On abstract differential equations with non instantaneous impulses

Eduardo Hernandez, Michelle Pierri, Donal O'Regan

DOI: http://dx.doi.org/10.12775/TMNA.2015.080


We introduce a class of abstract differential equation with non instantaneous impulses which extend and generalize some recent models considered in the literature. We study the existence of mild and classical solution and present some applications involving partial differential equations with non-instantaneous impulses.


Non-instantaneous impulses; impulsive differential equation; mild solution; partial differential equations with impulses

Full Text:



N. Abada, M. Benchohra, Mouffak and H. Hammouche, Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions, J. Differential Equations 246 (2009), no. 10, 3834-3863.

M. Benchohra, J. Henderson and S. Ntouyas, Impulsive differential equations and inclusions, Contemporary Mathemathics and its Applications, 2, Hindawi Publishing Corporation, New York, 2006.

J. Chu, J.J. Nieto, Impulsive periodic solutions of first-order singular differential equations, Bull. Lond. Math. Soc. 40 (2008), no. 1, 143-150.

Z. Fan and G. Li, Existence results for semilinear differential equations with nonlocal and impulsive conditions, J. Funct. Anal. 258 (2010), no. 5, 1709-1727.

D. Franco, E. Liz, J.J. Nietoand Y.V. Rogovchenko, A contribution to the study of functional differential equations with impulses, Math. Nachr. 218 (2000), 49-60.

M. Frigon and D. O'Regan, First order impulsive initial and periodic problems with variable moments, J. Math. Anal. Appl. 233 (1999), no. 2, 730-739.

M. Frigon and D. O'Regan, Existence results for first-order impulsive differential equations, J. Math. Anal. Appl. 193 (1995), no. 1, 96-113.

E. Hernandez and D. O'Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc. 141 (2013), no. 5, 1641-1649.

E. Hernandez, H. Henriquez and M. Rabello, Existence of solutions for a class of impulsive partial neutral functional differential equations, J. Math. Anal. Appl. 331 (2007), 1135-1158.

C. Kou, S. Zhang and S. Wu, Stability analysis in terms of two measures for impulsive differential equations, J. London Math. Soc. (2) 66 (2002), no. 1, 142-152.

V. Lakshmikantham, D.D. Bainov and P.S. Simeonov, Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics, vol. 6, World Scientific Publishing Co.; Inc., Teaneck, NJ, 1989.

Y. Liao and J. Wang, A note on stability of impulsive differential equations, Boundary Value Problems 2014, 2014:67. http://www.boundaryvalueproblems.com/content/2014/1/67

J.H. Liu, Nonlinear impulsive evolution equations,Dynam. Contin. Discrete Impuls. Systems 6 (1999), no. 1, 77-85.

A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, PNLDE Vol. 16, Birkhaauser Verlag, Basel, 1995.

J.J. Nieto and D. O'Regan, Variational approach to impulsive differential equations, Nonlinear Anal. Real World Appl. 10 (2009), no. 2, 680-690.

A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York-Berlin, 1983.

M. Pierri, D. O'Regan and V. Rolnik,Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses, Appl. Math. Comput. 219 (2013), no. 12, 6743-6749.

Y.V. Rogovchenko, Impulsive evolution systems: main results and new trends, Dynam. Contin. Discrete Impuls. Systems 3 (1997), no. 1, 57-88.

Y.V. Rogovchenko, Nonlinear impulse evolution systems and applications to population models, J. Math. Anal. Appl. 207 (1997), no. 2, 300-315.

A.M. Samoilenko and N.A. Perestyuk, Impulsive differential equations, With a preface by Yu.A. Mitropol'skii and a supplement by S.I. Trofimchuk. Transl. from the Russian by Y. Chapovsky. World Scientific Series on Nonlinear Science, Ser A. Monographs and Treatises, vol. 14, World Scientific Publishing Co., Inc., River Edge, NJ, 1995.

J.Wang and X. Li, Periodic BVP for integer/fractional order nonlinear differential equations with non-instantaneous impulses, J. Appl. Math. Comput., DOI: 10.1007/s12190-013-0751-4.

J. Wang and Z. Lin, A class of impulsive nonautonomous differential equations and Ulam-Hyers-Rassias stability, Math. Methods Appl. Sci., DOI: 10.1002/mma.3113.

J. Wang and Y. Zhang, Existence and stability of solutions to nonlinear impulsive differential equations in fi-normed spaces, Electron. J, Differential Equations 2014 (2014), No. 83, 1-10.

J. Wang and Y. Zhang, A class of nonlinear differential equations with fractional integrable impulses, Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 3001-3010.

J.S. Yu and X.H. Tang, Global attractivity in a delay population model under impulsive perturbations, Bull. London Math. Soc. 34 (2002), no. 3, 319-328.


  • There are currently no refbacks.

Partnerzy platformy czasopism