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Authors: TRINH VIET DUOC, NGUYEN NGOC HUY.
Received 19 September 2021; Accepted 20 January 2022. Published Online: 2024-03-20
Published Print: 2024-03-30
pp. 127-148
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In this paper we investigate the homogeneous linear differential equation v0(t) = A(t)v(t) and the semi-linear differential equation v0(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; E1), 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.
