Abstract
We establish a set of sufficient conditions for the existence of mild solution of a class of fractional mixed integro differential equation with not instantaneous impulses. The results are obtained by establishing two theorems by using semigroup theory, Banach fixed point theorem and Krasnoselskii’s fixed point theorem. Two examples are presented to validate the results of the theorems.
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Agarwal, R., Hristova, S., O’Regan, D.: Non-instantaneous impulses in caputo fractional differential equations. Fract. Calc. Appl. Anal. 20, 595–622 (2017)
Agarwal, R., O’Regan, D., Hristova, S.: Monotone iterative technique for the initial value problem for differential equations with non-instantaneous impulses. Appl. Math. Compt. 298, 45–56 (2017)
Anguraj, A., Latha Maheswari, M.: Existence of solutions for fractional impulsive neutral functional infinite delay integro-differential equations with nonlocal conditions. J. Nonlinear Sci. Appl. 20, 271–280 (2017)
Bai, L., Nieto, J.J.: Variational approach to differential equations with not instantaneous impulses. Appl. Math. Lett. 73, 44–48 (2017)
Bai, L., Nieto, J.J., Wang, X.: Variational approach to non-instantaneous impulsive nonlinear differential equations. J. Nonlinear Sci. Appl. 10, 2440–2448 (2017)
Bragdi, M., Hazi, M.: Existence and uniqueness of solutions of fractional quasilinear mixed integro-differential equations with nonlocal condition in Banach spaces. Electron. J. Qual. Theory Differ. Equ. 2012(51), 1–16 (2012)
Chang, Y.K., Kavitha, V., Arjunan, M.M.: Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order. Nonlinear Anal. 71(11), 5551–5559 (2009)
El-Sayed, A.M.A.: Fractional order evolution equations. J. Fract. Calc. 7(1), 995 (1995)
Fec, M., Zhou, Y., Wang, J.R.: On the concept and existence of solution for impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 17(7), 3050–3060 (2012)
Gautam, G.R., Dabas, J.: Mild solution for fractional functional integro-differential equation with not instantaneous impulse. Malaya J. Math. 2(3), 428–437 (2012)
Hernández, E., O’Regan, D.: On a new class of abstract impulsive differential equations. Proc. Am. Math. Soc. 141(5), 1641–1649 (2013)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Kumar, P., Pandey, D.N., Bahuguna, D.: On a new class of abstract impulsive functional differential equations of fractional order. J. Nonlinear Sci. Appl. 7(2), 102–114 (2014)
Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, Amsterdam (1993)
Mophou, Gisèle M.: Existence and uniqueness of mild solutions to impulsive fractional differential equations. Nonlinear Anal. 72(3), 1604–1615 (2010)
N’Guérékata, G.M.: A Cauchy problem for some fractional abstract differential equation with non local conditions. Nonlinear Anal. 70(5), 1873–1876 (2009)
Pierri, M., O’Regan, D., Rolnik, V.: Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses. Appl. Math. Comput. 219(12), 6743–6749 (2013)
Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198. Academic Press, New York (1998)
Ravichandran, C., Arjunan, M.M.: Existence results for abstract mixed type impulsive fractional semilinear evolution equations. Int. J. Math. Sci. Comput. 2(1), 14–21 (2012)
Samoilenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations, vol. 14. World Scientific, Singapore (1995)
Suganya, S., Baleanu, D., Kalamani, P., Arjunan, M.M.: On fractional neutral integro-differential systems with state-dependent delay and non-instantaneous impulses. Adv. Differ. Equ. 2015(1), 372 (2015)
Wang, J.R., Fečkan, M., Zhou, Y.: On the new concept of solutions and existence results for impulsive fractional evolution equations. Dyn. Partial Differ. Equ. 8(4), 345–361 (2011)
Wang, J.R., Zhou, Y., Lin, Z.: On a new class of impulsive fractional differential equations. Appl. Math. Comput. 242, 649–657 (2014)
Wang, J.R., Zhou, Y., Wei, W., Xu, H.: Nonlocal problems for fractional integrodifferential equations via fractional operators and optimal controls. Comput. Math. Appl. 62(3), 1427–1441 (2011)
Zhou, Y., Jiao, F.: Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl. 59(3), 1063–1077 (2010)
Zhou, Y., Wang, J.R., Zhang, L.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2016)
Acknowledgements
The first author is grateful to North Eastern Regional Institute of Science and Technology, Nirjuli, Arunachal Pradesh, India for granting leave for three years to pursue PhD and to Indian Institute of Technology Guwahati, Guwahati, India for providing opportunity to carry out research. Both authors express their gratitude to the esteemed reviewers for their careful reading of the manuscript and the insightful comments, and also to the Editor for allowing revision of the manuscript which has now definitely reached a much better form.
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Borah, J., Bora, S.N. Existence of Mild Solution for Mixed Volterra–Fredholm Integro Fractional Differential Equation with Non-instantaneous Impulses. Differ Equ Dyn Syst 30, 185–196 (2022). https://doi.org/10.1007/s12591-018-0410-1
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DOI: https://doi.org/10.1007/s12591-018-0410-1