# Existence results for impulsive semilinear fractional differential inclusions with delay in Banach spaces

Hadda Hammouche; Kaddour Guerbati; Mouffak Benchohra; Nadjat Abada

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2013)

- Volume: 33, Issue: 2, page 149-170
- ISSN: 1509-9407

## Access Full Article

top## Abstract

top## How to cite

topHadda Hammouche, et al. "Existence results for impulsive semilinear fractional differential inclusions with delay in Banach spaces." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 33.2 (2013): 149-170. <http://eudml.org/doc/270480>.

@article{HaddaHammouche2013,

abstract = {In this paper, we introduce a new concept of mild solution of some class of semilinear fractional differential inclusions of order 0 < α < 1. Also we establish an existence result when the multivalued function has convex values. The result is obtained upon the nonlinear alternative of Leray-Schauder type.},

author = {Hadda Hammouche, Kaddour Guerbati, Mouffak Benchohra, Nadjat Abada},

journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},

keywords = {fractional calculus; caputo fractional derivative; multivalued map; differential inclusions; mild solution; fixed point},

language = {eng},

number = {2},

pages = {149-170},

title = {Existence results for impulsive semilinear fractional differential inclusions with delay in Banach spaces},

url = {http://eudml.org/doc/270480},

volume = {33},

year = {2013},

}

TY - JOUR

AU - Hadda Hammouche

AU - Kaddour Guerbati

AU - Mouffak Benchohra

AU - Nadjat Abada

TI - Existence results for impulsive semilinear fractional differential inclusions with delay in Banach spaces

JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization

PY - 2013

VL - 33

IS - 2

SP - 149

EP - 170

AB - In this paper, we introduce a new concept of mild solution of some class of semilinear fractional differential inclusions of order 0 < α < 1. Also we establish an existence result when the multivalued function has convex values. The result is obtained upon the nonlinear alternative of Leray-Schauder type.

LA - eng

KW - fractional calculus; caputo fractional derivative; multivalued map; differential inclusions; mild solution; fixed point

UR - http://eudml.org/doc/270480

ER -

## References

top- [1] S. Abbas, M. Benchohra and G.M. N'Guérékata, Topics in Fractional Differential Equations, Springer, New York, 2012. doi: 10.1007/978-1-4614-4036-9
- [2] R.P. Agarwal, M. Benchohra and S. Hamani, Boundary value problems for fractional differential equations, Adv. Stud. Contemp. Math. 16 (2008), 181-196. Zbl1152.26005
- [3] A. Anguraj, P. Karthikeyan and G.M. N'Guérékata, Nonlocal Cauchy problem for some fractional abstract differential equations in Banach spaces, Commun. Math. Anal. 6 (2009). Zbl1167.34387
- [4] D. Baleanu, K. Diethelm, E. Scalas and J.J. Trujillo, Fractional Calculus Models and Numerical Methods, World Scientific Publishing, New York, 2012. Zbl1248.26011
- [5] M. Belmekki and M. Benchohra, Existence results for fractional order semilinear functional differential equations, Proc. A. Razmadze Math. Inst. 146 (2008), 9-20. Zbl1175.26006
- [6] M. Belmekki, M. Benchohra and L. Górniewicz, Semilinear functional differential equations with fractional order and infinite delay, Fixed Point Th. 9 (2008), 423-439. Zbl1162.26302
- [7] M. Benchohra, J.R. Graef and S. Hamani, Existence results for boundary value problems with nonlinear fractional differential equations, Appl. Anal. 87 (2008), 851-863. doi: 10.1080/00036810802307579 Zbl1198.26008
- [8] M. Benchohra, J. Henderson, S.K. Ntouyas and A. Ouahab, Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl. 338 (2008), 1340-1350. doi: 10.1016/j.jmaa.2007.06.021 Zbl1209.34096
- [9] M. Benchohra, S. Litimein and G. N'Guérékata, On fractional integro-differential inclusions with state-dependent delay in Banach spaces, Appl. Anal. 92 (2013), 335-350. doi: 10.1080/00036811.2011.616496
- [10] C. Chen and M. Li, On fractional resolvent operator functions Semigroup Forum. 80 (2010), 121-142. doi: 10.1007/s00233-009-9184-7 Zbl1185.47040
- [11] C-Cuevas and J-C. de Souza, S-asymptotically W-periodic solutions of semilinear fractional integro-differential equations, Appl. Math. Lett. 22 (2009), 865-870. doi: 10.1016/j.aml.2008.07.013
- [12] C-Cuevas and J-C. de Souza, Existence of S-asymptotically W-periodic solutions of fractional order functional integro-differential equations with infinite delay, Nonlinear Anal. 72 (2010), 1680-1689. doi: 10.1016/j.na.2009.09.007
- [13] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21593-8 Zbl1025.47002
- [14] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. Zbl0998.26002
- [15] A.A. Kilbas, Hari M. Srivastava and Juan J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006. Zbl1092.45003
- [16] V. Lakshmikantham, S. Leela and J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, 2009. Zbl1188.37002
- [17] A. Lasota and Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), 781-786. Zbl0151.10703
- [18] M. Li and Q. Zheng, On spectral inclusions and approximations of α-times resolvent families, Semigroup Forum. 69 (2004), 356-368. Zbl1096.47516
- [19] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993. Zbl0789.26002
- [20] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
- [21] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993.
- [22] V.E. Tarasov, Fractional Dynamics. Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Heidelberg, 2010. Zbl1214.81004