2011/08/17 by Bolis Basit, Basit, Bolis, Hans Günzler +1
Mathematics · #43A60 Secondary 43A99 #47A10 #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Functional Analysis (math.FA) #Meromorphic and Entire Functions #Nonlinear Differential Equations Analysis #Primary 47D06
paper · pdf · doi:10.48550/arxiv.1108.3392
openalex publication_date 2011/08/17 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
We prove that there is xϕ∈ X for which (*)(d u(t))/(dt)= A u(t) + ϕ(t) , u(0)=x has on \r a mild solution u∈ Cub (\r,X) (that is bounded and uniformly continuous) with u(0)=xϕ, where A is the generator of a holomorphic C0-semigroup (T(t))t≥ 0 on X with sup t≥ 0 ||T(t)|| < ∞, ϕ∈ L∞ (\r,X) and i sp (ϕ)∩ σ(A)=∅. As a consequence it is shown that if \n is the space of almost periodic AP, almost automorphic AA, bounded Levitan almost periodic LAPb, certain classes of recurrent functions RECb and ϕ∈ L∞ (\r,X) such that Mh ϕ:=(1/h)∫0h ϕ(⋅+s) ds ∈ \n for each h >0, then u∈ \n∩ Cub. These results seem new and generalize and strengthen several recent Theorems.