Ordinary differential equations
On the orbital Hausdorff dependence of differential equations with non-instantaneous impulses
Comptes Rendus. Mathématique, Volume 356 (2018) no. 2, pp. 150-171.

In this article, we investigate the orbital Hausdorff continuous dependence of the solutions to integer order and fractional nonlinear non-instantaneous differential equations. The concept of orbital Hausdorff continuous dependence is used to characterize the relations of solutions corresponding to the impulsive points and junction points in the sense of the Hausdorff distance. Then, we establish sufficient conditions to guarantee this specific continuous dependence on their respective trajectories. Finally, two examples are given to illustrate our theoretical results.

Nous étudions ici la dépendance orbitale de Hausdorff continue des solutions des équations différentielles d'ordre entier ou fractionnaire, non linéaires avec impulsion non instantanée. Le concept de dépendance orbitale de Hausdorff continue est utilisé pour évaluer la distance de Hausdorff entre les solutions correspondant aux points d'impulsion et de jonction. Nous montrons ensuite des conditions suffisantes garantissant cette dépendance continue spécifique sur leurs trajectoires respectives. Finalement, nous donnons deux exemples qui illustrent nos résultats théoriques.

Accepted:
Published online:
DOI: 10.1016/j.crma.2018.01.001
Yang, Dan 1; Wang, JinRong 1; O'Regan, Donal 2

1 Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China
2 School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland
@article{CRMATH_2018__356_2_150_0,
author = {Yang, Dan and Wang, JinRong and O'Regan, Donal},
title = {On the orbital {Hausdorff} dependence of differential equations with non-instantaneous impulses},
journal = {Comptes Rendus. Math\'ematique},
pages = {150--171},
publisher = {Elsevier},
volume = {356},
number = {2},
year = {2018},
doi = {10.1016/j.crma.2018.01.001},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2018.01.001/}
}
TY  - JOUR
AU  - Yang, Dan
AU  - Wang, JinRong
AU  - O'Regan, Donal
TI  - On the orbital Hausdorff dependence of differential equations with non-instantaneous impulses
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 150
EP  - 171
VL  - 356
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2018.01.001/
DO  - 10.1016/j.crma.2018.01.001
LA  - en
ID  - CRMATH_2018__356_2_150_0
ER  - 
%0 Journal Article
%A Yang, Dan
%A Wang, JinRong
%A O'Regan, Donal
%T On the orbital Hausdorff dependence of differential equations with non-instantaneous impulses
%J Comptes Rendus. Mathématique
%D 2018
%P 150-171
%V 356
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2018.01.001/
%R 10.1016/j.crma.2018.01.001
%G en
%F CRMATH_2018__356_2_150_0
Yang, Dan; Wang, JinRong; O'Regan, Donal. On the orbital Hausdorff dependence of differential equations with non-instantaneous impulses. Comptes Rendus. Mathématique, Volume 356 (2018) no. 2, pp. 150-171. doi : 10.1016/j.crma.2018.01.001. http://www.numdam.org/articles/10.1016/j.crma.2018.01.001/

[1] Abbas, S.; Benchohra, M. Uniqueness and Ulam stabilities results for partial fractional differential equations with not instantaneous impulses, Appl. Math. Comput., Volume 257 (2015), pp. 190-198

[2] Agarwal, R.P.; Benchohra, M.; Hamani, S. A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., Volume 109 (2010), pp. 973-1033

[3] Agarwal, R.; O'Regan, D.; Hristova, S. Stability with initial time difference of Caputo fractional differential equations by Lyapunov functions, Z. Anal. Anwend., Volume 36 (2017), pp. 49-77

[4] Agarwal, R.; O'Regan, D.; Hristova, S. Monotone iterative technique for the initial value problem for differential equations with non-instantaneous impulses, Appl. Math. Comput., Volume 298 (2017), pp. 45-56

[5] Akhmet, M.U.; Alzabut, J.; Zafer, A. Perron's theorem for linear impulsive differential equations with distributed delay, J. Comput. Appl. Math., Volume 193 (2006), pp. 204-218

[6] Bai, L.; Nieto, J.J. Variational approach to differential equations with not instantaneous impulses, Appl. Math. Lett., Volume 73 (2017), pp. 44-48

[7] Bainov, D.D.; Simeonov, P.S. Theory of Impulsive Differential Equations, Series on Advances in Mathematics for Applied Sciences, vol. 28, World Scientific, Singapore, 1995

[8] Benchohra, M.; Henderson, J.; Ntouyas, S.K. Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, 2006

[9] Chen, P.; Li, Y.; Yang, H. Perturbation method for nonlocal impulsive evolution equations, Nonlinear Anal. Hybrid Syst., Volume 8 (2013), pp. 22-30

[10] Colao, V.; Muglia, L.; Xu, H.K. An existence result for a new class of impulsive functional differential equations with delay, J. Math. Anal. Appl., Volume 441 (2016), pp. 668-683

