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.
[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
[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.