Year 2018,
, 128 - 137, 30.09.2018
Said Abbas
Amaria Arara
Mouffak Benchohra
,
Fatima Mesri
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 âAT Computational Fluid Dynamics, Reaction Engineering
and Molecular Properties (F. Keil, W. Mackens, H. Voss and J. Werther, eds.), SpringerâASVerlag, 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 Mathematics:
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-ASVerlag, 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. PetraGs, B. M. Vinagre, P. O’AZLeary 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.
Random evolution equations in Frechet spaces
Year 2018,
, 128 - 137, 30.09.2018
Said Abbas
Amaria Arara
Mouffak Benchohra
,
Fatima Mesri
Abstract
This paper deals with the existence of random mild solutions for some classes of first and second order functional evolution equations with random effects in Frechet spaces. The technique used is a generalization of the classical Darbo fixed point theorem for Frechet spaces associated with the concept of measure of noncompactness
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 âAT Computational Fluid Dynamics, Reaction Engineering
and Molecular Properties (F. Keil, W. Mackens, H. Voss and J. Werther, eds.), SpringerâASVerlag, 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 Mathematics:
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-ASVerlag, 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. PetraGs, B. M. Vinagre, P. O’AZLeary 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.