Exponential dichotomy and invariant manifolds of semi-linear differential equations on the line

Abstract

In this paper we investigate the homogeneous linear differential equation vi(t) = A(t)v(t) and the semi-linear differential equation vi(t) = A(t)v(t) + g(t, v(t)) in Banach space X, in which A : R → L(X) is a strongly continuous function, g : R × X → X is continuous and satisfies ϕ-Lipschitz condition. The first we characterize the exponential dichotomy of the associated evolution family with the homogeneous linear differential equation by space pair (E, E∞), this is a Perron type result. Applying the achieved results, we establish the robustness of exponential dichotomy. The next we show the existence of stable and unstable manifolds for the semi-linear differential equation and prove that each a fiber of these manifolds is differentiable submanifold of class C1. Mathematics Subject Classification (2010): 34C45, 34D09, 34D10. Received 14 June 2021; Accepted 09 September 2022