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 and uniqueness of solutions of initial value problems for nonlinear fractional differential equations. (English) Zbl 1201.34008

Considered are existence and uniqueness of solutions of the following initial value problems

D α x(t)=f(t,D β x(t)),0<t1;x (k) =n k ,k=0,1,,m-1,

where m-1<α<m, n-1<β<n (m,n,m-1>n), D α stands for the Caputo derivative of order α and f is a continuous function defined on [0,1]×. The proofs are achieved by means of the contraction mapping principle.

MSC:
34A08Fractional differential equations
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
References:
[1]Lin, W.: Global existence theory and chaos control of fractional differential equations, J. math. Anal. appl. 332, 709-726 (2007) · Zbl 1113.37016 · doi:10.1016/j.jmaa.2006.10.040
[2]Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, (2006)
[3]Diethelm, K.; Freed, A. D.: On the solution of nonlinear fractional order differential equations used in the modelling of viscoplasticity, Scientific computing in chemical engineering II–computational fluid dynamics and molecular properties, 217-224 (1999)
[4]Lakshmikantham, V.: Theory of fractional functional differential equations, Nonlinear anal. 69, 3337-3343 (2008) · Zbl 1162.34344 · doi:10.1016/j.na.2007.09.025
[5]Lakshmikantham, V.; Vatsala, A. S.: Basic theory of fractional differential equations, Nonlinear anal. 69, 2677-2682 (2008) · Zbl 1161.34001 · doi:10.1016/j.na.2007.08.042
[6]Lakshmikantham, V.; Vatsala, A. S.: General uniqueness and monotone iterative technique for fractional differential equations, Appl. math. Lett. 21, 828-834 (2008) · Zbl 1161.34031 · doi:10.1016/j.aml.2007.09.006
[7]Benchohra, M.; Henderson, J.; Ntoyuas, S. K.; Ouahab, A.: Existence results for fractional order functional differential equations with infinite delay, J. math. Anal. appl. 338, 1340-1350 (2008) · Zbl 1209.34096 · doi:10.1016/j.jmaa.2007.06.021
[8]Caputo, M.: Linear models of dissipation whose Q is almost frequency independent (Part II), Geophys. J. R. astron. Soc. 13, 529-539 (1967)
[9]Daftardar-Gejji, V.; Jaffari, H.: Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives, J. math. Anal. appl. 328, 1026-1033 (2007) · Zbl 1115.34006 · doi:10.1016/j.jmaa.2006.06.007
[10]Delbosco, D.; Rodino, L.: Existence and uniqueness for a nonlinear fractional differential equation, J. math. Anal. appl. 204, 609-625 (1996) · Zbl 0881.34005 · doi:10.1006/jmaa.1996.0456
[11]El-Sayed, W. G.; El-Sayed, A. M. A.: On the functional integral equations of mixed type and integro-differential equations of fractional orders, Appl. math. Comput. 154, 461-467 (2004) · Zbl 1061.45004 · doi:10.1016/S0096-3003(03)00727-6
[12]Kilbas, A. A.; Marzan, S. A.: Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions, Differ. uravn. 41, No. 1, 82-86 (2005) · Zbl 1160.34301 · doi:10.1007/s10625-005-0137-y
[13]Bonilla, B.; Rivero, M.; Rodriguez-Germa, L.; Trujillo, J. J.: Fractional differential equations as alternative models to nonlinear differential equations, Appl. math. Comput. 187, 79-88 (2007) · Zbl 1120.34323 · doi:10.1016/j.amc.2006.08.105
[14]Jaradat, O. K.; Al-Omari, A.; Momani, S.: Existence of the mild solution for fractional semilinear initial value problem, Nonlinear anal. 69, 3153-3159 (2008) · Zbl 1160.34300 · doi:10.1016/j.na.2007.09.008
[15]Li, C.; Deng, W.: Remarks on fractional derivatives, Appl. math. Comput. 187, 777-784 (2007) · Zbl 1125.26009 · doi:10.1016/j.amc.2006.08.163
[16]Kosmatov, N.: Integral equations and initial value problems for nonlinear differential equations of fractional order, Nonlinear anal. 70, 2521-2529 (2009) · Zbl 1169.34302 · doi:10.1016/j.na.2008.03.037