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