AP_r solutions for linear functional difference equations with infinite delay with/without Favard's property

Abstract

In this paper, we first consider Favard's type theorem that the linear functional difference equation (LFDE) with infinite delay has a unique $A{P}_r$ solution, $r\in [1,2]$, if it has at least one bounded solution and the bounded solutions of the homogeneous equation in hull satisfy the separation among bounded solution. If there are bounded solutions which are non-separated, an almost periodic solution does not exist [Ortega and M. Tarallo, Almost periodic linear differential equations with non-separated solutions, J. Funct. Anal. 237 (2006), pp. 402–426]. We second consider without Favard's property that the LFDE with infinite delay has a unique $A{P}_r$ solution, if the every non trivial bounded solution of the homogeneous equation in hull satisfies homoclinic to zero.

Keywords:

• $A{P}_r$ solutions
• linear $A{P}_r$ discrete system
• uniformly stable in B
• homoclinic to zero
• Favard separate condition
• without Favard's property

AMS (MOS) 2000 Subject classifications:

• 39A10
• 39A11

Acknowledgments

The author wishes to thank referees and the Editor for their many useful comments concerning the content of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).