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

Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia

IMPACT FACTOR 2017: 2.865
5-year IMPACT FACTOR: 3.323

CiteScore 2017: 3.06

SCImago Journal Rank (SJR) 2017: 1.967
Source Normalized Impact per Paper (SNIP) 2017: 1.954

Mathematical Citation Quotient (MCQ) 2017: 0.98

See all formats and pricing
More options …
Volume 16, Issue 4

Application of measure of noncompactness to a Cauchy problem for fractional differential equations in Banach spaces

Asadollah Aghajani / Ehsan Pourhadi / Juan Trujillo
Published Online: 2013-09-13 | DOI: https://doi.org/10.2478/s13540-013-0059-y


This paper is devoted to study the existence of solutions of a Cauchy type problem for a nonlinear fractional differential equation, via the techniques of measure of noncompactness. The investigation is based on a new fixed point result which is a generalization of the well known Darbo’s fixed point theorem. The main result is less restrictive than those given in the literature. Some illustrative examples are given.

MSC: Primary 47H10; Secondary 34A08

Keywords: fixed point; measure of noncompactness; fractional differential equation

  • [1] A. Aghajani, J. Banaś, N. Sabzali, Some generalizations of Darbo fixed point theorem and applications. Bull. Belg. Math. Soc. Simon Stevin, 20, No 2 (2013), 345–358. Google Scholar

  • [2] A. Aghajani, Y. Jalilian, J.J. Trujillo, On the existence of solutions of fractional integro-differential equations. Fract. Calc. Appl. Anal. 15, No 1 (2012), 44–69; DOI: 10.2478/s13540-012-0005-4; http://link.springer.com/article/10.2478/s13540-012-0005-4. CrossrefGoogle Scholar

  • [3] B. Ahmad, J.J. Nieto, Anti-periodic fractional boundary value problems with nonlinear term depending on lower order derivative. Fract. Calc. Appl. Anal. 15, No 3 (2012), 451–462; DOI: 10.2478/s13540-012-0032-1; http://link.springer.com/article/10.2478/s13540-012-0032-1. CrossrefGoogle Scholar

  • [4] B. Ahmad, J.J. Nieto, A. Alsaedi, M. El-Shahed, A study of nonlinear Langevin equation involving two fractional orders in different intervals. Nonlinear Anal., Real World Appl. 13 (2012), 599–606. http://dx.doi.org/10.1016/j.nonrwa.2011.07.052CrossrefGoogle Scholar

  • [5] B. Ahmad, S.K. Ntouyas, Nonlinear fractional differential equations and inclusions of arbitrary order and multi-strip boundary conditions. Electron. J. Differential Equations, 2012 (2012), Article # 98, 1–22. Google Scholar

  • [6] W.M. Ahmad, R. El-Khazali, Fractional-order dynamical models of love. Chaos Solitons Fractals 33 (2007), 1367–1375. http://dx.doi.org/10.1016/j.chaos.2006.01.098Web of ScienceCrossrefGoogle Scholar

  • [7] R.R. Akhmerov. M.I. Kamenskii, A.S. Potapov, A.E. Rodkina, B.N Sadovskii, Measures of Noncompactness and Condensing Operators. Birkhäuser Verlag, Basel-Boston-Berlin (1992). http://dx.doi.org/10.1007/978-3-0348-5727-7CrossrefGoogle Scholar

  • [8] J. Bai, X.-C. Feng, Fractional-order anisotropic diffusion for image denoising. IEEE Trans. Image Process. 16 (2007), 2492–2502. http://dx.doi.org/10.1109/TIP.2007.904971Web of ScienceCrossrefGoogle Scholar

  • [9] D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus: Models and Numerical Methods. World Scientific, Singapore (2012). Google Scholar

  • [10] J. Banaś, K. Goebel, Measure of Noncompactness in Banach Spaces. Lect. Notes Pure Appl. Math., Vol. 60, Marcel Dekker, New York (1980). Google Scholar

  • [11] M. Belmekki, J.J. Nieto, R. Rodríguez-López, Existence of periodic solution for a nonlinear fractional differential equation. Bound. Value Probl. 2009 (2009), Article ID 324561, 18 pages. Google Scholar

