zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Existence of the mild solution for some fractional differential equations with nonlocal conditions. (English) Zbl 1180.34006

Summary: We are concerned in this paper with the existence of mild solutions to the Cauchy Problem for the fractional differential equation with nonlocal conditions:

D q x(t)=Ax(t)+t n f(t,x(t),Bx(t)),t[0,T],n + ,x(0)+g(x)=x 0 ,

where 0<q<1, A is the infinitesimal generator of a C 0 -semigroup of bounded linear operators on a Banach space X.

MSC:
34A08Fractional differential equations
34G20Nonlinear ODE in abstract spaces
References:
[1]Aizicovici, S., McKibben, M.: Existence results for a class of abstract nonlocal Cauchy problems. Nonlinear Anal. TMA 39, 649–668 (2000) · Zbl 0954.34055 · doi:10.1016/S0362-546X(98)00227-2
[2]Anguraj, A., Karthikeyan, P., N’Guérékata, G.M.: Nonlocal Cauchy problem for some fractional abstract differential equations in Banach spaces. Commun. Math. Anal. 6(1), 31–35 (2009)
[3]Byszewski, L.: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162, 494–505 (1991) · Zbl 0748.34040 · doi:10.1016/0022-247X(91)90164-U
[4]Deng, K.: Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions. J. Math. Anal. Appl. 179, 630–637 (1993) · Zbl 0798.35076 · doi:10.1006/jmaa.1993.1373
[5]Devi, J.V., Lakshmikantham, V.: Nonsmooth analysis and fractional differential equations. Nonlinear Anal. (in press)
[6]Ezzinbi, K., Liu, J.: Nondensely defined evolution equations with nonlocal conditions. Math. Comput. Model. 36, 1027–1038 (2002) · Zbl 1035.34063 · doi:10.1016/S0895-7177(02)00256-X
[7]Fan, Z.: Existence of nondensely defined evolution equations with nonlocal conditions. Nonlinear Anal. (in press)
[8]Hernández, E.: Existence of solutions to a second order partial differential equation with nonlocal condition. Electr. J. Differ. Equ. 2003(51), 1–10 (2003)
[9]Jaradat, O.K., Al-Omari, A., Momani, S.: Existence of the mild solution for fractional semilinear initial Calue problems. Nonlinear Anal. 69, 3153–3159 (2008) · Zbl 1160.34300 · doi:10.1016/j.na.2007.09.008
[10]Krasnoselskii, M.A.: Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon, Elmsford (1964)
[11]Lakshmikantham, V.: Theory of fractional differential equations. Nonlinear Anal. TMA 60(10), 3337–3343 (2008)
[12]Lakshmikantham, V., Vatsala, A.S.: Basic theory of fractional differential equations. Nonlinear Anal. TMA 69(8), 2677–2682 (2008) · Zbl 1161.34001 · doi:10.1016/j.na.2007.08.042
[13]Lakshmikantham, V., Vatsala, A.S.: Theory of fractional differential inequalities and applications. Commun. Appl. Anal. (in press)
[14]Liu, H., Chang, J.-C.: Existence for a class of partial differential equations with nonlocal conditions, Nonlinear Anal. (in press)
[15]Mophou, G.M., Nakoulima, O., N’Guérékata, G.M.: Existence results for some fractional differential equations with nonlocal conditions (submitted)
[16]N’Guérékata, G.M.: Existence and uniqueness of an integral solution to some Cauchy problem with nonlocal conditions. In: Differential and Difference Equations and Applications, pp. 843–849. Hindawi Publ. Corp., New York (2006)
[17]N’Guérékata, G.M.: A Cauchy Problem for some fractional abstract differential equation with nonlocal conditions. Nonlinear Anal. TMA (in press)
[18]Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
[19]Wei, L.: Global existence and chaos control of fractional differential equations. J. Math. Anal. Appl. 332, 709–726 (2007) · Zbl 1211.35120 · doi:10.1016/j.jmaa.2006.09.083
[20]Zhang, S.: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electr. J. Differ. Equ. 2006(36), 1–12 (2006) · Zbl 1134.39008 · doi:10.1155/ADE/2006/90479