Asymptotic Behavior of Solutions of Systems of Neutral and Convolution Equations

Abstract

Suppose J=[:, ) for some : # R or J=R and let X be a Banach space. We study asymptotic behavior of solutions on J of neutral system of equations with values in X. This reduces to questions concerning the behavior of solutions of convolution equations (*) H V 0=b, whereWe prove that if 0 is a bounded uniformly continuous solution of (*) with b from some translation invariant suitably closed class A, then 0 belongs to A, provided, for example, that det H has countably many zeros on R and c 0 / 3 X. In particular, if b is (asymptotically) almost periodic, almost automorphic or recurrent, 0 is too. Our results extend theorems of Bohr, Neugebauer, Bochner, Doss, Basit, and Zhikov and also, certain theorems of Fink, Madych, Staffans, and others. Also, we investigate bounded solutions of (*). This leads to an extension of the known classes of almost periodicity to larger classes called mean-classes. We explore mean-classes and prove that bounded solutions of (*) belong to mean-classes provided certain conditions hold. These results seem new even for the simplest difference equation 0(t+1)&0(t)=b(t) with J=X=R and b Stepanoff almost periodic. Academic Press

A spectral theory of continuous functions and the Loomis–Arendt–Batty–Vu theory on the asymptotic behavior of solutions of evolution equations

“…2 Remark 2.26. The above theorem has been stated and proved (see [2,4,6,7,[9][10][11]) under an additional assumption on the uniform continuity of f . This assumption is essential for the use of the techniques involving the theory of C 0 -groups.…”

“…In fact, in [24] Loomis proved the above corollary in a larger context but with additional assumption on the uniform continuity of u. The uniform continuity assumption is essential for the techniques used in subsequent extensions (see [2,4,6,7,[9][10][11]39]). …”

Generalized Esclangon-Landau Results and Applications to Linear Difference-differential Systems in Banach Spaces

“…6) In the sequel if there is no danger of confusion we omit •3,Xin referring to any of the above classes; for example we write AP instead of AP0,X). Everywhere in the following, if not stated explicitly otherwise, the Jl and X can be arbitrary.…”

“…In this paper, we adopt the following notation (note the difference from Refs. [3,6] (there £, T£ stands for £ub,T£ub defined below)):…”

“…[33] and the references there for weighted L^-spaces; for fL-results for A-classes see Ref. [6,Corollary 2.7]; further references can be found in Ref. [40, p. 1-40].…”