  • [12] M. Benchohra, J.R. Graef, F.Z. Mostafai, Weak solutions for nonlinear fractional differential equations on reflexive Banach spaces. Electron. J. Qual. Theory. 2010 (2010), Article # 54, 1–10. Google Scholar

  • [13] D. Bothe, Multivalued perturbations of m-accretive differential inclusions. Isreal J. Math. 108 (1998), 109–138. http://dx.doi.org/10.1007/BF02783044CrossrefGoogle Scholar

  • [14] R. Caponetto, G. Dongola, L. Fortuna, I. Petráš, Fractional Order Systems: Modeling and Control Applications. World Scientific, River Edge, NJ (2010). Google Scholar

  • [15] M. Caputo, F. Mainardi, A new dissipation model based on memory mechanism. Pure Appl. Geophys. 91 (1971), 134–147; Reprinted in: Fract. Calc. Appl. Anal. 10 (2007), 310–323. http://dx.doi.org/10.1007/BF00879562CrossrefGoogle Scholar

  • [16] A. Cernea, A note on the existence of solutions for some boundary value problems of fractional differential inclusions. Fract. Calc. Appl. Anal. 15, No 2 (2012), 183–194; DOI: 10.2478/s13540-012-0013-4; http://link.springer.com/article/10.2478/s13540-012-0013-4. CrossrefGoogle Scholar

  • [17] A. Chatterjee, Statistical origins of fractional derivatives in viscoelasticity. J. Sound Vib. 284 (2005), 1239–1245. http://dx.doi.org/10.1016/j.jsv.2004.09.019CrossrefGoogle Scholar

  • [18] J.-T. Chern, Finite Element Modeling of Viscoelastic Materials on the Theory of Fractional Calculus. Ph.D. thesis, Pennsylvania State University (1993). Google Scholar

  • [19] E. Cuesta, J. Finat Codes, Image processing by means of a linear integro-differential equation. In: M.H. Hamza (Ed.) Visualization, Imaging, and Image Processing 2003, Paper 91, ACTA Press, Calgary (2003). Google Scholar

  • [20] G. Darbo, Punti uniti in transformazioni a condominio non compatto. Rend. Sem. Math. Univ. Padova 24 (1955), 84–92. Google Scholar

  • [21] W. Deng, Short memory principle and a predictor-corrector approach for fractional differential equations. J. Comput. Appl. Math. 206 (2007), 174–188. http://dx.doi.org/10.1016/j.cam.2006.06.008CrossrefGoogle Scholar

  • [22] K. Diethelm, On the separation of solutions of fractional differential equations. Fract. Calc. Appl. Anal. 11, No 2 (2008), 259–268; http://www.math.bas.bg/~fcaa. Google Scholar

  • [23] K. Diethelm, M. Weilbeer, A numerical approach for Joulins model of a point source initiated flame. Fract. Calc. Appl. Anal. 7, No 1 (2004), 191–212. Google Scholar

  • [24] A.D. Freed, K. Diethelm, Fractional calculus in biomechanics: a 3D viscoelastic model using regularized fractional-derivative kernels with application to the human calcaneal fat pad. Biomech. Model. Mechanobiol. 5 (2006), 203–215. http://dx.doi.org/10.1007/s10237-005-0011-0CrossrefGoogle Scholar

  • [25] A.D. Freed, K. Diethelm, Y. Luchko, Fractional-Order Viscoelasticity (FOV): Constitutive Development Using the Fractional Calculus (First Annual Report). Technical Memorandum 2002-211914, NASA Glenn Research Center, Cleveland (2002). Google Scholar

  • [26] L. Gaul, P. Klein, S. Kempfle, Damping description involving fractional operators. Mech. Syst. Signal Process. 5 (1991), 81–88. http://dx.doi.org/10.1016/0888-3270(91)90016-XCrossrefGoogle Scholar

  • [27] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Vol. 204. Elsevier Science, Amsterdam (2006). http://dx.doi.org/10.1016/S0304-0208(06)80001-0Google Scholar

  • [28] A.A. Kilbas, J.J. Trujillo, Differential equations of fractional order. Methods, results and problems, I. Appl. Anal. 78 (2001), 153–192. http://dx.doi.org/10.1080/00036810108840931CrossrefGoogle Scholar

  • [29] A.A. Kilbas, J.J. Trujillo, Differential equations of fractional order. Methods, results and problems, II. Appl. Anal. 81 (2002), 435–493. http://dx.doi.org/10.1080/0003681021000022032CrossrefGoogle Scholar

