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Journal of Mathematics
Volume 2014 (2014), Article ID 989526, 9 pages
http://dx.doi.org/10.1155/2014/989526
Research Article

Parameter Dependence of Stable Invariant Manifolds for Delay Differential Equations under -Dichotomies

School of Mathematics, Jia Ying University, Meizhou, Guangdong 514015, China

Received 20 May 2014; Accepted 14 September 2014; Published 27 October 2014

Academic Editor: Alfred Peris

Copyright © 2014 Lijun Pan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We obtain the existence of stable invariant manifolds for the nonlinear equation provided that the linear delay equation admits a nonuniform -dichotomy and is a sufficiently small Lipschitz perturbation. We show that the stable invariant manifolds are dependent on parameter . Namely, the stable invariant manifolds are Lipschitz in the parameter . In addition, we also show that nonuniform -contraction persists under sufficiently small nonlinear perturbations.