Mittag-Leffler stability for non-instantaneous impulsive Caputo fractional differential equations with delays

References

[1] Abbas, S.—Benchohra, M.: Stability results for fractional differential equations with state-dependent delay and not instantaneous impulses, Math. Slovaca 67(4) (2017), 875–894.10.1515/ms-2017-0017Search in Google Scholar

[2] Agarwal, Ravi—O’regan, D.—Hristova, S.: Stability by Lyapunov functions of nonlinear differential equations with non-instantaneous impulses, J. Appl. Math. Comput. 53(2) (2015), 147–168.10.1007/s12190-015-0961-zSearch in Google Scholar

[3] Agarwal, R.—Hristova, S.—O’regan, D.: A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations, Fract. Calc. Appl. Anal. 19(2) (2016), 290–318.10.1515/fca-2016-0017Search in Google Scholar

[4] Anguraj, A.—Kanjanadevi, S.: Non-instantaneous impulsive fractional neutral differential equations with state-dependent delay, Progr. Fract. Differ. Appl. 3(3) (2017), 207–218.10.18576/pfda/030304Search in Google Scholar

[5] BAI, L.—Nieto J. J.: Variational approach to differential equations with not instantaneous impulses, Appl. Math. Lett. 73 (2017), 44–48.10.1016/j.aml.2017.02.019Search in Google Scholar

[6] Bai, L.—Nieto, J. J.—Wang, X.: Variational approach to non-instantaneous impulsive nonlinear differential equations, J. Nonlinear Sci. Appl. 10 (2017), 2440–2448.10.22436/jnsa.010.05.14Search in Google Scholar

[7] Chen Y. Q.: Ubiquitous fractional order controls? IFAC Proceedings Volumes 39(11) (2006), 481–492.10.3182/20060719-3-PT-4902.00081Search in Google Scholar

[8] Colao, V.—Mugla, L.—Xu, H.-K.: Existence of solutions for a second-order differential equation with noninstantaneous impulses and delay, Ann. Mat. Pura Appl. 19 (2016), 697–716.10.1007/s10231-015-0484-0Search in Google Scholar

[9] Das, Sh.: Functional Fractional Calculus, Springer-Verlag Berlin Heidelberg, 2011.10.1007/978-3-642-20545-3Search in Google Scholar

[10] Diethelm, K.: The Analysis of Fractional Differential Equations, Springer-Verlag Berlin Heidelberg, 2010.10.1007/978-3-642-14574-2Search in Google Scholar

[11] Devi, J. V.—Mc Rae, F. A.—Drici, Z.: Variational Lyapunov method for fractional differential equations, Comput. Math. Appl. 64 (2012), 2982–2989.10.1016/j.camwa.2012.01.070Search in Google Scholar

[12] Fečkan, M.—Zhou, Y.—Wang, J.: On the concept and existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 3050–3060.10.1016/j.cnsns.2011.11.017Search in Google Scholar

[13] Hammouch, Z.—Mekkaoui, T.: Control of a new chaotic fractional-order system using Mittag-Leffler stability, Nonlinear Stud. 22(4) (2015), 1–13.Search in Google Scholar

[14] Hernansez, E.—O’regan, D.: On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc. 141 (2013), 1641–1649.10.1090/S0002-9939-2012-11613-2Search in Google Scholar

[15] Hristova, S.—Stamova, I.: On the Mittag-Leffler stability of impulsive fracional neural networks with finite delays, Int. J. Pure Appl. Math. 109(1) (2016), 105–117.10.12732/ijpam.v109i1.8Search in Google Scholar

[16] Lakshmikantham, V.—Bainov, D.—Simeonov, P.: Theory of Impulsive Differential Equations, World Scientiffic, Singapore, 1989.10.1142/0906Search in Google Scholar

[17] Li, Y.—Xu, Ch.: Boundary value problems of fractional order differential equation with integral boundary conditions and not instantaneous impulses, J. Funct. Spaces 2015 (2015), Art. ID 954925.10.1155/2015/954925Search in Google Scholar

[18] Pandey, D. N.—Das, S.—Sukavanam, N.: Existence of solutions for a second order neutral differential equation with state dependent delay and not instantaneous impulses, Int. J. Nonlinear Sci. 18(2) (2014), 145–155.Search in Google Scholar

[19] Li, Y.—Chen, Y.—Podlubny, I.: Mittag-Leffler stability of fractional order nonlinear dynamic systems, Automatica 45 (2009), 1965–1969.10.1016/j.automatica.2009.04.003Search in Google Scholar

[20] Podlubny, I.: Fractional Differential Equations, Academic Press: San Diego, 1999.Search in Google Scholar

[21] Sadati, S. J.—Baleanu, D.—Ranjbar, A.—Ghaderi, R.—AbbeljawaD(Maraaba), R.: Mittag-Leffler stability theorem for fractional nonlinear systems with delay, Abstr. Appl. Anal. 2010 (2010), Art. ID 108651, 7 pp.10.1155/2010/108651Search in Google Scholar

[22] Samoilenko, A. M.—Perestyuk, N. A.: Impulsive Differential Equations, World Scientiffic, Singapore, 1995.10.1142/2892Search in Google Scholar

[23] Sayavand, K.: A Study on existence and global asymptotical Mittag-Leffler stability of fractional Black-Sholes europian option pricing equation, J. Hyperstruct. 3(2) (2014), 126–138.Search in Google Scholar

[24] Stamova, I.—Stamov, G.: Functional and Impulsive Differential Equations of Fractional Order: Qualitative Analysis and Applications, CRC Press, 2017.10.1201/9781315367453Search in Google Scholar

[25] Suganya, S.—Baleanu, D.—Kalamani, P. et al.: On fractional neutral integro-differential systems with state-dependent delay and non-instantaneous impulses, Adv. Difference Equ. 2015:372 (2015).10.1186/s13662-015-0709-ySearch in Google Scholar

[26] Wang, J.—Fečkan, M.: A general class of impulsive evolution equations, Topol. Methods Nonlinear Anal. 46 (2015), 915–933.10.12775/TMNA.2015.072Search in Google Scholar

[27] Wang, J.—Fečkan, M.—Zhou, Y.: A survey on impulsive fractional differential equations, Fract. Calc. Appl. Anal. 19(4) (2016), 806–831.10.1515/fca-2016-0044Search in Google Scholar

[28] Wang, J.—LI, X.: A Uniform method to UlamHyers stability for some linear fractional equations, Mediterr. J. Math. 13 (2016), 625–635.10.1007/s00009-015-0523-5Search in Google Scholar

[29] Wang, J.—Zhang, Y.: Ulam-Hyers-Mittag-Leffler stability of fractional-order delay differential equations, Optimization 63(8) (2014), 1181–1190.10.1080/02331934.2014.906597Search in Google Scholar

[30] Yang, D.—Wang, J.—O’regan, D.: A class of nonlinear non-instantaneous impulsive differential equations involving parameters and fractional order, Appl. Math. Comput. 321 C (2018), 654–671.10.1016/j.amc.2017.11.025Search in Google Scholar

[31] Yang, D.—Wang, J.—O’regan, D.: On the orbital Hausdorff dependence of differential equations with non-instantaneous impulses, Comptes Rendus Mathematique, Ser.I 356(2) (2018), 150–171.10.1016/j.crma.2018.01.001Search in Google Scholar