Almost Periodicity for a Nonautonomous Discrete Dispersive Population Model

ABSTRACT

In this article, study the almost periodic profile of solutions for nonautonomous difference equations in Banach spaces. We apply our results in population dynamics.

KEYWORDS:

• Almost periodic sequences
• integro-difference equations (IDE)
• nonautonomous difference equations

MATHEMATICS SUBJECT CLASSIFICATION:

• 39A05
• 39A10
• 39A24
• 39A60

Acknowledgements

This work was completed when Claudio Cuevas was visiting the University of La Frontera (August 2014–January 2015). He is grateful to the Evolution Equations and Applications Group and the Department of Mathematics and Statistics for providing a stimulating atmosphere to work.

Notes

¹We observe that in the infinite dimensional framework the situation is much worse, and the available results are more restricted to perturbation type arguments. Until now, there is no unified theory for nonautonomous evolution equations and many questions remain unanswered.

²For a given family of closed linear operators {A(t):t∈ℝ} on 𝕏, the existence of an evolution family U(t,s) associated with it is not always guaranteed. However, if the family A(t) satisfies the Acquistapace-Terreni conditions, then the family of linear operators A(t) has an evolution family associated with it such that U(t,s)X⊆D(A(t)) for all t,s∈ℝ. Moreover for t>s, (t,s)→U(t,s)∈ℬ(X) is continuously differentiable in t with ∂_tU(t,s) = A(t)U(t,s), see [1].

³In the case of that T(n) in (1.2) is not dependent on n, in [2] the authors deals with existence and stability of solutions by using recent characterization of maximal regularity. It is well known that the study of maximal regularity is very useful for treating semilinear and quasilinear problems.

⁴A unified exposition about this theory and some applications we refer the reader to the relevant book by Corduneanu [6].

⁵Time scale is a recent theory which started to be developed by Stefan Hilger on his doctoral thesis (see [13]). This theory represents a useful tool for applications to economics, population models, quantum physics among others.

⁶Almost automorphic dynamics have been given a notable amount of attention in recent years with respect to the study of almost periodically forced monotone systems. We observe that there have been extensive studies on periodically forced second-order oscillations. In engineering applications, periodic oscillations are referred as harmonic oscillations and almost periodic ones are interpreted as harmonic oscillations covered with small “noise”. Accordingly, almost automorphic oscillations can be regarded as harmonic ones covered with big “noise” (see [26]).

⁷In this case, we say that the members of family ℱ are separated in D.