Mild Solutions for Nonlocal Impulsive Fractional Semilinear Differential Inclusions with Delay in Banach Spaces

Ahmed Gamal Ibrahim, Nawal Abdulwahab Al Sarori

Mathematics Department, Faculty of Science, King Faisal University, Al-Ahsa, KSA.

DOI: 10.4236/am.2013.47A008                

Abstract

In this paper, we give various existence results concerning the existence of mild solutions for nonlocal impulsive differential inclusions with delay and of fractional order in Caputo sense in Banach space. We consider the case when the values of the orient field are convex as well as nonconvex. Our obtained results improve and generalize many results proved in recent papers.

Keywords

Fractional Differential Inclusions; Impulsive Semilinear Functional Differential Inclusions; The Infinitesimal Generator of a Semigroup; Nonlocal Conditions; Mild Solutions

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	W. H. Glocke and T. F. Nonnemacher, “A Fractional CalCulus Approach of Self-Similar Protein Dynamics,” Biophysical Journal, Vol. 68, No. 1, 1995, pp. 46-53. doi:10.1016/S0006-3495(95)80157-8
[2]	R. Hilfer, “Applications of Fractional Calculus in Physics,” World Scientific, Singapore City, 1999.
[3]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, “Theory and Applications of Fractional Differential Equations, in: North Holland Mathematics Studies, 204,” Elsevier Science, Publishers BV, Amsterdam, 2006.
[4]	K. S. Miller and B. Ross, “An Introduction to the Fractional Calculus and Differential Equations,” John Wiley, New York, 1993.
[5]	A. M. A. El-Sayed and A. G. Ibrahim, “Multivalued DifFerential Inclusions,” Applied Mathematics and Computation, Vol. 68, No. 1, 1995, pp. 15-25. doi:10.1016/0096-3003(94)00080-N
[6]	B. Ahmed, “Existence Results for Fractional Differential Inclusions with Separated Boundary Conditions,” Bulletin of the Korean Mathematical Society, Vol. 47, No. 4, 2010, pp. 805-813.
[7]	R. P. Agarwal, M. Benchohra and B. A. Slimani, “Existence Results for Differential Equations with Fractional Order and Impulses,” Mem. Differential Equations, Math. Phys., Vol. 44, 2008, pp. 1-21.
[8]	M. Belmekki and M. Benchohra, “Existence Results for Fractional Order Semilinear Functional Differential Equations with Nondense Domain,” Nonlinear Analysis, Vol. 72, No. 2, 2010, pp. 925-932. doi:10.1016/j.na.2009.07.034
[9]	A. Ouahab, “Fractional Semilinear Differential Inclusions,” Computer and Mathematics with Applications, Vol. 64, No. 10, 2012, pp. 3235-3252.
[10]	J.-R. Wang, M. Feckan and Y. Zhou, “On the New Concept of Solutions and Existence Results for Impulsive Fractional Evolutions,” Dynamics of PDE, Vol. 8, No. 4, 2011, pp. 345-361.
[11]	J. R. Wang and Y. Zhou, “Existence and Controllability Results for Fractional Semilinear Differential Inclusions,” Nonlinear Analysis: Real World Applications, Vol. 12, No. 6, 2011, pp. 3642-3653. doi:10.1016/j.nonrwa.2011.06.021
[12]	Z. Zhang and B. Liu, “Existence of Mild Solutions for Fractional Evolutions Equations,” Journal of Fractional Calculus and Applications, Vol. 2, No. 10, 2012, pp. 1-10.
[13]	Y. Zhou and F. Jiao, “Nonlocal Cauchy Problem for Fractional Evolution Equations,” Nonlinear Analysis: RWA, Vol. 11, No. 5, 2010, pp. 4465-4475.
[14]	Y. Zhou and F. Jiao, “Nonlocal Cauchy Problem for Fractional Natural Evolution Equations,” Computer and Mathematics with Applications, Vol. 59, No. 3, 2010, pp. 10631077. doi:10.1016/j.camwa.2009.06.026
[15]	Z. Agur, L. Cojocaru, G. Mazaur, R. M. Anderson and Y. L. Danon, “Pulse Mass Measles Vaccination across Age Shorts,” Proceedings of the National Academy of Sciences of the United States of America, Vol. 90, No. 24, 1993, pp. 11698-11702. doi:10.1073/pnas.90.24.11698
[16]	G. Ballinger and X. Liu, “Boundedness for Impulsive Delay Differential Equations and Applications in Populations Growth Models,” Nonlinear Analysis, Vol. 53, No. 7-8, 2003, pp. 1041-1062. doi:10.1016/S0362-546X(03)00041-5
