Communications in Mathematical Analysis

Boundary Value Problems for a Class of Fractional Differential Equations Depending on First Derivative

D. Foukrach, T. Moussaoui, and S. K. Ntouyas

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This paper deals with the existence and uniqueness results for nonlinear and double perturbed BVPs for fractional differential equations with firstorder dependence derivative. Our approach is based on fixed point theorems and monotone iterative technique. Some illustrative examples are also presented.

Article information

Commun. Math. Anal., Volume 15, Number 2 (2013), 15-28.

First available in Project Euclid: 9 August 2013

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Zentralblatt MATH identifier

Primary: 26A33 34A08: Fractional differential equations 34A12: Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions

Existence and uniqueness fractional differential equations fixed point theorems monotone iterative technique


Foukrach, D.; Moussaoui, T.; Ntouyas, S. K. Boundary Value Problems for a Class of Fractional Differential Equations Depending on First Derivative. Commun. Math. Anal. 15 (2013), no. 2, 15--28.

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  • R. P. Agarwal, M. Benchohra and S. Hamani, Boundary value problems for fractional differential equations, Georgian Math. J. 16 (2009), 401-411.
  • R. P. Agarwal and B. Ahmad, Existence of solutions for impulsive anti-periodic boundary value problems of fractional semilinear evolution equations, Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 18 (2011), 457-470.
  • B. Ahmad and S. Sivasundaram, Existence and uniqueness results for nonlinear boundary value problems of fractional differential equations with separated boundary conditions, Commun. Appl. Anal. 13 (2009), 121-228.
  • B. Ahmad and J.J. Nieto, Existence of solutions for nonlocal boundary value problems of higher order nonlinear fractional differential equations, Abstr. Appl. Anal. 2009, Art. ID 494720, 9 pp.
  • Z. Bai, On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Anal. 72 (2010), 916-924.
  • Z. Bai and H. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. 311 (2005), 495-505.
  • M. Benchohra, S. Djebali and T. Moussaoui, Boundary value problems for doubly perturbed first order ordinary differential systems, Electron. J. Differ. Equ., Vol. 2006 (2006), No. 11, pp. 1-10.
  • M. Benchohra, S. Hamani and S.K. Ntouyas, Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Anal. 71 (2009), 2391-2396.
  • D.W. Boyd and J.S.W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969), 458-464.
  • M. El-Shahed, Positive solution for boundary value problems of nonlinear fractional differential equation, Abstr. Appl. Anal., vol (2007), Article ID 10368, 8 pages.
  • S. Hamani, M. Benchohra and J. R. Graef, Existence results for boundary value problems with nonlinear fractional inclusions and integral conditions, Electron. J. Differ. Equ. Vol. 2010 (2010), No. 20, pp. 1-16.
  • A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.
  • N. Kosmatov, A singular boundary value problem for nonlinear differential equations of fractional order, J. Appl. Math. Comput. 29 (2009), 125-135.
  • T. Moussaoui and S.K. Ntouyas, Existence and uniqueness of solutions of a boundary value problem of fractional order, Commun. Math. Anal. 12 (2012), 64-75.
  • S.K. Ntouyas and P. Ch. Tsamatos, A Fixed point theorem of Krasonoselskii Nonlinear alternative type with applications to functional integral equation, Diff. Eqn. Dyn. Syst. 7 (1999), 139-146.
  • I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  • J. Sabatier, O.P. Agrawal and J.A.T. Machado (Eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, 2007.
  • S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993.
  • S. Stanek, The existence of positive solutions of singular fractional boundary value problems, Comput. Math. Appl. 62 (2011), 1379-1388.
  • S. Zhang, Positive solutions to singular boundary value problem for nonlinear fractional differential equation, Comput. Math. Appl. 59 (2010), 1300-1309.