zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

47D06One-parameter semigroups and linear evolution equations
35R11Fractional partial differential equations
34A08Fractional differential equations
26A33Fractional derivatives and integrals (real functions)
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) · Zbl 0784.45006
[3] Arendt, W.; Batty, C.; Hieber, M.; Neubrander, F.: Vector-valued Laplace transforms and Cauchy problems, Monogr. math. 96 (2001) · Zbl 0978.34001
[4] Arendt, W.: Vector-value Laplace transforms and Cauchy problems, Israel J. Math. 59, 327-352 (1987) · Zbl 0637.44001 · doi:10.1007/BF02774144
[5] Chen, C.; Li, M.: On fractional resolvent operator functions, Semigroup forum 80, 121-142 (2010) · Zbl 1185.47040 · doi:10.1007/s00233-009-9184-7
[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 1068.35037
[9] Zaslavsky, G. M.: Fractional kinetic equation for Hamiltonian chaos, Phys. D 76, 110-122 (1994) · Zbl 1194.37163 · doi:10.1016/0167-2789(94)90254-2
[10] Orsingher, E.; Beghin, L.: Fractional diffusion equations and processes with randomly varying time, Ann. probab. 37, 206-249 (2009) · Zbl 1173.60027 · doi:10.1214/08-AOP401
[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 · doi:10.1016/j.jfa.2010.07.007
[13] Li, M.; Zh, Q.: On spectral inclusions and approximation of ${\alpha}$-times resolvent families, Semigroup forum 69, 356-368 (2004) · Zbl 1096.47516 · doi:10.1007/s00233-004-0128-y
[14] Brezis, H.: Opérateurs maximaux monotones et semi-groupes de contrations dans LES espaces de Hilbert, Math. stud. 5 (1973) · Zbl 0252.47055
[15] Yosida, K.: Functional analysis, (1980) · Zbl 0435.46002