Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 9 (2014), 79 -- 92


Mouffak Benchohra and Jamal Eddine Lazreg

Abstract. In this paper, we establish the existence and uniqueness of solution for a class of initial value problem for implicit fractional differential equations with Caputo fractional derivative. The arguments are based upon the Banach contraction principle, Schauder' fixed point theorem and the nonlinear alternative of Leray-Schauder type. As applications, two examples are included to show the applicability of our results.

2010 Mathematics Subject Classification: 26A33; 34A08.
Keywords: Initial value problem; Caputo's fractional derivative; implicit fractional differential equations; fractional integral; existence, Gronwall's lemma; fixed point.

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  1. R. P. Agarwal, S. Arshad, D. O'Regan and V. Lupulescu, Fuzzy fractional integral equations under compactness type condition, Fract. Calc. Appl. Anal. 15 (2012), 572-590. MR2974320.

  2. R.P Agarwal, M. Benchohra and S. Hamani, Boundary value problems for fractional differential equations, Adv. Stud. Contemp. Math. 16 (2) (2008), 181-196. MR2572663(2010j:34003). Zbl 1179.26011.

  3. S. Abbes, M. Benchohra and G M. N'Guérékata, Topics in Fractional Differential Equations, Springer-Verlag, New York, 2012. MR2962045. Zbl 1273.35001.

  4. S. Abbes, M. Benchohra and A N. Vityuk, On fractional order derivatives and Darboux problem for implicit differential equations, Frac. Calc. Appl. Anal. 15, (2012), 168-182. MR2897771. Zbl 06194280.

  5. A. Babakhani and V. Daftardar-Gejji, Existence of positive solutions for multi-term non-autonomous fractional differential equations with polynomial coefficients, Electron. J. Differential Equations 2006, No. 129, 12 pp. MR2255244(2007e:34008). Zbl 1116.26003.

  6. A. Babakhani and V. Daftardar-Gejji, Existence of positive solutions for N-term non-autonomous fractional differential equations, Positivity 9 (2) (2005), 193-206. MR2189743(2006g:34068). Zbl 1111.34006.

  7. D. Baleanu, K. Diethelm, E. Scalas, and J.J. Trujillo, Fractional Calculus Models and Numerical Methods, World Scientific Publishing, New York, 2012. Zbl 1248.26011.

  8. D. Baleanu, Z.B. Güvenç and J.A.T. Machado, New Trends in Nanotechnology and Fractional Calculus Applications, Springer, New York, 2010. MR2605606(2011a:93004). Zbl 1196.65021.

  9. M. Belmekki and M. Benchohra, Existence results for fractional order semilinear functional differential equations, Proc. A. Razmadze Math. Inst. 146 (2008), 9-20. MR2464039(2009h:34089). Zbl 1175.26006.

  10. M. Benchohra, F. Berhoun and G M. N'Guérékata, Bounded solutions for fractional order differential equations on the half-line, Bull. Math. Anal. Appl. 146 (4) (2012), 62-71. MR2955875.

  11. M. Benchohra, J.R. Graef and S. Hamani, Existence results for boundary value problems with nonlinear fractional differential equations, Appl. Anal. 87 (7) (2008), 851-863. Zbl 1198.26008.

  12. M. Benchohra, S. Hamani and S.K. Ntouyas, Boundary value problems for differential equations with fractional order, Surv. Math. Appl. 3 (2008), 1-12. MR2532767. Zbl 1157.26301.

  13. M. Benchohra, J. Henderson, S.K. Ntouyas and A. Ouahab, Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl. 338 (2) (2008), 1340-1350. MR2386501(2008m:34182). Zbl 1209.34096.

  14. A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003. MR1987179(2004d:58012). Zbl 1025.47002.

  15. A. Guezane-Lakoud, R. Khaldi, Solvability of a fractional boundary value problem with fractional integral condition, Nonlinear Anal. 75 (2012), 2692-2700. MR2870948(2012j:34038). Zbl 1256.34003.

  16. D. Henry, Geometric Theory of Semilinear Parabolic Partial Differential Equations, Springer-Verlag, Berlin/New York, 1989.

  17. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. MR1890104(2002j:00009). Zbl 0998.26002.

  18. M. Hu and L. Wang, Existence of solutions for a nonlinear fractional differential equation with integral boundary condition, Int. J. Math. Comp. Sc., 7(1) (2011).

  19. A.A. Kilbas and S. A. Marzan, Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions, Diff. Equat. 41 (2005), 84-89. MR2213269(2006k:34010). Zbl 1160.34301.

  20. A.A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006. MR2218073(2007a:34002). Zbl 1092.45003.

  21. V. Lakshmikantham, S. Leela and J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, 2009.

  22. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993. MR1219954(94e:26013). Zbl 0789.26002.

  23. S. A. Murad and S. Hadid, An existence and uniqueness theorem for fractional differential equation with integral boundary condition, J. Frac. Calc. Appl. 3 (6), (2012), 1-9.

  24. M. D. Ortigueira, Fractional Calculus for Scientists and Engineers. Lecture Notes in Electrical Engineering, 84. Springer, Dordrecht, 2011. MR2768178(2012b:26003). Zbl 1251.26005.

  25. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. MR1658022(99m:26009). Zbl 0924.34008.

  26. X. Su and L. Liu, Existence of solution for boundary value problem of nonlinear fractional differential equation, Appl. Math. 22 (3) (2007) 291-298. MR2351068(2008f:34010). Zbl 1150.34005.

  27. V.E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of particles, Fields and Media, Springer, Heidelberg; Higher Education Press, Beijing, 2010. MR2796453(2012f:74008). Zbl 1214.81004.

  28. H. Ye, J. Gao, and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl. 328 (2007), 1075-1081. MR2290034. Zbl 1120.26003.

Mouffak Benchohra
Laboratory of Mathematics, University of Sidi Bel-Abbès
P. O. 89, Sidi Bel-Abbès 22000, Algérie.

Department of Mathematics, King Abdulaziz University
P.O. Box 80203, Jeddah 21589, Saudi Arabia.


Jamal Eddine Lazreg
Laboratory of Mathematics, University of Sidi Bel-Abbès
P. O. 89, Sidi Bel-Abbès 22000, Algérie.