Existence of an Almost Periodic Solution in a Difference Equation with Infinite Delay

Abstract

In order to obtain the existence of an almost periodic functional difference equation xðn þ 1Þ ¼ f ðn; x n Þ; n [ Z þ and where x n is defined by x n ðsÞ ¼ xðn þ sÞ for s [ Z 2 ; on an axiomatic phase space B, we consider a certain stability property, which is referred to as BS-stable under disturbances from V( f ) with respect to K, this stability implies r-stable under disturbances from V( f ) with respect to compact set K.

Periodic and asymptotically periodic solutions of systems of nonlinear difference equations with infinite delay

“…Also, the authors of [17] analyzed the asymptotic behavior of positive solutions of second order nonlinear difference systems, while the authors of [18] studied the classification and the existence of positive solutions of the system of Volterra nonlinear difference equations. Periodicity of the solutions of difference equations has been handled by [2], [6]- [11]. In [7] and [8], the authors focused on a system of Volterra difference equations of the form…”

Regularity of Difference Equations on Banach Spaces

“…In fact, properties of the solutions have been studied in several contexts. For example, invariant manifolds theory [140], convergence theory [50,56,57], discrete maximal regularity [59], asymptotic behavior [51,58,76,82,141,142], exponential dichotomy [36,167], robustness [36], stability [83,84,150], and periodicity [2,15,40,60,63,97,136,139,151,[170][171][172][173]178].…”

Global attractivity and almost periodic solution of a discrete mutualism model with delays

“…First, we give the definitions of the terminologies involved. Definition () A sequence x : Z → R is called an almost periodic sequence if the ϵ ‐translation set of x is a relatively dense set in Z for all ϵ > 0, that is, for any given ϵ > 0, there exists an integer l ( ϵ ) > 0 such that each interval of length l ( ϵ ) contains an integer τ ∈ E { ϵ , x } with τ is called an ϵ ‐translation number of x ( n ). Definition () Let D be an open subset of R m , f : Z × D → R m . f ( n , x ) is said to be almost periodic in n uniformly for x ∈ D if for any ϵ > 0 and any compact set S ⊂ D , there exists a positive integer l = l ( ϵ , S ) such that any interval of length l contains an integer τ for which τ is called an ϵ ‐translation number of f ( n , x ). Definition () The hull of f , denoted by H ( f ), is defined by …”