Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 17 (2022), 113 -- 138

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ASYMPTOTICALLY AUTOMORPHIC SOLUTIONS OF ABSTRACT FRACTIONAL EVOLUTION EQUATIONS WITH NON-INSTANTANEOUS IMPULSES

Noreddine Rezoug, Mouffak Benchohra and Khalil Ezzinbi

Abstract. In this paper, we study the existence of asymptotically automorphic mild solutions of fractional evolution equations with non-instantaneous impulses. The main results are based upon some properties of sectorial operators, and Krasnoselkii fixed point theorem. A new set of sufficient conditions is formulated which guarantees the existence of the solution of the non-instantaneous system. An example is also discussed to illustrate the efficiency of the obtained results.

2020 Mathematics Subject Classification: 34G20
Keywords: Asymptotically almost automorphic, abstract fractional equations, sectorial operator, fixed point theorem, non-instantaneous impulses.

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References

  1. S. Abbas, M. Benchohra, J.R. Graef and J. Henderson, Implicit Fractional Differential and Integral Equations: Existence and Stability, De Gruyter, Berlin, 2018. MR3791511. Zbl 1390.34002.

  2. S. Abbas, M. Benchohra, G. M. N'Guérékata, Topics in Fractional Differential Equations, Springer-Verlag, New York, 2012. MR2962045. Zbl 1273.35001.

  3. S. Abbas, M. Benchohra, G. M. N'Guérékata, Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2015. MR3309582. Zbl 1314.34002.

  4. R. P. Agarwal, S. Hristova, D. O'Regan, Non-Instantaneous Impulses in Differential Equations, Springer, New York, 2017. MR3726876. Zbl 1426.34001.

  5. A. Anguraj and S. Kanjanadevi, Existence results for fractional non-instantaneous impulsive integro-differential equations with nonlocal conditions, Dynam. Cont. Disc. Ser. A 23 (2016), 429-445. MR3592986. Zbl 1353.35301.

  6. D. Araya, C. Lizama, Almost automorphic mild solutions to fractional differential equations. Nonlinear Anal. 69 (2008), no. 11, 3692-3705. MR2463325. Zbl 1166.34033.

  7. D. Baleanu, Z. B. Güvenç, J. A. T. Machado, New Trends in Nanotechnology and Fractional Calculus Applications, Springer, New York, 2010. MR2605606. Zbl 1196.65021.

  8. M. Benchohra, J. Henderson and S. K. Ntouyas; Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, Vol. 2, New York, 2006. MR2322133. Zbl 1130.34003.

  9. S. Bochner, A new approach in almost-periodicity. Proc. Nat. Acad. Sci. USA 48 (1962), 2039-2043. MR0145283. Zbl 0112.31401.

  10. M. Benchohra. S. Litimein. Existence results for a new class of fractional integro-differential equations with state dependent delay. Mem. Differ. Equ. Math. Phys. 74 (2018), 27-38. 37, 16 pp.MR3832632. Zbl 1400.34127.

  11. T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces. Springer, Cham, 2013. MR3098423. Zbl 1279.43010.

  12. J. Cao, Z. Huang, G. M. N'Guérékata, Existence of asymptotically almost automorphic mild solutions for nonautonomous semilinear evolution equations. Electron. J. Differential Equations 2018, Paper No. 37, 16 pp. MR3762824. Zbl 1397.34101.

  13. C. Corduneanu, Integral Equations and Stability of Feedback Systems, Academic Press, New York, 1973. MR0358245. Zbl 0273.45001.

  14. E. Cuesta, Asymptotic bahaviour of the solutions of fractional integrodifferential equations and some time discretizations. Disc. Contin. Dyn. Syst. (Supplement)(2007), 277-285. MR2409222. Zbl 1126.35003.

  15. C. Cuevas, C. Lizama, Almost automorphic solutions to a class of semilinear fractional differential equations, Appl. Math. Lett. 21 (2008), 1315-1319. MR2464387. Zbl 1192.34006.

  16. T. Dianaga, G. M. N'Guérékata, Almost automorphic solutions to some classes of partial evolution equations, Appl. Math. Lett. 20 (2007), 462-466. MR2303379. Zbl 1169.35300.

  17. H. S. Ding, T. J. Xiao, J. Liang. Asymptotically almost automorphic solutions for some integrodifferential equations with nonlocal initial conditions. J. Math. Anal. Appl. 338 (2008), 141-151. MR2386405. Zbl 1142.45005.

  18. K. Ezzinbi, and G. M. N'Guérékata, Almost automorphic solutions for some partial functional differential equations. J. Math. Anal. Appl. 328 (1) (2007), 344-358. MR2138883. Zbl 1121.34081.

  19. J. A. Goldstein, G.M. N'Guérékata, Almost automorphic solutions of semilinear evolution equations, Proc. Amer. Math. Soc. 133 (2005), 2401-2408. MR2138883 Zbl 1073.34073.

  20. G. Ram Gautam and J. Dabas, Mild solutions for class of neutral fractional functional differential equations with not instantaneous impulses, Appl. Math. Comput. 259 (2015), 480-489. MR3338384. Zbl 1390.34221.

