zbMATH — the first resource for mathematics

Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives. (English) Zbl 1266.47066
In this paper, the authors study certain Cauchy-type problems of fractional differential equations with fractional differential conditions, involving Riemann-Liouville derivatives, in infinite-dimensional Banach spaces. They introduce a certain fractional resolvent and study some of its properties. Moreover, they prove that a homogeneous \(\alpha\)-order Cauchy problem is well posed if and only if its coefficient operator is the generator of an \(\alpha\)-order fractional resolvent, and they give sufficient conditions to guarantee the existence and uniqueness of weak solutions and strong solutions of a certain inhomogeneous \(\alpha\)-order Cauchy problem.

47D06 One-parameter semigroups and linear evolution equations
35R11 Fractional partial differential equations
34A08 Fractional ordinary differential equations and fractional differential inclusions
26A33 Fractional derivatives and integrals
Full Text: DOI
[1] E. Bazhlekova, Fractional evolution equations in Banach spaces, PhD thesis, University Press Facilities, Eindhoven University of Technology, 2001.
[2] Prüss, J., Evolutionary integral equations and applications, (1993), Birkhäuser Basel, Berlin · Zbl 0793.45014
[3] Arendt, W.; Batty, C.; Hieber, M.; Neubrander, F., Vector-valued Laplace transforms and Cauchy problems, Monogr. math., vol. 96, (2001), Birkhäuser Basel · Zbl 0978.34001
[4] Arendt, W., Vector-value Laplace transforms and Cauchy problems, Israel J. math., 59, 327-352, (1987) · Zbl 0637.44001
[5] Chen, C.; Li, M., On fractional resolvent operator functions, Semigroup forum, 80, 121-142, (2010) · Zbl 1185.47040
[6] Nigmatullin, R.R., To the theoretical explanation of the “universal response”, Phys. stat. solidi B, 123, 739-745, (1984)
[7] Heymans, N.; Podlubny, I., Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives, Rheol. acta, 45, 765-771, (2006)
[8] Eidelman, S.D.; Kochubei, A.N., Cauchy problems for fractional diffusion equations, J. differential equations, 199, 211-255, (2004) · Zbl 1129.35427
[9] Zaslavsky, G.M., Fractional kinetic equation for Hamiltonian chaos, Phys. D, 76, 110-122, (1994) · Zbl 1194.37163
[10] Orsingher, E.; Beghin, L., Fractional diffusion equations and processes with randomly varying time, Ann. probab., 37, 206-249, (2009) · Zbl 1173.60027
[11] Meerschaert, M.M.; Nane, E.; Vellaisamy, P., Fractional Cauchy problems on bounded domains, Ann. probab., 37, 979-1007, (2009) · Zbl 1247.60078
[12] Li, M.; Chen, C.; Li, Fu-Bo, On fractional powers of generators of fractional resolvent families, J. funct. anal., 259, 2702-2726, (2010) · Zbl 1203.47021
[13] Li, M.; Zh, Q., On spectral inclusions and approximation of α-times resolvent families, Semigroup forum, 69, 356-368, (2004) · Zbl 1096.47516
[14] Brezis, H., Opérateurs maximaux monotones et semi-groupes de contrations dans LES espaces de Hilbert, Math. stud., vol. 5, (1973), North-Holland Amsterdam · Zbl 0252.47055
[15] Yosida, K., Functional analysis, (1980), Springer-Verlag · Zbl 0152.32102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.