Yosida Distance and Existence of Invariant Manifolds in the Infinite-Dimensional Dynamical Systems

2023/01/28 by Xuan‐Quang Bui, Bui, Xuan-Quang, Nguyễn Văn Minh +1
Engineering · Mathematics · #34C45 #34D09 #34D20 #34G10 #37D10 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2301.12080

openalex publication_date 2023/01/28 · openalex created_date 2023/02/02 · openalex updated_date 2026/07/28

Abstract

We introduce a new concept of Yosida distance between two (unbounded) linear operators A and B in a Banach space \mathbbX defined as d_Y(A,B):=\limsup_{μ→ +∞} ‖ A_μ-B_μ‖, where A_μ and B_μ are the Yosida approximations of A and B, respectively, and then study the persistence of evolution equations under small Yosida perturbation. This new concept of distance is also used to define the continuity of the proto-derivative of the operator F in the equation u'(t)=Fu(t), where F \colon D(F)⊂ \mathbbX → \mathbbX is a nonlinear operator. We show that the above-mentioned equation has local stable and unstable invariant manifolds near an exponentially dichotomous equilibrium if the proto-derivative of F is continuous. The Yosida distance approach to perturbation theory allows us to free the requirement on the domains of the perturbation operators. Finally, the obtained results seem to be new.

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