Electron. J. Differential Equations, Vol. 2019 (2019), No. 25, pp. 1-22.

Stable manifolds for impulsive delay equations and parameter dependence

Dhirendra Bahuguna, Lokesh Singh

Abstract:
In this article, we establish the existence of Lipschitz stable invariant manifolds for the semiflows generated by the delay differential equation $x'= L(t)x_t + f(t,x_t,\lambda)$ with impulses at times $\{\tau_i\}_{i=1}^\infty $, assuming that the perturbation $f(t,x_t,\lambda)$ as well as the impulses are small and the corresponding linear delay differential equation admits a nonuniform exponential dichotomy. We also show that the obtained manifolds are Lipschitz in the parameter $\lambda$.

Submitted April 3, 2018. Published February 13, 2019.
Math Subject Classifications: 37D10, 34D09, 37D25, 35R10, 35R12.
Key Words: Delay impulsive equation; exponential dichotomy; stable invariant manifold.

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Dhirendra Bahuguna
Department of Mathematics and Statistics
Indian Institute of Technology
Kanpur-208016, India
email: dhiren@iitk.ac.in
Lokesh Singh
Department of Mathematics and Statistics
Indian Institute of Technology
Kanpur-208016, India
email: lokesh@iitk.ac.in

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