  • [30] K. Li, J. Peng, J. Gao, Nonlocal fractional semilinear differential equations in separable Banach spaces. Electron. J. Differential Equations 2013 (2013), Article # 7, 1–7. http://dx.doi.org/10.1155/2013/802324CrossrefGoogle Scholar

  • [31] J. Liang, Z. Liu, X. Wang, Solvability for a couple system of nonlinear fractional differential equations in a Banach space. Fract. Calc. Appl. Anal. 16, No 1 (2013), 51–63; DOI: 10.2478/s13540-013-0004-0; http://link.springer.com/article/10.2478/s13540-013-0004-0. CrossrefGoogle Scholar

  • [32] R. Metzler, W. Schick, H.-G. Kilian, T.F. Nonnenmacher, Relaxation in filled polymers: a fractional calculus approach. J. Chem. Phys. 103 (1995), 7180–7186. http://dx.doi.org/10.1063/1.470346CrossrefGoogle Scholar

  • [33] I. Podlubny, Fractional-Order Systems and Fractional-Order Controllers. Technical Report UEF-03-94, Institute for Experimental Physics, Slovak Acad. Sci. (1994). Google Scholar

  • [34] I. Podlubny, L. Dorcak, J. Misanek, Application of fractional-order derivatives to calculation of heat load intensity change in blast furnace walls. Trans. Tech. Univ. Košice 5 (1995), 137–144. Google Scholar

  • [35] S. Shaw, M.K. Warby, J.R. Whiteman, A comparison of hereditary integral and internal variable approaches to numerical linear solid elasticity. In: Proc. of the XIII Polish Conf. on Computer Methods in Mechanics, Poznan (1997). Google Scholar

  • [36] L. Song, S.Y. Xu, J.Y. Yang, Dynamical models of happiness with fractional order. Commun. Nonlinear Sci. Numer. Simulat. 15 (2010), 616–628. http://dx.doi.org/10.1016/j.cnsns.2009.04.029CrossrefGoogle Scholar

  • [37] G. Wang, D. Baleanu, L. Zhang, Monotone iterative method for a class of nonlinear fractional differential equations. Fract. Calc. Appl. Anal. 15, No 2 (2012), 244–252; DOI: 10.2478/s13540-012-0018-z; http://link.springer.com/article/10.2478/s13540-012-0018-z. CrossrefGoogle Scholar

  • [38] J.R. Wang, Y. Zhou, M. Fečkan, Abstract Cauchy problem for fractional differential equations. Nonlinear Dynam. 71, No 4 (2013), 685–700; DOI 10.1007/s11071-012-0452-9. http://dx.doi.org/10.1007/s11071-012-0452-9CrossrefWeb of ScienceGoogle Scholar

About the article

Published Online: 2013-09-13

Published in Print: 2013-12-01

Citation Information: Fractional Calculus and Applied Analysis, Volume 16, Issue 4, Pages 962–977, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.2478/s13540-013-0059-y.

Export Citation

© 2013 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.

Xianghu Liu, JinRong Wang, and Yong Zhou
Journal of Dynamical and Control Systems, 2018
Shivaji Tate and H. T. Dinde
Mediterranean Journal of Mathematics, 2017, Volume 14, Number 2
Mohamed Jleli and Bessem Samet
Advances in Difference Equations, 2017, Volume 2017, Number 1
JinRong Wang, Michal Fĕckan, and Yong Zhou
Applied Mathematics and Computation, 2017, Volume 296, Page 257
Chung-Sik Sin and Liancun Zheng
Fractional Calculus and Applied Analysis, 2016, Volume 19, Number 3
M. Mursaleen
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2017, Volume 111, Number 2, Page 587
Ravi P. Agarwal, Asma, Vasile Lupulescu, and Donal O’Regan
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2017, Volume 111, Number 1, Page 257
J. Losada, J.J. Nieto, and E. Pourhadi
Journal of Computational and Applied Mathematics, 2017, Volume 312, Page 2
Sadia Arshad, Vasile Lupulescu, and Donal O’Regan
Fractional Calculus and Applied Analysis, 2014, Volume 17, Number 1

Comments (0)

Please log in or register to comment.
Log in