New results for infinite functional differential inclusions with impulses effect and sectorial operators in Banach spaces

Year 2021, Volume: 5 Issue: 3, 382 - 392, 30.09.2021

Nawal Alsarori , Kirtiwant Ghadle

https://doi.org/10.31197/atnaa.844234

Abstract

This article aims to use Bohnenblust Karlin’s fixed point theorem to obtain new results for the impulsive
inclusions with infinite delay in Banah space given by the form
(P)
8><
>:
cD®
t x(t )¡ Ax(t ) 2 F(t ,xt ), t 2 J , t 6Æ ti ,
¢x(ti ) Æ Ii (x(t¡
i )), i Æ 1, ...,m,
x(t ) ƪ(t ), t 2 (¡1,0].
where cD® is theCaputo derivative. We examine the casewhen themultivalued function F is an upperCarathéodory
and the linear part is sectorial operator defined on Banach space. Also, we provide an example to elaborate the
outcomes.

Keywords

Fractional impulsive differential inclusions, Nonlocal conditions, Fixed point theorems

References

• [1] R. P. Agarwal, B. Ahmad, A. Alsaedi and N. Shahzad; Existence and dimension of the set of mild solutions to semilinear fractional differential inclusions, Advances in Difference Equations 2012, 2012:74.
• [2] K. Aissani and M. Benchohra, Impulsive fractional differential inclusions with infinite delay, Electron. J. Differential Equations, 2013 (265) (2013), 1-13.
• [3] N. A. Alsarori, K. P. Ghadle, On the mild solution for nonlocal impulsive fractional semilinear differential inclusion in Banach spaces, J.Math.Modeling, Vol. 6, No. 2, 2018, pp. 239-258 .
• [4] N. A. Alsarori, K. P. Ghadle, Differential inclusion of fractional order with Impulse effects in Banach spaces, Nonlinear Functional Analysis and Applications, Vol. 25, No. 1 (2020), pp. 101-116.
• [5] Z. Agur, L. Cojocaru, G. Mazaur, R. M. Anderson, Y. L. Danon, Pulse mass measles vaccination across age shorts, Proc. Natl. Acad. Sci. USA, 90 (1993) 11698-11702.
• [6] R. Bader, M. Kamenskii, V. Obukhowskii, On some class of operator inclusions with lower semicontinuous nonlinearities, nonlinear Analysis. Journal of the Juliusz schauder center 17 (2001), 143-156.
• [7] E. Bajlekova, Fractional evolution equations in Banach spaces (Ph.D. thesis), Eindhoven University of Technology, 2001.
• [8] J. M. Ball, Initial boundary value problems for an extensible beam, J.Math. Anal. Appl., 42 (1973) 16-90.
• [9] G. Ballinger, X. Liu, Boundedness for impulsive delay differential equations and applications in populations growth models, Nonlinear Anal., 53 (2003) 1041-1062.
• [10] J. Banas, K. Goebel, Measure of Noncompactness in Banach Spaces. Lect. Notes Pure Appl. Math., vol. 60. Dekker, New York (1980).
• [11] M. Benchohra, J. Henderson, S. Ntouyas, Impulsive Differential Equations and Inclusions Hindawi, Philadelphia (2007).
• [12] D. Bothe,Multivalued perturbation of m-accerative differential inclusions, Isreal J.Math., 108 (1998) 109-138.
• [13] A. Bressan, G. Coombo, Extensions and selections of maps with decomposable values, Studia Mathematica, Vol.39.(2000)117-126.
• [14] L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J.Math. Anal. Appl., 162 (1991) 494-505.
• [15] T. Cardinali, P. Rubbioni, Impulsive mild solution for semilinear differential inclusions with nonlocal conditions in Banach spaces, Nonlinear Anal., 75 (2012) 871-879.
• [16] E. A. Ddas, M. benchohra, S. hamani, Impulsive fractional differential inclusions involving The Caputo fractional derivative, Fractional Calculus and Applied Analysis, 12 (2009) 15-36.
• [17] Z. Fan, Impulsive problems for semilinear differential equations with nonlocal conditions, Nonlinear Anal., 72 (2010) 1104-1109.
• [18] W. E. Fitzgibbon, Global existence and Boundedness of solutions to the extensible beam equation, SIAM J. Math. Anal., 13(5) (1982) 739-745.
• [19] W. H. Glocke, T. F. Nonnemacher, A fractional calculus approach of self-similar protein dynamics, Biophys. J.68 (1995) 46-53.
• [20] A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
• [21] H. R. Heinz, On the Behavior of measure of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal., 7 (1983) 1351-1371.
• [22] J. Henderson, A. Ouahab, Impulsive differential inclusions with fractional order, Compu. Math. with Appl., 59 (2010) 1191-1226.
• [23] F. Hiai, H. Umegaki, Integrals, conditional expectation, and martingales of multivalued functions, J. ofMultivariate Analysis,Vol 7(1977)149-182.
• [24] R. Hilfer, Applications of Fractional Calculus in Physics,World Scientific, Singapore (1999).
• [25] A. G. Ibrahim, N. A. Alsarori, Mild solutions for nonlocal impulsive fractional semilinear differential inclusions with delay in Banach spaces, AppliedMathematics, 4 (2013) 40-56.
• [26] O. K. Jaradat, A. Al-Omari, S. Momani, Existence of the mild solution for fractional semi-linear initial value problems, Nonlinear Anal. TMA, 69 (2008) 3153-3159.
• [27] M. Kamenskii, V. Obukhowskii , P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter Saur. Nonlinear Anal. Appl.,Walter Berlin-New 7 (2001).
• [28] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, in:North HollandMathematics Studies, 204. Elsevier Science. Publishers BV, Amesterdam (2006).
• [29] A. Lasota, 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.
• [30] K. Li, J. Peng, J. Gao, Nonlocal fractional semilinear differential equations in separable Banach spaces. Electron. J. Differ. Equ., 7 (2013).
• [31] T. Lian, C. Xue, S.Deng,Mild solution to fractional differential inclusionswith nonlocal conditions, Boundary Value problems, (2016) 2016:219.
• [32] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations John Wiley, New York (1993).
• [33] G. M.Mophou, Existence and uniquness of mild solution to impulsive fractional differential equations, Nonlinear Anal.TMA, 72 (2010) 1604-1615.
• [34] A.D.Onofrio, On pulse vaccination strategy in the SIR epidemic modelwith vertical transmission, Appl. Lett., 18 (2005) 729-732.
• [35] A. Ouahab, Fractional semilinear differential inclusions, Comput.Math. Appl. 64 (2012) 32353252.
• [36] A. Pazy, Semigroups of linear operators and applications to partial differential equations, AppliedMathematical Sciences, Springer-Verlag, New York, (1983).
• [37] X. B. Shu, Y. Z. Lai, Y. Chen; The existence of mild solutions for impulsive fractional partial differential equations, Nonlinear Anal. TMA 74 (2011), 2003-2011.
• [38] J. Wang, M. Feckan, Y. Zhou, On the new concept of solutions and existence results for impulsive fractional evolutions, Dynamics of PDE, Vol 8, No.4 (2011) 345-361.
• [39] J. Wang, Y. Zhou, Existence and controllability results for fractional semilinear differential inclusions, Nonlinear Anal., RealWorld Appl., 12 (2011) 3642-3653.
• [40] Y. Zhou, F. Jiao, Nonlocal Cauchy problem for fractional netural evolution equations, Compu.Math. Appl., 59 (2010) 1063-1077.