[11] Diblík, J. Positive solutions of nonlinear delayed differential equations with impulses, Appl. Math. Lett., Volume 72 (2017), pp. 16-22

[12] Dishliev, A.; Dishlieva, K.; Nenov, S. Specific Asymptotic Properties of the Solutions of Impulsive Differential Equations: Methods and Applications, Academic Publication, 2012

[13] Dishlieva, K.; Antonov, A. Hausdorff Metric and Differential Equations with Variable Structure and Impulses, Technical University of Sofia, Bulgaria, 2015

[14] Fan, Z.; Li, G. Existence results for semilinear differential equations with nonlocal and impulsive conditions, J. Funct. Anal., Volume 258 (2010), pp. 1709-1727

[15] Gautam, G.R.; Dabas, J. Mild solutions for class of neutral fractional functional differential equations with not instantaneous impulses, Appl. Math. Comput., Volume 259 (2015), pp. 480-489

[16] Hernández, E.; O'Regan, D. On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc., Volume 141 (2013), pp. 1641-1649

[17] Hernández, E.; Pierri, M.; O'Regan, D. On abstract differential equations with non instantaneous impulses, Topol. Methods Nonlinear Anal., Volume 46 (2015), pp. 1067-1085

[18] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 2006

[19] Leiva, H. Controllability of semilinear impulsive nonautonomous systems, Int. J. Control, Volume 88 (2015), pp. 585-592

[20] Liu, S.; Debbouche, A.; Wang, J. On the iterative learning control for stochastic impulsive differential equations with randomly varying trial lengths, J. Comput. Appl. Math., Volume 312 (2017), pp. 47-57

[21] Muslim, M.; Kumar, A.; Fečkan, M. Existence, uniqueness and stability of solutions to second order nonlinear differential equations with non-instantaneous impulses, J. King Saud Univ. (2016) | DOI

[22] Pierri, M.; Henríquez, H.R.; Prokczyk, A. Global solutions for abstract differential equations with non-instantaneous impulses, Mediterr. J. Math., Volume 34 (2016), pp. 1685-1708

[23] Pierri, M.; O'Regan, D.; Rolnik, V. Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses, Appl. Math. Comput., Volume 219 (2013), pp. 6743-6749

[24] Samoilenko, A.M.; Perestyuk, N.A. Impulsive Differential Equations, World Scientific, 1995

[25] Sendov, B. Hausdorff Approximations, Springer Science and Business, Media, 1990

[26] Stamova, I.; Stamov, G. Functional and Impulsive Differential Equations of Fractional Order: Qualitative Analysis and Applications, CRC Press, 2017

[27] Sun, J.; Chu, J.; Chen, H. Periodic solution generated by impulses for singular differential equations, J. Math. Anal. Appl., Volume 404 (2013), pp. 562-569

[28] Wang, J. Stability of noninstantaneous impulsive evolution equations, Appl. Math. Lett., Volume 73 (2017), pp. 157-162

[29] Wang, J.; Fečkan, M. A general class of impulsive evolution equations, Topol. Methods Nonlinear Anal., Volume 46 (2015), pp. 915-934

[30] Wang, J.; Fečkan, M.; Tian, Y. Stability analysis for a general class of non-instantaneous impulsive differential equations, Mediterr. J. Math., Volume 14 (2017)

[31] Wang, J.; Fečkan, M.; Zhou, Y. A survey on impulsive fractional differential equations, Fract. Calc. Appl. Anal., Volume 19 (2016), pp. 806-831

[32] Wang, J.; Zhou, Y.; Fečkan, M. Nonlinear impulsive problems for fractional differential equations and Ulam stability, Comput. Math. Appl., Volume 64 (2012), pp. 3389-3405

[33] Wang, J.; Zhou, Y.; Lin, Z. On a new class of impulsive fractional differential equations, Appl. Math. Comput., Volume 242 (2014), pp. 649-657

[34] Yang, D.; Wang, J. Non-instantaneous impulsive fractional-order implicit differential equations with random effects, Stoch. Anal. Appl., Volume 35 (2017), pp. 719-741

[35] Yang, D.; Wang, J.; O'Regan, D. Asymptotic properties of the solutions of nonlinear non-instantaneous impulsive differential equations, J. Franklin Inst., Volume 354 (2017), pp. 6978-7011

[36] Yuan, X.; Xia, Y.H.; O'Regan, D. Nonautonomous impulsive systems with unbounded nonlinear terms, Appl. Math. Comput., Volume 245 (2014), pp. 391-403

[37] Zhang, G.L.; Song, M.H.; Liu, M.Z. Exponential stability of the exact solutions and the numerical solutions for a class of linear impulsive delay differential equations, J. Comput. Appl. Math., Volume 285 (2015), pp. 32-44

Cited by Sources:

The authors acknowledge the National Natural Science Foundation of China (11661016), Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), and Unite Foundation of Guizhou Province ([2015]7640).