2023/10/07 by Xuan‐Quang Bui, Bui, Xuan-Quang, Nguyễn Văn Minh +1
Computer Science · Mathematics · #34C45 #34D09 #34D20 #34G10 #37D10 #Advanced Mathematical Modeling in Engineering #Differential Equations and Numerical Methods #Dynamical Systems (math.DS) #FOS: Mathematics #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2310.04873
openalex publication_date 2023/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is concerned with the roughness of exponential dichotomies under unbounded perturbations of a class of linear partial functional differential equations u'(t)=Au(t)+But, where A is a linear operator on a Banach space \mathbbX and B is a linear operator from C([-r,0],\mathbbX) into \mathbbX, where r>0 is a given constant. To quantify the size of unbounded perturbations, we introduce the Yosida distance between linear operators U and V, defined by dY(U,V):=\limsupμ→ +∞ ‖ Uμ-Vμ‖, where Uμ and Vμ are the Yosida approximations of U and V, respectively. We show that if dY(A, A1) and dY(B, B1) are sufficiently small, then the perturbed equation u'(t)=A1u(t)+B1ut also admits an exponential dichotomy whenever \eqrefpfde-000-1star admits one. The proofs are based on estimates of the Yosida distance between the generators of the solution semigroups associated with \eqrefpfde-000-1star and \eqrefpfde-000-2star in the phase space C([-r,0],\mathbbX), without assuming any relation between their domains.