Smooth Stable Manifold for Delay Differential Equations with Arbitrary Growth Rate - Lokesh Singh (Axioms, 2021)
In this article, we construct a <semantics>C1</semantics> stable invariant manifold for the delay differential equation <semantics>x′=Ax(t)+Lxt+f(t,xt)</semantics> assuming the <semantics>ρ</semantics>-nonuniform exponential dichotomy for the corresponding solution operator. We also assume that the <semantics>C1</semantics> perturbation, <semantics>f(t,xt)</semantics>, and its derivative are sufficiently small and satisfy smoothness conditions. To obtain the invariant manifold, we follow the method developed by Lyapunov and Perron. We also show the dependence of invariant manifold on the perturbation <semantics>f(t,xt)</semantics>.