Recurrent solutions of neutral differential-difference systems

TL;DR

This paper investigates the properties of recurrent solutions in linear neutral differential-difference systems, establishing conditions under which solutions are recurrent and extending classical results to these systems.

Contribution

It introduces subclasses of recurrent functions and extends classical theorems to the context of neutral differential-difference systems.

Findings

Recurrent solutions are characterized under certain conditions.

Subclasses of recurrent functions with specific properties are defined.

Classical results are extended to the setting of neutral differential-difference systems.

Abstract

Results of Bohr-Neugebauer type are obtained for recurrent functions : If $y$ is a bounded uniformly continuous solution of a linear neutral difference-differential system with recurrent right-hand side, then $y$ is recurrent if $c_0 \not \subset X$ ; also analogues and extensions to half lines are given. For this, various subclasses "$rec$" are introduced which are linear (the set REC of all recurrent functions is not), invariant, closed etc. Also, analogues of the Bohl-Bohr-Amerio-Kadets and Esclangon- Landau results for REC are obtained.