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)
Integral equations and initial value problems for nonlinear differential equations of fractional order. (English) Zbl 1169.34302
The paper is concerned with developing existence and uniqueness theory for certain nonlinear integral and integro-differential equations of Volterra and Abel type that arise when the Caputo fractional derivative is used. Following preliminaries, in which the necessary definitions are introduced, the paper presents a series of theorems on existence and uniqueness of solutions under appropriate conditions.

MSC:
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
45D05Volterra integral equations
26A33Fractional derivatives and integrals (real functions)
47N20Applications of operator theory to differential and integral equations
45J05Integro-ordinary differential equations
WorldCat.org
Full Text: DOI
References:
[1] Agarwal, R. P.; O’regan, D.: Infinite interval problems for differential difference and integral equations. (2001)
[2] 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
[3] Caputo, M.: Linear models of dissipation whose Q is almost frequency independent (Part II). Geophys. J. R. astron. Soc. 13, 529-539 (1967)
[4] 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
[5] Delbosco, D.; Rodino, L.: Existence and uniqueness for a nonlinear fractional differential equation. J. math. Anal. appl. 204, 609-625 (1996) · Zbl 0881.34005
[6] 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)
[7] 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
[8] El-Sayed, A. M. A.; Gaber, M.: On the finite Caputo and finite rietz derivatives. Electron. J. Theor. phys. 13, No. 12, 81-95 (2006)
[9] Jaradat, O. K.; Al-Omari, A.; Momani, S.: Existence of the mild solution for fractional semilinear initial value problems. Nonlinear anal. (2007) · Zbl 1160.34300
[10] 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
[11] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North Holland Mathematics Studies, vol. 204, Elsevier · Zbl 1092.45003
[12] Kilbas, A. A.; Trujillo, J. J.: Differential equations of fractional order: methods, results and problems. Appl. anal. 78, 153-192 (2001) · Zbl 1031.34002
[13] Lakshmikantham, V.: Theory of fractional functional differential equations. Nonlinear anal. (2007)
[14] Lakshmikantham, V.; Vatsala, A. S.: Basic theory of fractional differential equations. Nonlinear anal. (2007) · Zbl 1159.34006
[15] Li, C.; Deng, W.: Remarks on fractional derivatives. Appl. math. Comput. 187, 777-784 (2007) · Zbl 1125.26009
[16] Lin, W.: Global existence theory and chaos control of fractional differential equations. J. math. Anal. appl. 332, 709-726 (2007) · Zbl 1113.37016
[17] Podlubny, I.: Fractional differential equations, mathematics in sciences and applications. (1999) · Zbl 0924.34008
[18] Samko, S. G.; Kilbas, A. A.; Mirichev, O. I.: Fractional integral and derivatives (Theory and applications). (1993)
[19] Zeidler, E.: Nonlinear functional analysis and applications, I: Fixed point theorems. (1986) · Zbl 0583.47050