We introduce a class of pairs of operators defining a linear homogeneous degenerate evolution fractional differential equation in a Banach space. Reflexive Banach spaces are represented as the direct sums of the phase space of the equation and the kernel of the operator at the fractional derivative. In a sector of the complex plane containing the positive half-axis, we construct an analytic family of resolving operators that degenerate only on the kernel. The results are used in the study of the solvability of initial-boundary value problems for partial differential equations containing fractional time-derivatives and polynomials in the Laplace operator with respect to the spatial variable.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. N. Gerasimov, “A generalization of linear laws of deformation and its application to problems of internal friction” [in Russian], Akad. Nauk SSSR. Prikl. Mat. Mekh. 12, 251–260 (1948).
M. Caputo, “Linear model of dissipation whose Q is almost frequency independent. II,” Geophys. J. Int. 13, No. 5, 529–539 (1967).
E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces, Eindhoven Univ. Press, Eindhovern (2001).
J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel (1993).
G. A. Sviridyuk and V. E. Fedorov, “On the identities of analytic semigroups of operators with kernels,” Sib. Mat. J. 39, No. 3, 522–533 (1998).
G. A. Sviridyuk and V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht (2003).
V. E. Fedorov, “Holomorphic resolving semigroups of Sobolev-type equations in locally convex spaces,” Sb. Math. 195, No. 8, 1205–1234 (2004).
V. E. Fedorov and A. Debbouche, “A class of degenerate fractional evolution systems in Banach spaces,” Differ. Equ. 49, No. 12, 1569–1576 (2013).
V. E. Fedorov and D. M. Gordievskikh, “Resolving operators of degenerate evolution equations with fractional derivative with respect to time,” Russ. Math. 59, No. 1, 60–70 (2015).
V. E. Fedorov, D. M. Gordievskikh, and M. V. Plekhanova, “Equations in Banach spaces with a degenerate operator under a fractional derivative” Differ. Equ. 51, No. 10, 1360–1368 (2015).
M. Kostić, “Abstract time-fractional equations: existence and growth of solutions,” Fract. Calc. Appl. Anal. 14, No. 2, 301–316 (2011).
I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, San Diego, CA (1999).
M. V. Plekhanova and V. E. Fedorov, “An optimal control problem for a class of degenerate equations” [in Russian], Izv. Akad. Nauk Teor. Sist. Upr. No. 5, 40-44 (2004).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sibirskii Zhurnal Chistoi i Prikladnoi Matematiki 16, No. 2, 2016, pp. 93-107.
Rights and permissions
About this article
Cite this article
Fedorov, V.E., Romanova, E.A. & Debbouche, A. Analytic in a Sector Resolving Families of Operators for Degenerate Evolution Fractional Equations. J Math Sci 228, 380–394 (2018). https://doi.org/10.1007/s10958-017-3629-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-017-3629-4