In this paper, we concern on the periodic boundary value problem for a class of semilinear impulsive fractional evolution equations in an ordered Banach space E. First, we establish the existence results of mild solutions for the associated linear periodic boundary value problem. Next, we obtain the existence results of mild solutions by using the monotone iterative technique with L-quasi-upper and lower solutions, the results are new and extend some previously known results. Finally, two examples are also given to illustrate the main results.
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Agarwal, R.P., Benchohra, M., Hamani, S.: A survey on existence result for boundary value problem of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109, 973–1033 (2010)
Ahmad, B., Sivasundaram, S.: Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations. Nonlinear Anal. Hybrid Syst. 3, 251–258 (2009)
Zhou, Y., Jiao, F.: Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl. 59, 1063–1077 (2010)
Chen, P., Li, Y., Chen, Q., Feng, B.: On the initial value problem of fractional evolution equations with noncompact semigroup. Comput. Math. Appl. 67, 1108–1115 (2014)
Bai, Z., Lü, H.: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 311, 495–505 (2005)
Agarwal, P., Al-Mdallal, Q.M., Cho, Y.J., Jain, S.: Fractional differential equations for the generalized Mittag-Leffler function. Adv. Differ. Equ. 2018, 58 (2018)
Sitho, S., Ntouyas, S.K., Agarwal, P., Tariboon, J.: Noninstantaneous impulsive inequalities via conformable fractional calculus. J. Inequal. Appl. 2018, 261 (2018)
Agarwal, P., El-Sayed, A.A.: Non-standard finite difference and Chebyshev collocation methods for solving fractional diffusion equation. Phys. A: Stat. Mech. Appl. 500, 40–49 (2018)
Li, Y., Liu, Z.: Monotone iterative technique for addressing impulsive integro-differential equations in Banach spaces. Nonlinear Anal. 66, 83–92 (2007)
Mu, J., Li, Y.X.: Monotone iterative technique for impulsive fractional evolution equations. J. Inequal. Appl. 2011, 125 (2011)
Du, S., Lakshmikantham, V.: Monotone iterative technique for differential equtions in Banach spaces. J. Anal. Math. Anal. 87, 454–459 (1982)
Zhao, J., Wang, R.: Mixed monotone iterative technique for fractional impulsive evolution equations. Miskolc Math. Notes 17(1), 683–696 (2016)
Chen, P., Mu, J.: Monotone iterative method for semilinear impulsive evolution equations of mixed type in Banach spaces. Electron. J. Differ. Equ. 2010(149), 1–13 (2010)
Chen, P., Li, Y.: Mixed monotone iterative technique for a class of semilinear impulsive evolution equations in Banach spaces. Nonlinear Anal. 74, 3578–3588 (2011)
Li, B.L., Gou, H.D.: Monotone iterative method for the periodic boundary value problems of impulsive evolution equations in Banach spaces. Chaos Solitons Fractals 110, 209–215 (2018)
Wang, L., Wang, Z.: Monotone iterative technique for parameterized BVPs of abstract semilinear evolution equations. Comput. Math. Appl. 46, 1229–1243 (2003)
EI-Gebeily, M.A., ORegan, D., Nieto, J.J.: A monotone iterative technique for stationary and time dependent problems in Banach spaces. J. Comput. Appl. Math. 233, 2359–2404 (2010)
Wang, J., Fec̆kan, M., Zhou, Y.: On the new concept of solutions and existence results for impulsive fractional evolution equations. Dyn. Partial Differ. Equ. 8, 345–361 (2011)
Mu, J.: Extremal mild solutions for impulsive fractional evolution equations with nonlocal initial conditions. Bound. Value Probl. 2012, 71 (2012)
Li, Y.X.: The positive solutions of abstract semilinear evolution equations and their applications. Acta Math. Sin. 39(5), 666–672 (1996). (in Chinese)
Guo, D.J., Sun, J.X.: Ordinary Differential Equations in Abstract Spaces. Shandong Science and Technology, Jinan (1989). (in Chinese)
Deiling, K.: Nonlinear Functional Analysis. Springer, New York (1985)
Ye, H., Gao, J., Ding, Y.: A generalized Gronwall inequality and its applications to a fractional differential equation. J. Math. Anal. Appl. 328, 1075–1081 (2007)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)
The authors would like to thank the referees for their useful suggestions which have significantly improved the paper.
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This work is supported by the National Natural Science Foundation of China (Grant No. 11661071).
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Gou, H., Li, Y. Existence of mild solutions for impulsive fractional evolution equations with periodic boundary conditions. J. Pseudo-Differ. Oper. Appl. 11, 425–445 (2020). https://doi.org/10.1007/s11868-019-00278-2
- Monotone iterative technique
- Coupled L-quasi-upper and lower solutions
- Periodic boundary conditions
- Analytic semigroup
Mathematics Subject Classification