Asymptotic properties of the solutions of nonlinear non-instantaneous impulsive differential equations

Abstract In this article, we investigate asymptotic properties of solutions, continuous dependence and stability, of integer order and fractional order nonlinear non-instantaneous impulsive differential equations (IDEs). We introduce the concept of continuous dependence and stability of solutions to integer order and fractional order non-instantaneous impulsive Cauchy problems (ICPs) and establish sufficient conditions to guarantee that the solutions of both the original and the perturbed non-instantaneous ICPs are close to each other in a certain sense. Finally, examples are given to illustrate our results.

[1]  JinRong Wang,et al.  On a new class of impulsive fractional differential equations , 2014, Appl. Math. Comput..

[2]  Martin Bohner,et al.  Impulsive differential equations: Periodic solutions and applications , 2015, Autom..

[3]  Michal Fečkan,et al.  Stability Analysis for a General Class of Non-instantaneous Impulsive Differential Equations , 2017 .

[4]  Mouffak Benchohra,et al.  Uniqueness and Ulam stabilities results for partial fractional differential equations with not instantaneous impulses , 2015, Appl. Math. Comput..

[5]  Donal O'Regan,et al.  On abstract differential equations with non instantaneous impulses , 2015 .

[6]  Donal O'Regan,et al.  On a new class of abstract impulsive differential equations , 2012 .

[7]  Michal Fečkan,et al.  Random Noninstantaneous Impulsive Models for Studying Periodic Evolution Processes in Pharmacotherapy , 2016 .

[8]  Michelle Pierri,et al.  Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses , 2013, Appl. Math. Comput..

[9]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[10]  Michal Fečkan,et al.  A survey on impulsive fractional differential equations , 2016 .

[11]  Jaydev Dabas,et al.  Mild solutions for class of neutral fractional functional differential equations with not instantaneous impulses , 2015, Appl. Math. Comput..

[12]  D. O’Regan,et al.  Stability by Lyapunov functions of Caputo fractional differential equations with non-instantaneous impulses , 2015, 1512.05772.

[13]  A. Samoilenko,et al.  Impulsive differential equations , 1995 .

[14]  Donal O'Regan,et al.  Nonautonomous impulsive systems with unbounded nonlinear terms , 2014, Appl. Math. Comput..

[15]  Xianhua Tang,et al.  Sign-changing solutions for asymptotically linear Schr\"odinger equation in bounded domains , 2016 .

[16]  Yong-sheng Ding,et al.  A generalized Gronwall inequality and its application to a fractional differential equation , 2007 .

[17]  JinRong Wang,et al.  On the iterative learning control for stochastic impulsive differential equations with randomly varying trial lengths , 2017, J. Comput. Appl. Math..

[18]  Xuping Zhang.  EXISTENCE OF MILD SOLUTIONS TO PARTIAL DIFFERENTIAL EQUATIONS WITH NON-INSTANTANEOUS IMPULSES , 2016 .

[19]  Michal Fečkan,et al.  Periodic Solutions for Nonlinear Evolution Equations with Non-instantaneous Impulses , 2014 .

[20]  D. Bainov,et al.  Integral Inequalities and Applications , 1992 .

[21]  Ying Li,et al.  Periodic Solutions for a Class of Nonautonomous Differential System with Impulses and Time-varying Delays , 2011 .

[22]  Zuomao Yan,et al.  The optimal control of a new class of impulsive stochastic neutral evolution integro-differential equations with infinite delay , 2016, Int. J. Control.

[23]  Gang Li,et al.  Existence results for semilinear differential equations with nonlocal and impulsive conditions , 2010 .

[24]  M. Z. Liu,et al.  Exponential stability of the exact solutions and the numerical solutions for a class of linear impulsive delay differential equations , 2015, J. Comput. Appl. Math..

[25]  He Yang,et al.  Perturbation method for nonlocal impulsive evolution equations , 2013 .

[26]  R. Agarwal,et al.  A Survey on Existence Results for Boundary Value Problems of Nonlinear Fractional Differential Equations and Inclusions , 2010 .

[27]  Haibo Chen,et al.  Periodic solution generated by impulses for singular differential equations , 2013 .

[28]  Agacik Zafer,et al.  Perron's theorem for linear impulsive differential equations with distributed delay , 2006 .

[29]  V. Lakshmikantham,et al.  Theory of Impulsive Differential Equations , 1989, Series in Modern Applied Mathematics.

[30]  M. Benchohra,et al.  Impulsive differential equations and inclusions , 2006 .

[31]  Hernán R. Henríquez,et al.  Global Solutions for Abstract Differential Equations with Non-Instantaneous Impulses , 2016 .

[32]  Michal Fečkan,et al.  A General Class of Impulsive Evolution Equations , 2015 .

[33]  D. O’Regan,et al.  Stability by Lyapunov like functions of nonlinear differential equations with non-instantaneous impulses , 2017 .