2007/11/13 by Nguyen Van Minh, Nguyễn Văn Minh, Gaston N'guerekata +5
Computer Science · Mathematics · #34G10 #47D05 #47H20 #Advanced Mathematical Modeling in Engineering #Differential Equations and Numerical Methods #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Differential Equations Analysis #math.DS #math.FA #msc:34G10 #msc:47D05 #msc:47H20
paper · pdf · doi:10.48550/arxiv.0711.2000
18 pages
openalex publication_date 2007/11/13 · arxiv created 2009/02/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the existence and uniqueness of bounded solutions of periodic evolution equations of the form u'=A(t)u+εH(t,u)+f(t), where A(t) is, in general, an unbounded operator depending 1-periodically on t, H is 1-periodic in t, ε is small, and f is a bounded and continuous function that is not necessarily uniformly continuous. We propose a new approach to the spectral theory of functions via the concept of "circular spectrum" and then apply it to study the linear equations u'=A(t)u+f(t) with general conditions on f. For small ε we show that the perturbed equation inherits some properties of the linear unperturbed one. The main results extend recent results in the direction, saying that if the unitary spectrum of the monodromy operator does not intersect the circular spectrum of f, then the evolution equation has a unique mild solution with its circular spectrum contained in the circular spectrum of f.