Abstract
We consider resolving operators of a fractional linear differential equation in a Banach space with a degenerate operator under the derivative. Under the assumption of relative p-boundedness of a pair of operators in this equation, we find the form of resolving operators and study their properties. It is shown that solution trajectories to the equation fill up a subspace of a Banach space. We obtain necessary and sufficient conditions for relative p-boundedness of a pair of operators in terms of families of resolving operators for degenerate fractional differential equation. Abstract results are illustrated by examples of the Cauchy problem for degenerate finite-dimensional system of fractional differential equations and of initial boundary-value problem for a fractional equation with respect to the time containing polynomials of Laplace operators with respect to spatial variables.
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References
Bajlekova, E. G. “Fractional Evolution Equations in Banach Spaces,” (PhD Thesis, Eindhoven University of Technology, University Press Facilities, 2001).
Samko, S. G., Kilbas, A. A., Marichev, O. I. Fractional Integrals and Derivatives: Theory and Applications (OPA, Amsterdam, 1993).
Podlubny, I. Fractional Differential Equations (Academic Press, San Diego, Boston, 1999).
Kilbas, A. A., Srivastava, H. M., Trujillo, J. J. Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, Boston, Heidelberg, 2006).
Favini, A., Yagi, A. Degenerate Differential Equations in Banach Spaces (Marcel Dekker Inc., New York, Basel, Hong Kong, 1999).
Sveshnikov, A. G., Al’shin, A. B., Korpusov, M. O., Pletner, Yu. D. Linear and Nonlinear Equations of Sobolev Type (Fizmatlit, Moscow, 2007).
Uchaikin, V. V. Method of Fractional Derivatives (Artishok Publishers, Ul’yanovsk, 2008).
Tarasov, V. E. Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media (Springer, New York, 2011).
Balachandran, K., Kiruthika, S. “Existence of Solutions of Abstract Fractional Integrodifferential Equations of Sobolev Type,” Comput. Math. Appl. 64, No. 10, 3406–3413 (2012).
Li, F., Liang, J., Xu, H.-K. “Existence of Mild Solutions for Fractional Integrodifferential Equations of Sobolev Type with Nonlocal Conditions,” J. Math. Anal. Appl. 391, No. 2, 510–525 (2012).
Sviridyuk, G. A., Fedorov, V. E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators (VSP, Utrecht, Boston, 2003).
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Original Russian Text © V.E. Fedorov, D.M. Gordievskikh, 2015, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2015, No. 1, pp. 71–83.
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Fedorov, V.E., Gordievskikh, D.M. Resolving operators of degenerate evolution equations with fractional derivative with respect to time. Russ Math. 59, 60–70 (2015). https://doi.org/10.3103/S1066369X15010065
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DOI: https://doi.org/10.3103/S1066369X15010065