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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.

##### MSC:
 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
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##### References:
 [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
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