Parameter Dependence of Stable Invariant Manifolds for Delay Differential Equations under (μ, ν)-Dichotomies. (27th October 2014)

• Record Type: Journal Article
• Authors: Pan, Lijun
• Other Names: Peris Alfred Academic Editor.
• Is Part Of: Journal of mathematics. Volume 2014(2014)
• Publication Date: 2014-10-27
• Subjects: Mathematics -- Periodicals; Mathematics; Periodicals; 510
• DOI: 10.1155/2014/989526
• Languages: English
• ISSNs: 2314-4629

Abstract

Abstract : We obtain the existence of stable invariant manifolds for the nonlinear equation x ′ = L ( t ) x t + f ( t, x t, λ ) provided that the linear delay equation x ′ = L ( t ) x t admits a nonuniform ( μ, ν ) -dichotomy and f is a sufficiently small Lipschitz perturbation. We show that the stable invariant manifolds are dependent on parameter λ . Namely, the stable invariant manifolds are Lipschitz in the parameter λ . In addition, we also show that nonuniform ( μ, ν ) -contraction persists under sufficiently small nonlinear perturbations.