Abstract
We study the existence and uniqueness of mild and classical solutions for abstract impulsive differential equations with state-dependent time impulses and an example is presented.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aiello, W., Freedman, H.I., Wu, J.: Analysis of a model representing stage-structured population growth with state-dependent time delay. SIAM J. Appl. Math. 52(3), 855–869 (1992)
Bainov, D., Covachev, V.: Impulsive Differential Equations with a Small Parameter. Series on Advances in Mathematics for Applied Sciences, vol. 24. World Scientific Publishing Co., River Edge (1994)
Benchohra, M., Henderson, J., Ntouyas, S.: Impulsive Differential Equations and Inclusions. Contemporary Mathematics and Its Applications, vol. 2. Hindawi Publishing Corporation, New Delhi (2006)
Driver, R.D.: A functional-differential system of neutral type arising in a two-body problem of classical electrodynamics. In: LaSalle, J., Lefschtz, S. (eds.) International Symposium on Nonlinear Differential Equations and Nonlinear Mechanics, pp. 474–484. Academic Press, New York (1963)
Driver, R.D.: A neutral system with state-dependent delay. J. Differ. Equ. 54, 73–86 (1984)
Hakl, R., Pinto, M., Tkachenko, V., Trofimchuk, S.: Almost periodic evolution systems with impulse action at state-dependent moments. J. Math. Anal. Appl. 446(1), 1030–1045 (2017)
Hartung, F., Krisztin, T., Walther, H.-O., Wu, J.: Functional Differential Equations with State-Dependent Delays: Theory and Applications. Handbook of Differential Equations: Ordinary Differential Equations, vol. 3, pp. 435–545. Amsterdam (2006)
Hernández, E., Prokopczyk, A., Ladeira, L.: A note on partial functional differential equations with state-dependent delay. Nonlinear Anal. Real World Appl. 7(4), 510–519 (2006)
Hernández, E., Pierri, M., Goncalves, G.: Existence results for an impulsive abstract partial differential equation with state-dependent delay. Comput. Math. Appl. 52(3–4), 411–420 (2006)
Hernandez, E., Pierri, M., Wu, J.: \({{\bf C}}^{1+\alpha }\)-strict solutions and wellposedness of abstract differential equations with state dependent delay. J. Differ. Equ. 261(12), 6856–6882 (2016)
Krisztin, T., Rezounenkob, A.: Parabolic partial differential equations with discrete state-dependent delay: classical solutions and solution manifold. J. Differ. Equ. 260(5), 4454–4472 (2016)
Kosovalic, N., Chen, Y., Wu, J.: Algebraic-delay differential systems: \( C^{0}\)-extendable submanifolds and linearization. Trans. Am. Math. Soc. 369(5), 3387–3419 (2017)
Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. Series in Modern Applied Mathematics, vol. 6. World Scientific Publishing Co., Teaneck (1989)
Li, X., Wu, J.: Stability of nonlinear differential systems with state-dependent delayed impulses. Autom. J. IFAC 64, 63–69 (2016)
Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems, PNLDE, vol. 16. Birkhäauser, Basel (1995)
Lv, Y., Rong, Y., Yongzhen, P.: Smoothness of semiflows for parabolic partial differential equations with state-dependent delay. J. Differ. Equ. 260, 6201–6231 (2016)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences. Springer, New York-Berlin (1983)
Rezounenko, A.V.: A condition on delay for differential equations with discrete state-dependent delay. J. Math. Anal. Appl. 385(1), 506–516 (2012)
Rezounenko, A.V., Wu, J.: A non-local PDE model for population dynamics with state-selective delay: local theory and global attractors. J. Comput. Appl. Math. 190(1–2), 99–113 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Azevedo, K.A.G. Existence and Uniqueness of Solution for Abstract Differential Equations with State-Dependent Time Impulses. Mediterr. J. Math. 16, 42 (2019). https://doi.org/10.1007/s00009-019-1308-z
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-019-1308-z