  21. E. Hernandez and D. O'Regan, On a new class of abstract impulsive differential equations. Proc. Amer. Math. Soc. 141(2013), 1641-1649. MR3151715. Zbl 1266.34101.

  22. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. MR1890104. Zbl 0998.26002.

  23. V. Kavitha, S. Abbas, R. Murugesu, Asymptotically almost automorphic solutions of fractional order neutral integro-differential equations. Bull. Malays. Math. Sci. Soc. 39 (2016), no. 3, 1075-1088. MR3515067. Zbl 1347.34114.

  24. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier, Amsterdam, 2006. MR2218073. Zbl 1092.45003.

  25. M. A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations. A Pergamon Press Book The Macmillan Co., New York 1964. MR0159197. Zbl 0111.30303.

  26. P. P. Kumar, N. Dwijendra, D. Bahuguna. On a new class of abstract impulsive functional differential equations of fractional order. J. Nonlinear Sci. Appl. 7(2014), 102-114. MR3151715. Zbl 11312.34117.

  27. J. Liang, J. Zhang, T. Xiao. Composition of pseudo almost automorphic and asymptotically almost automorphic functions. J. Math. Anal. Appl. 340 (2008), no. 2, 1493-1499. MR2390946. Zbl 1134.43001.

  28. L. Mahto, S. Abbas, PC-almost automorphic solution of impulsive fractional differential equations. Mediterr. J. Math. 12 (2015), no. 3, 771--790. MR3376811. Zbl 1325.34014.

  29. I. Mishra, D. Bahuguna, S. Abbas, Existence of almost automorphic solutions of neutral functional differential equation, Nonlinear Dyn. Syst. Theory 11 no. 2 (2011), 165-172. MR2817174. Zbl 1237.34135.

  30. G. M. N'Guérékata, Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer Academic, New York, London, Moscow, 2001. MR1880351. Zbl 1040.34068.

  31. G. M. N'Guérékata, Topics in Almost Automorphy, Springer, New York, Boston, Dordrecht, London, Moscow, 2005. MR2107829. Zbl 1073.43004.

  32. G. M. N'Guérékata, Spectral Theory for Bounded Functions and Applications to Evolution Equations. Mathematics Research Developments. Nova Science Publishers, Inc., New York, 2017. MR3676744. Zbl 1455.34001.

  33. G. M. N'Guérékata, A. Pankov, Integral operators in spaces of bounded, almost periodic and almost automorphic functions. Differ. Integral Equ. 21 (2008), 1155-1176. MR2482500. Zbl 1224.45021.

  34. M. D. Otigueira, Fractional Calculus for Scientists and Engineers, Lecture Notes in Electrical Engineering, 84, Springer, Dordrecht, 2011. MR2768178. Zbl 1251.26005.

  35. D. N. Pandey, S. Das and N. Sukavanam, Existence of solution for a second-order neutral differential equation with state dependent delay and non-instantaneous impulses, Int. J. Nonlinear Sci.18(2) (2014), 145-155. MR3271134. Zbl 1394.34167.

  36. M. Pierri, D. O'Regan, V. Rolnik. Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses. Appl. Math. Comput. 219 (2013), no. 12, 6743-6749. MR3432212. Zbl 1293.34019.

  37. A.M. Samoilenko and N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. MR1355787. Zbl 0837.34003.

  38. S. Suganya, D. Baleanu, P. Kalamani and M. Mallika Arjunan, On fractional neutral integro-differential systems with state-dependent delay and non-instantaneous impulses, Adv. Difference Equ. 2015(1) (2015), 1-39. MR3432212. Zbl 1422.34224.

  39. T. J. Xiao, X. X. Zhu, J. Liang. Pseudo-almost automorphic mild solutions to nonautonomous differential equations and applications. Nonlinear Anal. 70 (2009), no. 11, 4079-4085. MR2515324. Zbl 1175.34076.

  40. Z. Zhao, Y. Chang, J. Nieto, Almost automorphic and pseudo-almost automorphic mild solutions to an abstract differential equation in Banach spaces. Nonlinear Anal. Theory Methods Appl. 72 (2010), 1886-1894. MR2577585. Zbl 1232.34087.

  41. X. J. Zheng, C. Z. Ye, H. S. Ding, Asymptotically almost automorphic solutions to nonautonomous semilinear evolution equations. Afr. Diaspora J. Math. 12 (2011), no. 2, 104--112. MR2847307. Zbl 1241.430005.

  42. Y. Zhou. Basic Theory of Fractional Differential Equations. New Jersey: World Scientific, 2014.MR3287248. Zbl 1372.00017.



Noreddine Rezoug
Department of Mathematics, University of Relizane, Relizane, 48000, Algeria.
e-mail: noreddinerezoug@yahoo.fr

Mouffak Benchohra
Laboratory of Mathematics, University of Sidi Bel Abbès
PO Box 89, Sidi Bel Abbès 22000, Algeria.
e-mail: benchohra@univ-sba.dz

Khalil Ezzinbi
Faculty of Sciences Semlalia, Department of Mathematics, Cadi Ayyad University,
PO Box 2390, Marrakesh, Morocco.
e-mail: ezzinbi@uca.ma

http://www.utgjiu.ro/math/sma