We establish conditions for the existence of piecewise continuous almost periodic solutions of a system of impulsive differential equations with exponentially dichotomous linear part. The robustness of exponential dichotomy and exponential contraction are investigated for linear systems with small perturbations of the right-hand sides and the points of impulsive action.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Halanay and D. Wexler Qualitative Theory of Systems with Impulses [Russian translation], Nauka, Moscow (1971).
A. M. Samoilenko and N. A. Perestyuk Impulsive Differential Equations, World Scientific, Singapore (1995).
G. T. Stamov, “Almost periodic solutions of impulsive differential equations,” Lect. Notes Math., 2047 (2012).
Y. M. Myslo and V. I. Tkachenko, “Almost periodic solutions of Mackey–Glass equations with pulse action,” Nonlin. Oscillat., 15, No. 4, 537–546 (2012).
A. M. Samoilenko and S. I. Trofimchuk, “Almost periodic impulse systems,” Different. Equat., 29, No. 5, 684–691 (1993).
M. U. Akhmetov and N. A. Perestyuk, “Periodic and almost periodic solutions of strongly nonlinear impulse systems,” J. Appl. Math. Mech., 56, No. 6, 829–837 (1992).
D. Henry, “Geometric theory of semilinear parabolic equations,” Lect. Notes Math., 840 (1981).
S.-N. Chow and H. Leiva, “Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces,” J. Different. Equat., 120, 429–477 (1995).
V. A. Pliss and G. R. Sell, “Robustness of exponential dichotomies in infinite-dimensional dynamical systems,” J. Dynam. Different. Equat., 11, 471–513 (1999).
I. Kmit, L. Recke, and V. I. Tkachenko, “Robustness of exponential dichotomies of boundary-value problems for the general first-order hyperbolic systems,” Ukr. Math. J., 65, No. 2, 236–251 (2013).
V. I. Tkachenko, “On multi-frequency systems with impulses,” Nelin. Kolyv., No. 1, 107–116 (1998).
V. I. Tkachenko, “On the exponential dichotomy of pulse evolution systems,” Ukr. Math. J., 46, No. 4, 441–448 (1994).
L. Barreira and C. Valls, “Robustness for impulsive equations,” Nonlin. Anal., 72, 2542–2563 (2010).
R. Naulin and M. Pinto, “Splitting of linear systems with impulses,” Rocky Mountain J. Math., 29, No. 3, 1067–1084 (1999).
Author information
Authors and Affiliations
Additional information
Published in Neliniini Kolyvannya, Vol. 17, No. 4, pp. 546–557, October–December, 2014.
Rights and permissions
About this article
Cite this article
Tkachenko, V.I. Exponential Dichotomy and Existence of Almost Periodic Solutions of Impulsive Differential Equations. J Math Sci 212, 490–502 (2016). https://doi.org/10.1007/s10958-015-2677-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-015-2677-x