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

Trinh Viet Duoc, Nguyen Ngoc Huy

Abstract


In this paper we investigate the homogeneous linear differential equation \(v'(t)=A(t)v(t)\) and the semi-linear differential equation \(v'(t)=A(t)v(t)+g(t,v(t))\) in Banach space \(X\), in which \(A:\mathbb{R} \to \mathcal{L}(X)\) is a strongly continuous function, \(g:\mathbb{R} \times X\to X\) is continuous and satisfies \(\varphi\)-Lipschitz condition. The first we characterize the exponential dichotomy of the associated evolution family with the homogeneous linear differential equation by space pair \((\mathcal{E},\mathcal{E}_{\infty})\), 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 \(C^1\).

Keywords


Exponential dichotomy, invariant manifolds, semi-linear differential equations

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References


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DOI: http://dx.doi.org/10.24193/subbmath.2024.1.09

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