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Boundary Value Problems For Caputo-Hadamard Fractional Differential Equations

Year 2018, Volume: 2 Issue: 3, 138 - 145, 30.09.2018
https://doi.org/10.31197/atnaa.419517

Abstract

In this paper, we investigate the existence of  solutions of a boundary value problem for Caputo-Hadamard  fractional differential equations.
Our analysis relies on   classical fixed point theorems.  Examples are given to illustrate our theoretical results.

References

  • [1] R. P Agarwal, M. Benchohra and S. Hamani, A survey on existence results for boundary value problems for nonlinear fractional differential equations and inclusions, Acta Applicandae Math. 109, No. 3 (2010), 973-1033.
  • [2] R. P. Agarwal, M. Meehan and D. O’Regan, Fixed point theory and applications, Cambridge Tracts in Mathematics 141 Cambridge University Press, Cambridge, UK, (2001).
  • [3] M. Benchohra, J. R. Graef and S. Hamani, Existence results for boundary value problems of nonlinear fractional differential equations with integral conditions, Appl. Anal. 87, No. 7 (2008), 851-863.
  • [4] M. Benchohra and S. Hamani, Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Anal. 71 (2009), 2391-2396.
  • [5] M. Benchohra, S. Hamani and S. K. Ntouyas, Boundary value problems for differential equations with fractional order, Surv. Math. Appl. 3 (2008), 1-12.
  • [6] W. Benhamida, J. R. Graef, and S. Hamani, Boundary value problems for fractional differential equations with integral and anti-periodic conditions in a Banach space, Prog. Frac. Differ. Appl. 4, No. 2 (2018), 1-7.
  • [7] W. Benhamida, J. R. Graef and S. Hamani, Boundary value problems for Hadamard fractional differential equations with nonlocal multi-point boundary conditions, (to appear).
  • [8] W. Benhamida, S. Hamani, and J. Henderson, A boundary value problem for fractional differential equations with Hadamard derivative and nonlocal conditions, PanAmerican Math. J. 26 (2016), 1-11.
  • [9] P. L. Butzer, A. A. Kilbas and J. J. Trujillo, Composition of Hadamard-type fractional integration operators and the semigroup property, J. Math. Anal. Appl. 269 (2002), 387-400.
  • [10] P. L. Butzer, A. A. Kilbas and J. J. Trujillo, Fractional calculus in the Mellin setting and Hadamard-type fractional integrals, J. Math. Anal. Appl. 269 (2002), 1-27.
  • [11] P. L. Butzer, A. A. Kilbas and J. J. Trujillo, Mellin transform analysis and integration by parts for Hadamard-type fractional integrals, J. Math. Anal. Appl. 270 (2002), 1-15.
  • [12] Z. Cui, P. Yu and Z. Mao, Existence of solutions for nonlocal boundary value problems of nonlinear fractional differential equations, Adv. Dynam. Sys. Appl. 7 (2012), 31-40.
  • [13] D. Delbosco and L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl. 204 (1996), 609-625.
  • [14] K. Diethelm and A. D. Freed, On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity, Scientifice Computing in Chemical Engineering II. . Computational Fluid Dynamics, Reaction Engi- neering and Molecular Properties (F. Keil, W. Mackens, H. Voss and J. Werther, eds.), Springer--Verlag, Heidelberg, 1999, pp. 217-224.
  • [15] K. Diethelm and N. J. Ford Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002), 229-248.
  • [16] K. Diethelm and G. Walz, Numerical solution of fractional order differential equations by extrapolation, Numer. Algorithms 16 (1997), 231-253.
  • [17] A. M. A. EL-Sayed and E. O. Bin-Taher, Positive solutions for a nonlocal multi-point boundary-value problem of fractional and second order, Electron. J. Differential Equations, Number 64, (2013), 1-8.
  • [18] A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York, 2003.
  • [19] J. Hadamard, Essai sur l’etude des fonctions donnees par leur development de Taylor, J. Mat. Pure Appl. Ser. 8 (1892), 101-186. [20] F. Jarad, D. Baleanu and T. Abdeljawad, Caputo-type modification of the Hadamard fractional derivatives, Adv. Differ. Equ. 2012, No.1 (2012),1-8.
  • [21] A. A. Kilbas and S. A. Marzan, Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions, Differential Equations 41 (2005), 84-89.
  • [22] 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.
  • [23] V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract Spaces, International Series in Mathemat- ics: Theory, Methods and Applications, 2, Pergamon Press, Oxford, UK, 1981.
  • [24] F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, Fractals and Fractional Calculus in Continuum Mechanics (A. Carpinteri and F. Mainardi, eds.), Springer-Verlag, Wien, 1997, pp. 291 348.
  • [25] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993. [26] I. Podlubny, I. Petras, B. M. Vinagre, P. O’Leary and L. Dorcak, Analogue realizations of fractional-order controllers. Fractional order calculus and its applications, Nonlinear Dynam. 29 (2002), 281-296.
  • [27] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993.
  • [28] P. Thiramanus, S. K. Ntouyas and J. Tariboon, Existence and uniqueness results for Hadamard- type fractional differential equations with nonlocal fractional integral boundary conditions, Abstr. Appl. Anal. (2014), Art. ID 902054, 9 pp.
Year 2018, Volume: 2 Issue: 3, 138 - 145, 30.09.2018
https://doi.org/10.31197/atnaa.419517

