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Year 2018, Volume: 2 Issue: 3, 128 - 137, 30.09.2018
https://doi.org/10.31197/atnaa.451341

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 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, Volume: 2 Issue: 3, 128 - 137, 30.09.2018
https://doi.org/10.31197/atnaa.451341

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.
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Said Abbas This is me

Amaria Arara This is me

Mouffak Benchohra

Fatima Mesri This is me

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

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