Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia

IMPACT FACTOR 2018: 3.514
5-year IMPACT FACTOR: 3.524

CiteScore 2018: 3.44

SCImago Journal Rank (SJR) 2018: 1.891
Source Normalized Impact per Paper (SNIP) 2018: 1.808

Mathematical Citation Quotient (MCQ) 2018: 1.08

See all formats and pricing
More options …
Volume 14, Issue 4


On the existence and uniqueness and formula for the solution of R-L fractional cauchy problem in ℝn

Dariusz Idczak / Rafal Kamocki
Published Online: 2011-09-29 | DOI: https://doi.org/10.2478/s13540-011-0033-5


In this paper we obtain results on the existence and uniqueness of a solution to a fractional nonlinear Cauchy problem containing the Riemann-Liouville derivative, in a fractional counterpart of the set of ℝn-valued absolutely continuous functions. We also derive a Cauchy formula for the solution to the linear problem of such a type.

MSC: Primary 34A08, 26A33

Keywords: Riemann-Liouville derivative; fractional Cauchy problem; existence and uniqueness of a solution; Cauchy formula

  • [1] R.P. Agarwal. V. Lakshmikantham, J.J. Nieto, On the concept of solution for fractional differential equations with uncertainty. Nonlinear Analysis 72 (2010), 2859–2862. http://dx.doi.org/10.1016/j.na.2009.11.029CrossrefGoogle Scholar

  • [2] M.A. Al-Bassam, Some existence theorems on differential equations of generalized order. J. Reine Angew. Math. 218, No 1 (1965), 70–78. http://dx.doi.org/10.1515/crll.1965.218.70CrossrefGoogle Scholar

  • [3] J.H. Barrett, Differential equations of non-integer order. Canad. J. Math. 6, No 4 (1954), 529–541. http://dx.doi.org/10.4153/CJM-1954-058-2CrossrefGoogle Scholar

  • [4] M. Belmekki, J.J. Nieto, R. Rodriguez-Lopez, Existence of periodic solution for a nonlinear fractional differential equation. Boundary Value Problems, vol. 2009, Article ID 324561. Google Scholar

  • [5] A. Bielecki, Une remarque sur l’application de la methode de Banach-Cocciopoli-Tichonov dans la theorie de l’equation s = f(x, y, z, p, q). Bull. Acad. Pol. Sci. 4 (1956), 265–268. Google Scholar

  • [6] B. Bonilla, M. Rivero, L. Rodriguez-Germa, J.J. Trujillo, Fractional differential equations as alternative models to nonlinear differential equations. Applied Mathematics and Computation 187 (2007), 79–88. http://dx.doi.org/10.1016/j.amc.2006.08.105CrossrefGoogle Scholar

  • [7] D. Delbosco, L. Rodino, Existence and uniqueness for a nonlinear fractional differential equations. J. Math. Anal. Appl. 204, No 2 (1996), 609–625. http://dx.doi.org/10.1006/jmaa.1996.0456CrossrefGoogle Scholar

  • [8] Z.F.A. El-Raheem, Modification of the application of a contraction mapping method on a class of fractional differential equation. Applied Mathematics and Computation 137 (2003), 371–374. http://dx.doi.org/10.1016/S0096-3003(02)00136-4CrossrefGoogle Scholar

  • [9] N. Hayek, J. Trujillo, M. Rivero, B. Bonilla, J.C. Moreno, An extension of Picard-Lindeloff theorem to fractional differential equations. Appl. Anal. 70, No 3–4 (1999), 347–361. Google Scholar

  • [10] N. Heymans, I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheol. Acta 45 (2006), 765–771. http://dx.doi.org/10.1007/s00397-005-0043-5CrossrefGoogle Scholar

  • [11] G. Jumarie, An approach via fractional analusis to non-linearity induced by coarse-graining in space. Nonlinear Analysis: Real World Applications 11 (2010), 535–546. http://dx.doi.org/10.1016/j.nonrwa.2009.01.003CrossrefGoogle Scholar

  • [12] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006). Google Scholar

  • [13] N. Kosmatov, Integral equations and initial value problems for nonlinear differential equations of fractional order. Nonlinear Analysis 70 (2009), 2521–2529. http://dx.doi.org/10.1016/j.na.2008.03.037CrossrefGoogle Scholar

  • [14] Y.F. Luchko, M. Rivero, J.J. Trujillo, M.P. Velasco, Fractional models, non-locality, and complex systems. Computers and Mathematics with Applications 59 (2010), 1048–1056. http://dx.doi.org/10.1016/j.camwa.2009.08.015CrossrefGoogle Scholar

  • [15] J.J. Nieto, Maximum principles for fractional differential equations derived from Mittag-Leffler functions. Appl. Math. Lett. 23, No 10 (2010), 1248–1251. http://dx.doi.org/10.1016/j.aml.2010.06.007CrossrefGoogle Scholar

  • [16] E. Pitcher, W.E. Sewell, Existence theorems for solutions of differential equations of non-integral order. Bull. Amer.Math. Soc. 44, No 2 (1938), 100–107. http://dx.doi.org/10.1090/S0002-9904-1938-06695-5CrossrefGoogle Scholar

  • [17] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives — Theory and Applications. Gordon and Breach, Amsterdam (1993). Google Scholar

  • [18] Z. Shuqin, Monotone iterative method for initial value problem involving Riemann-Liouville fractional derivatives. Nonlinear Analysis 71 (2009), 2087–2093. http://dx.doi.org/10.1016/j.na.2009.01.043CrossrefGoogle Scholar

  • [19] Z. Wei, O. Li, J. Che, Initial value problems for fractional differential equations involving Riemann-Lioville sequential fractional derivative. J. Math. Anal. Appl. 367 (2010), 260–272. http://dx.doi.org/10.1016/j.jmaa.2010.01.023CrossrefGoogle Scholar

About the article

Published Online: 2011-09-29

Published in Print: 2011-12-01

Citation Information: Fractional Calculus and Applied Analysis, Volume 14, Issue 4, Pages 538–553, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.2478/s13540-011-0033-5.

Export Citation

© 2011 Diogenes Co., Sofia. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Rafał Kamocki
Nonlinear Analysis: Modelling and Control, 2019, Volume 24, Number 2, Page 279
Kishor D. Kucche, Ashwini D. Mali, and J. Vanterler da C. Sousa
Computational and Applied Mathematics, 2019, Volume 38, Number 2
Rafał Kamocki
Journal of Computational and Applied Mathematics, 2016, Volume 308, Page 39
Rafał Kamocki and Marek Majewski
Optimal Control Applications and Methods, 2015, Volume 36, Number 6, Page 953
Massimiliano Ferrara, Giovanni Molica Bisci, and Binlin Zhang
Discrete and Continuous Dynamical Systems - Series B, 2014, Volume 19, Number 8, Page 2483
Rafał Kamocki
Applied Mathematics and Computation, 2014, Volume 235, Page 94
Dariusz Idczak, Andrzej Skowron, and Stanisław Walczak
Abstract and Applied Analysis, 2013, Volume 2013, Page 1
Dariusz Idczak and Rafał Kamocki
Multidimensional Systems and Signal Processing, 2015, Volume 26, Number 1, Page 193
Rafal Kamocki
Mathematical Methods in the Applied Sciences, 2014, Volume 37, Number 11, Page 1668
Asadollah Aghajani, Yaghoub Jalilian, and Juan Trujillo
Fractional Calculus and Applied Analysis, 2012, Volume 15, Number 1

Comments (0)

Please log in or register to comment.
Log in