Existence and controllability of fractional evolution inclusions with impulse and sectorial operator

Year 2022, Volume: 5 Issue: 3, 235 - 249, 30.09.2022

Nawal Alsarori , Kirtiwant Ghadle

https://doi.org/10.53006/rna.1018780

Abstract

Many evolutionary operations fromdiverse fields of engineering and physical sciences go through
abrupt modifications of state at specific moments of time among periods of non-stop evolution.
These operations are more conveniently modeled via impulsive differential equations and inclusions.
In this work, firstly we address the existence of mild solutions for nonlocal fractional impulsive
semilinear differential inclusions related to Caputo derivative in Banach spaces when the
linear part is sectorial. Secondly, we determine the enough, conditions for the controllability of
the studied control problem. We apply effectively fixed point theorems, contraction mapping,
multivalued analysis and fractional calculus. Moreover, we enhance our results by introducing an
illustrative examples.

Keywords

Controllability, Contraction mapping, Impulsive fractional differential inclusions, Fixed point theorem, Sectorial operators

References

• [1] N. Abada, M. Benchohra and H. Hammouche, Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions, J. Differential Equations 10 (2009), 3834-3863
• [2] R. P. Agarwal, S. Baghli and M. Benchohra, Controllability for semilinear functional and neutral functional evolution equations with infinite delay in Fréchet spaces, Appl.Math. Optim., 60 (2009), 253-274.
• [3] D. Aimene, D. Baleanu and D. Seba, Controllability of semilinear impulsive Atangana-Baleanu fractional differential equations with delay, Chaos, Solitons and Fractals, 128 (2019), 51-57.
• [4] D. Aimene, D. Seba and K. Laoubi, Controllability of impulsive fractional functional evolution equations with infinite state-dependent delay in Banach spaces, Math Meth Appl Sci. (2019), 116.
• [5] N. A. Alsarori, K. P. Ghadle, On the mild solution for nonlocal impulsive fractional semilinear differential inclusion in Banach spaces, J.Math.Modeling, 2 (2018), 239-258 .
• [6] N. A. Alsarori, K. P. Ghadle, Differential inclusion of fractional order with Impulse effects in Banach spaces, Nonlinear Functional Analysis and Applications, 1 (2020), 101-116.
• [7] N. Alsarori, K. Ghadle, S. Sessa, Saleh, S. Alabiad, New Study of Existence and Dimension of the Set of Solutions for Nonlocal Impulsive Differential Inclusions with Sectorial Operator. Symmetry, (2021), 13, 491.
• [8] J. P. Aubin, H. Frankoeska, Set-valued Analysis, Birkhäuser, Boston, Basel, Berlin (1990).
• [9] E. Bajlekova, Fractional evolution equations in Banach spaces (Ph.D. thesis), Eindhoven University of Technology, 2001.
• [10] H.F. Bohnenblust, S. Karlin, On a Theorem of Ville, in: Contributions to the Theory of Games, vol. I, Princeton University Press, Princeton, NJ, (1950), 155160.
• [11] 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.
• [12] C. Castaing, M. Valadier : Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics 580. Springer- Verlag, Berlin (1977).
• [13] D. N. Chalishajar and F. S. Acharya, Controllability of Neutral Impulsive Differential Inclusions with Non-Local Conditions; AppliedMathematics, 2 (2011) 1486-1496.
• [14] H. Covitz and S. B.Nadler, Multi-valued contraction mappings in generalized metric spaces, Israel Journal ofMathematics, vol. 8, pp. 511, 1970.
• [15] Z. Fan, Impulsive problems for semilinear differential equations with nonlocal conditions, Nonlinear Anal., 72 (2010) 1104-1109.
• [16] M. Haase, The Functional Calculus for Sectorial Operators, of Operator Theory: Advances and Applications, Birkhauser, Basel, Switzerland, (2006).
• [17] Q. Haiyong, Z. Chenghui, Li. Tongxing and C. Ying, Controllability of abstract fractional differential evolution equations with nonlocal conditions; Journal ofMathematics and Computer Science, 17 (2017) 293-300.
• [18] 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.
• [19] F. Hiai, H. Umegaki, Integrals, conditional expectation, and martingales of multivalued functions, J. ofMultivariate Analysis, 7 (1977), 149-182.
• [20] R. Hilfer, Applications of Fractional Calculus in Physics,World Scientific, Singapore (1999).
• [21] S. Hu, N. Papageorgiou, Handbook ofMultivalued Analysis, Theory I. Kluwer, Dordrecht. (1997).
• [22] 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).
• [23] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, in:North Holland Mathematics Studies, 204. Elsevier Science. Publishers BV, Amesterdam (2006).
• [24] 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.
• [25] K. Li, J. Peng, J. Gao, Nonlocal fractional semilinear differential equations in separable Banach spaces. Electron. J. Differ. Equ., 7 (2013).
• [26] T. Lian, C. Xue, S. Deng, Mild solution to fractional differential inclusions with nonlocal conditions, Boundary Value problems, (2016) 2016:219.
• [27] Z. Liu and X. Li, Approximate controllability of fractional evolution systems with RiemannLiouville fractional derivatives. SIAM J. Control Optim., 53,(2015) 19201933.
• [28] Y. Luo, Existence for Semilinear Impulsive Differential InclusionsWithout Compactness, Journal of Dynamical and Control Systems, (2020) 26:663672.
• [29] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations John Wiley, New York (1993).
• [30] A. Ouahab, Fractional semilinear differential inclusions, Comput. Math. Appl. 64 (2012) 32353252.
• [31] 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.
• [32] X. B. Shu and Y. Shi, A study on the mild solution of impulsive fractional evolution equations; AppliedMathematics and Computation 273 (2016), 465-476.
• [33] 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.
• [34] J.Wang, Y. Zhou, Existence and controllability results for fractional semilinear differential inclusions, Nonlinear Anal., RealWorld Appl., 12 (2011), 3642-3653.
• [35] J. Wang, M. Feckan and Y. Zhou, Controllability of Sobolev type fractional evolution systems. Dyn. Partial Differ. Equ., 11 (2014), 7187.
• [36] Z. Yan and F. Lu, The optimal control of a new class of impulsive stochastic neutral evolution integro differential equations with infinite delay. Int. J. Control, 89 (2016), 15921612.
• [37] D. Zhang, Y. Liang, Existence and controllability of fractional evolution equation with sectorial operator and impulse, Adv. in Diff. Eq., (2018) 2018:219.