APr solutions for linear functional difference equations with infinite delay with/without Favard's property. Issue 3 (3rd March 2020)

• Record Type: Journal Article
• Authors: Hamaya, Yoshihiro
• Is Part Of: Journal of difference equations and applications. Volume 26:Issue 3(2020)
• Page Start: 328
• Page End: 352
• Publication Date: 2020-03-03
• Subjects: APr solutions -- linear APr discrete system -- uniformly stable in B -- homoclinic to zero -- Favard separate condition -- without Favard's property; 39A10 -- 39A11; Difference equations -- Periodicals; 515.625
• DOI: 10.1080/10236198.2020.1729142
• Languages: English
• ISSNs: 1023-6198

Abstract

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 ∈ [ 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.