Abstract

References

  • [1] R. P Agarwal, M. Benchohra and S. Hamani, A survey on existence results for boundary value problems for nonlinear fractional differential equations and inclusions, Acta Applicandae Math. 109, No. 3 (2010), 973-1033.
  • [2] R. P. Agarwal, M. Meehan and D. O’Regan, Fixed point theory and applications, Cambridge Tracts in Mathematics 141 Cambridge University Press, Cambridge, UK, (2001).
  • [3] M. Benchohra, J. R. Graef and S. Hamani, Existence results for boundary value problems of nonlinear fractional differential equations with integral conditions, Appl. Anal. 87, No. 7 (2008), 851-863.
  • [4] M. Benchohra and S. Hamani, Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Anal. 71 (2009), 2391-2396.
  • [5] M. Benchohra, S. Hamani and S. K. Ntouyas, Boundary value problems for differential equations with fractional order, Surv. Math. Appl. 3 (2008), 1-12.
  • [6] W. Benhamida, J. R. Graef, and S. Hamani, Boundary value problems for fractional differential equations with integral and anti-periodic conditions in a Banach space, Prog. Frac. Differ. Appl. 4, No. 2 (2018), 1-7.
  • [7] W. Benhamida, J. R. Graef and S. Hamani, Boundary value problems for Hadamard fractional differential equations with nonlocal multi-point boundary conditions, (to appear).
  • [8] W. Benhamida, S. Hamani, and J. Henderson, A boundary value problem for fractional differential equations with Hadamard derivative and nonlocal conditions, PanAmerican Math. J. 26 (2016), 1-11.
  • [9] P. L. Butzer, A. A. Kilbas and J. J. Trujillo, Composition of Hadamard-type fractional integration operators and the semigroup property, J. Math. Anal. Appl. 269 (2002), 387-400.
  • [10] P. L. Butzer, A. A. Kilbas and J. J. Trujillo, Fractional calculus in the Mellin setting and Hadamard-type fractional integrals, J. Math. Anal. Appl. 269 (2002), 1-27.
  • [11] P. L. Butzer, A. A. Kilbas and J. J. Trujillo, Mellin transform analysis and integration by parts for Hadamard-type fractional integrals, J. Math. Anal. Appl. 270 (2002), 1-15.
  • [12] Z. Cui, P. Yu and Z. Mao, Existence of solutions for nonlocal boundary value problems of nonlinear fractional differential equations, Adv. Dynam. Sys. Appl. 7 (2012), 31-40.
  • [13] D. Delbosco and L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl. 204 (1996), 609-625.
  • [14] K. Diethelm and A. D. Freed, On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity, Scientifice Computing in Chemical Engineering II. . Computational Fluid Dynamics, Reaction Engi- neering and Molecular Properties (F. Keil, W. Mackens, H. Voss and J. Werther, eds.), Springer--Verlag, Heidelberg, 1999, pp. 217-224.
  • [15] K. Diethelm and N. J. Ford Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002), 229-248.
  • [16] K. Diethelm and G. Walz, Numerical solution of fractional order differential equations by extrapolation, Numer. Algorithms 16 (1997), 231-253.
  • [17] A. M. A. EL-Sayed and E. O. Bin-Taher, Positive solutions for a nonlocal multi-point boundary-value problem of fractional and second order, Electron. J. Differential Equations, Number 64, (2013), 1-8.
  • [18] A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York, 2003.
  • [19] J. Hadamard, Essai sur l’etude des fonctions donnees par leur development de Taylor, J. Mat. Pure Appl. Ser. 8 (1892), 101-186. [20] F. Jarad, D. Baleanu and T. Abdeljawad, Caputo-type modification of the Hadamard fractional derivatives, Adv. Differ. Equ. 2012, No.1 (2012),1-8.
  • [21] A. A. Kilbas and S. A. Marzan, Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions, Differential Equations 41 (2005), 84-89.
  • [22] 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.
  • [23] V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract Spaces, International Series in Mathemat- ics: Theory, Methods and Applications, 2, Pergamon Press, Oxford, UK, 1981.
  • [24] F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, Fractals and Fractional Calculus in Continuum Mechanics (A. Carpinteri and F. Mainardi, eds.), Springer-Verlag, Wien, 1997, pp. 291 348.
  • [25] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993. [26] I. Podlubny, I. Petras, B. M. Vinagre, P. O’Leary and L. Dorcak, Analogue realizations of fractional-order controllers. Fractional order calculus and its applications, Nonlinear Dynam. 29 (2002), 281-296.
  • [27] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993.
  • [28] P. Thiramanus, S. K. Ntouyas and J. Tariboon, Existence and uniqueness results for Hadamard- type fractional differential equations with nonlocal fractional integral boundary conditions, Abstr. Appl. Anal. (2014), Art. ID 902054, 9 pp.
There are 26 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Samira Hamani

Wafaa Benhamida This is me

Johnny Henderson This is me

Publication Date September 30, 2018
Published in Issue Year 2018 Volume: 2 Issue: 3

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