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)
Continuity of solutions to discrete fractional initial value problems. (English) Zbl 1197.39002
Summary: We consider a fractional initial value problem (IVP) in the case where the order $\nu $ of the fractional difference satisfies $0<\nu \leq 1$. We show that solutions of this IVP satisfy a continuity condition both with respect to the order of the difference, $\nu $, and with respect to the initial conditions, and we deduce several important corollaries from this theorem. Thus, we address a complication that arises in the fractional case but not in the classical (integer-order) case.

39A10Additive difference equations
26A33Fractional derivatives and integrals (real functions)
45J05Integro-ordinary differential equations
Full Text: DOI
[1] Atici, F. M.; Eloe, P. W.: Initial value problems in discrete fractional calculus, Proc. amer. Math. soc. 137, No. 3, 981-989 (2009) · Zbl 1166.39005 · doi:10.1090/S0002-9939-08-09626-3
[2] Diethelm, K.; Ford, N.: Analysis of fractional differential equations, J. math. Anal. appl. 265, 229-248 (2002) · Zbl 1014.34003 · doi:10.1006/jmaa.2001.7194
[3] Agrawal, O. P.: Formulation of Euler--Lagrange equations for fractional variational problems, J. math. Anal. appl. 272, 368-379 (2002) · Zbl 1070.49013 · doi:10.1016/S0022-247X(02)00180-4
[4] Babakhani, A.; Daftardar-Gejji, V.: Existence of positive solutions of nonlinear fractional differential equations, J. math. Anal. appl. 278, 434-442 (2003) · Zbl 1027.34003 · doi:10.1016/S0022-247X(02)00716-3
[5] Bai, Z.; Lü, H.: Positive solutions for boundary value problem of nonlinear fractional differential equation, J. math. Anal. appl. 311, 495-505 (2005) · Zbl 1079.34048 · doi:10.1016/j.jmaa.2005.02.052
[6] 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
[7] Xu, X.; Jiang, D.; Yuan, C.: Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation, Nonlinear anal. 71, 4676-4688 (2009) · Zbl 1178.34006 · doi:10.1016/j.na.2009.03.030
[8] Oldham, K.; Spanier, J.: The fractional calculus: theory and applications of differentiation and integration to arbitrary order, (2002) · Zbl 0292.26011
[9] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008
[10] , Reviews of nonlinear dynamics and complexity (2008) · Zbl 1141.37001
[11] Atici, F. M.; Eloe, P. W.: Fractional q-calculus on a time scale, J. nonlinear math. Phys. 14, No. 3, 333-344 (2007) · Zbl 1157.81315
[12] Atici, F. M.; Eloe, P. W.: A transform method in discrete fractional calculus, Int. J. Difference equ. 2, No. 2, 165-176 (2007)
[13] F.M. Atici, P.W. Eloe, Two-point boundary value problems for finite fractional difference equations, J. Differ. Equ. Appl., in press (doi:10.1080/10236190903029241). · Zbl 1189.39004
[14] C.S. Goodrich, Solutions to a discrete right-focal fractional boundary value problem, Int. J. Difference Equ. (submitted for publication).
[15] C.S. Goodrich, Some new existence results for fractional difference equations (submitted for publication). · Zbl 1215.39004 · doi:10.1504/IJDSDE.2011.038499
[16] C.S. Goodrich, On a fractional boundary value problem with fractional boundary conditions (submitted for publication). · Zbl 1266.39006
[17] C.S. Goodrich, On a discrete fractional three-point boundary value problem (submitted for publication).
[18] C.S. Goodrich, A comparison result for a discrete fractional boundary value problem (submitted for publication).
[19] C.S. Goodrich, Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions (submitted for publication). · Zbl 1211.39002 · doi:10.1016/j.camwa.2010.10.041
[20] Bohner, M.; Peterson, A.: Dynamic equations on time scales: an introduction with application, (2001) · Zbl 0978.39001