[17]	A. D. Onofrio, “On Pulse Vaccination Strategy in the SIR Epidemic Model with Vertical Transmission,” Applied Mathematics Letters, Vol. 18, No. 7, 2005, pp. 729-732. doi:10.1016/j.aml.2004.05.012
[18]	M. Benchohra, J. Henderson and S. Ntouyas, “Impulsive Differential Equations and Inclusions,” Hindawi Publishing, New York, Egypt, 2007.
[19]	J. M. Ball, “Initial Boundary Value Problems for an Extensible Beam,” Journal of Mathematical Analysis and Applications, Vol. 42, No. 1, 1973, pp. 16-90. doi:10.1016/0022-247X(73)90121-2
[20]	W. E. Fitzgibbon, “Global Existence and Boundedness of Solutions to the Extensible Beam Equation,” SIAM Journal on Mathematical Analysis, Vol. 13, No. 5, 1982, pp. 739-745. doi:10.1137/0513050
[21]	R. A. Al-Omair and A. G. Ibrahim, “Existence of Mild Solutions of a Semilinear Evolution Differential Inclusions with Nonlocal Conditions,” EJDE, Vol. 42, 2009, pp. 1-11.
[22]	T. Cardinali and P. Rubbioni, “Impulsive Mild Solution for Semilinear Differential Inclusions with Nonlocal Conditions in Banach Spaces,” Nonlinear Analysis, Vol. 75, No. 2, 2012, pp. 871-879. doi:10.1016/j.na.2011.09.023
[23]	Z. Fan, “Impulsive Problems for Semilinear Differential Equations with Nonlocal Conditions,” Nonlinear Analysis, Vol. 72, No. 2, 2010, pp. 1104-1109. doi:10.1016/j.na.2009.07.049
[24]	G. M. Mophou, “Existence and Uniqueness of Mild Solution to Impulsive Fractional Differential Equations,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 72, No. 3-4, 2010, pp. 1604-1615.
[25]	O. K. Jaradat, A. Al-Omari and S. Momani, “Existence of the Mild Solution for Fractional Semi-Linear Initial Value problems,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 69, No. 1, 2008, pp. 3153-3159. doi:10.1016/j.na.2007.09.008
[26]	E. A. Ddas, M. Benchohra and S. Hamani, “Impulsive Fractional Differential Inclusions Involving the Caputo Fractional Derivative,” Fractional Calculus & Applied Analysis, Vol. 12, No. 1, 2009, pp. 15-36.
[27]	J. Henderson and A. Ouahab, “Impulsive Differential Inclusions with Fractional Order,” Computers & Mathematics with Applications, Vol. 59, No. 3, 2010, pp. 11911226. doi:10.1016/j.camwa.2009.05.011
[28]	A. Pazy, “Semigroups of Linear Operators and Applications to Partial Differential Equations,” Springer Verlag, New York, 1983.
[29]	J. P. Aubin and H. Frankoeska, “Set-Valued Analysis,” Birkhauser, Boston, Basel, Berlin, 1990.
[30]	C. Castaing and M. Valadier, “Convex Analysis and Measurable Multifunctions,” Lecture Notes in Mathematics, Springer Verlag, Berlin and New York, 1977.
[31]	S. Hu and N. S. Papageorgiou, “Handbook of Multivalued Analysis. Vol. I: Theory in Mathematics and Its Applications, Vol. 419,” Kluwer Academic Publisher, Dordrecht, 1979.
[32]	S. Hu and N. S. Papageorgiou, “Handbook of Multivalued Analysis. Vol. II: Theory in Mathematics and Its ApPlications, Vol. 500, Kluwer Academic Publisher, Dordrecht, 2000.
[33]	M. Kamenskii, V. Obukhowskii and P. Zecca, “Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces,” De Gruyter Series in Nonlinear Analysis and Applications 7, Walter, Berlin, 2001.
[34]	H. F. Bohnenblust and S. Karlin, “On a Theorem of Ville, in: Contribution to the Theory of Games,” Princeton University Press, Princeton, 1950, pp. 155-160.
[35]	H. Covitz and S. B. Nadler, “Multivalued Contraction Mapping in Generalized Metric Space,” Israel Journal of Mathematics, Vol. 8, No. 1, 1970, pp. 5-11.
[36]	A. Granass and J. Dugundij, “Fixed Point Theorems,” Springer-Verlag, New York, 2003.
[37]	F. Hiai and H. Umegaki, “Integrals, Conditional Expectation, and Martingales of Multivalued Functions,” Journal of Multivariate Analysis, Vol. 7, No. 1, 1977, pp. 149182.
[38]	A. Bressan and G. Coombo, “Extensions and Selections of Maps with Decomposable Values,” Studia Mathematica, Vol. 90, No. 1, 1988, pp. 69-86.