI

[10] G. V. Demidenko, S. V. Uspenskii, Partial differential equations and systems not solvable with respect to the highest-order derivative, Marcel Dekker, Inc., New York–Basel–Hong Kong, 2003, 239 pp.

[11] Sveshnikov A. G., Al'shin A. B., Korpusov M. O., Pletner Yu. D., Linear and Nonlinear Equation of Sobolev Type, FizMatLit Publ., M., 2007, 736 pp. (in Russ.)

[12] N. Sidorov, B. Loginov, A. Sinithyn, M. Falaleev, Lyapunov–Shmidt Methods in Nonlinear Analysis and Applications, Kluwer Academic Publishers, Dordrecht–Boston–London, 2002, 548 pp.

[13] Bergh J., Lofstrom J., Interpolation Spaces. An Introduction, Springer-Verlag, Berlin–Heidelberg–New York, 1976, 207 pp.

[14] Al-Delfi J. K., “Quasi-Sobolev Spaces $l_p^m$”, Bulletin of South Ural State University. Series of “Mathematics. Mechanics. Physics”, 5:1 (2013), 107–109 (in Russ.)

[15] Sviridyuk G. A., Al-Delfi J. K., “The Laplace Quasi-Operator in the Quasi-Banach Spaces”, Differential Equations. Function Spaces. Approximation Theory, Abstracts of International Conference Dedicated to the 105th Anniversary of the Birthday of S. L. Sobolev (Novosibirsk, 2013), 247 (in Russ.)

[16] Al-Delfi J. K., “The Laplace' Quasi-operator in Quasi-Sobolev spaces”, Vestnik Samarskogo gosudarstvennogo tekhnicheskogo universiteta. Seriya: Fiziko-matematicheskie nauki, 2013, no. 2(31), 13–16 (in Russ.) | DOI

[17] Sviridyuk G. A., Zagrebina S. A., “The Showalter–Sidorov problem as a Phenomena of the Sobolev type Equations”, Bulletin of Irkutsk State University. Series: “Mathematics”, 3:1 (2010), 104–125 (in Russ.)

[18] Sviridyuk G. A., Zagrebina S. A., “Nonclassical Models of Mathematical Physics”, Bulletin of the South Ural State University. Series “Mathematical Modelling, Programming Computer Software”, 40 (299) (2012), 7–18 (in Russ.)