Existence Criteria for Katugampola Fractional Type Impulsive Differential Equations with Inclusions

Yıl 2019, Cilt: 2 Sayı: 1, 51 - 63, 20.04.2019

Murugaiya Janaki , Kuppusamy Kanagarajan , Elsayed Mohammed Elsayed

https://doi.org/10.33187/jmsm.434266

Cited By: 1

Öz

In this paper, we consider the existence and uniqueness of solutions to the impulsive differential equations with inclusions involving Katugampola fractional derivative. With the help of properties of Katugampola fractional calculus and fixed point methods, we derive existence and uniqueness results. Finally, an example is given to illustrate our theoretical results.

Anahtar Kelimeler

Katugampola fractional derivative, Differential inclusion, Impulses; Existence, Uniqueness; Fixed point.

Kaynakça

• [1] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
• [2] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 2006.
• [3] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
• [4] R. P. Agarwal, M. Benchohra, S. Hamani, Boundary value problems for differential inclusions with fractional order, Adv. Cont. Math., 12(2) (2008), 181–196.
• [5] J. P. Aubin, A. Cellina, Differential Inclusions, Springer-Verlag, Berlin-Heidelberg, New York, 1984.
• [6] M. Benchohra, S. Hamani, Nonlinear boundary value problems for differential Inclusions with Caputo fractional derivative, Topol. Methods Nonlinear Anal., 321 (2008), 115–130.
• [7] K. Diethelm, N. J. Ford, Analysis of fractional differential equations, J. Math. Anal.Appl., 265 (2002), 229–248.
• [8] A. A. Kilbas, S. A. Marzan, Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions, J. Differential Equations, 41 (2005), 84–89.
• [9] D. Vivek, K. Kanagarajan, S. Harikrishnan, Theory and analysis of impulsive type pantograph equations with Katugampola fractional derivative, J. Vibration Testing and System Dynamics, 2(1) (2018), 9–20.
• [10] U. N. Katugampola, Existence and uniqueness results for a class of generalized fractional differential equations, (2016), arXiv:1411.5229v2[math.CA].
• [11] U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6(4) (2014), 1–15.
• [12] U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput., 218(3) (2011), 860–865.
• [13] M. Benchohra, B. A. Slimani, Impulsive fractional differential equations, Electron. J. Differ. Equ., 10 (2009), 1–11.
• [14] J. R. Graef, J. Henderson, A. Ouahab, Impulsive Differential Inclusions: A Fixed Point Approach, De Gruyter, Berlin/Boston, 2013.
• [15] J. Henderson, J. Ouahab, Impulsive differential inclusions with fractional order, Comput. Math. Appl., 59 (2010), 1191–1226.
• [16] V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
• [17] E. E. Ndiyo, Existence result for solution of second order impulsive differential inclusion to dynamic evolutionary process, Amer. J. Appl. Math., 7(2) (2017), 89–92.
• [18] M. Benchohra, J. Henderson, S. K. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, 2, New York, 2006.
• [19] M. Benchohra, J. J. Nieto, B. A. Slimani, Existence of solutions to differential inclusions with fractional order and impulses, Electron. J. Differ. Equ., 80 (2010), 1–18.
• [20] K. Deimling, Multivalued Differential Equations, Walter De Gruyter, Berlin-New York, 1992.
• [21] Sh. Hu, N. Papageorgiou, Handbook of Multivalued Analysis, Theory I, Kluwer, Dordrecht, 1997.
• [22] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer, Dordrecht, The Netherlands,1991.
• [23] H. Covitz, S. B. Nadler Jr, Multivalued contraction mappings in generalized metric spaces, Israel J. Math., 8 (1970), 5–11.
• [24] A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
• [25] A. Bressan, G. Colombo, Extensions and selections of maps with decomposable values, Studia Math., 90 (1988), 69–86.
• [26] C. Castaing, M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics 580, Springer-Verlag, Berlin-Heidelberg-New York, 1977.
• [27] A. Fryszkowski, Fixed Point Theory for Decomposable Sets, Topol. Fixed Point Theory Appl., 2. Kluwer Academic Publishers, Dordrecht, 2004.
• [28] M. Frigon, A. Granas, Theoremes dexistence pour des inclusions differentielles sans convexite, Comptes Rendus Acad. Sci., Paris, Ser. I, 310 (1990), 819–822.