Geodesic
Fundamental operator functions of integro-differential operators under spectral or polynomial boundedness
Ufa mathematical journal, Tome 12 (2020) no. 2, pp. 56-71 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We study a Cauchy problem for a degenerate higher order integro-differential equation in Banach spaces. The operator kernel of the integral part of the equation is a linear combination of the operator coefficients of its differential part, which corresponds to the physical meaning of some technological processes. The solution is constructed in the space of generalized functions (distributions) in Banach spaces using the methods of the theory of fundamental operands. The convolutional representation of the original equation leads to a further active use of the convolutional technique and its properties. For the considered equations, the corresponding fundamental operator functions are constructed. By means of this operator, a unique generalized solution to the original Cauchy problem in the class of distributions with a left-bounded support is recovered. The analysis of the resulting generalized solution allows us to study the solvability problem in the classical sense. The fundamental operator function is constructed in terms of the theory of semigroups of operators with kernels. Abstract results are illustrated by examples of initial-boundary value problems from visco-elasticity theory.
Keywords: Banach space, generalized function, fundamental operator-function, integro-differential operator, spectral boundedness, polynomial boundedness.
Mots-clés : distribution 
@article{UFA_2020_12_2_a6,
     author = {M. V. Falaleev},
     title = {Fundamental operator functions of integro-differential operators under spectral or polynomial boundedness},
     journal = {Ufa mathematical journal},
     pages = {56--71},
     year = {2020},
     volume = {12},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2020_12_2_a6/}
}
TY  - JOUR
AU  - M. V. Falaleev
TI  - Fundamental operator functions of integro-differential operators under spectral or polynomial boundedness
JO  - Ufa mathematical journal
PY  - 2020
SP  - 56
EP  - 71
VL  - 12
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/UFA_2020_12_2_a6/
LA  - en
ID  - UFA_2020_12_2_a6
ER  - 
%0 Journal Article
%A M. V. Falaleev
%T Fundamental operator functions of integro-differential operators under spectral or polynomial boundedness
%J Ufa mathematical journal
%D 2020
%P 56-71
%V 12
%N 2
%U http://geodesic.mathdoc.fr/item/UFA_2020_12_2_a6/
%G en
%F UFA_2020_12_2_a6
M. V. Falaleev. Fundamental operator functions of integro-differential operators under spectral or polynomial boundedness. Ufa mathematical journal, Tome 12 (2020) no. 2, pp. 56-71. http://geodesic.mathdoc.fr/item/UFA_2020_12_2_a6/

[1] V. S. Vladimirov, Generalized functions in mathematical physics, Mir Publishers, M., 1979

[2] V. S. Vladimirov, Equations of mathematical physics, Pure Appl. Math., 3, Marcel Dekker, New York, 1971 | MR | MR | Zbl

[3] A.A. Zamyshlyaeva, Higher order linear equations of Sobolev type, Izd. centr YuURGU, Chelyabinsk, 2012 (in Russian)

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

[5] M. V. Falaleev, E. Yu. Grazhdantseva, “Fundamental operator functions of singular differential operators under spectral boundedness conditions”, Differ. Equ., 42:6 (2006), 819–825 | DOI | MR | Zbl

[6] M. V. Falaleev, S. S. Orlov, “Degenerated integro-differential equations of special kind in Banach spaces and its applications”, Vestnik YuURGU. Ser. Matem. Model. Progr., 7:4 (2011), 100–110 (in Russian) | Zbl

[7] M. V. Falaleev, “Linear models in theory of viscoelasticity of Sobolev type”, Vestnik YuURGU. Ser. Matem. Model. Progr., 6:4 (2013), 101–107 (in Russian) | Zbl

[8] N. Sidorov, B. Loginov, A. Sinitsyn, M. Falaleev, Lyapunov-Schmidt Methods in Nonlinaear Analysis and Applications, Kluwer Acad. Publ., Dordrecht, 2002 | MR

[9] G. A. Sviridyuk, V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht, 2003 | MR | Zbl

[10] M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. Ferreira, “Existence and Uniform Decay for a NonLinear Viscoelastic Equation with Strong Damping”, Math. Meth. Appl. Sci., 24 (2001), 1043–1053 | DOI | MR | Zbl