Matrix measure on time scales and almost periodic analysis of the impulsive Lasota–Wazewska model with patch structure and forced perturbations

Abstract

In this paper, a new impulsive Lasota–Wazewska model with patch structure and forced perturbed terms is proposed and analyzed on almost periodic time scales. For this, we introduce the concept of matrix measure on time scales and obtain some of its properties. Then, sufficient conditions are established which ensure the existence and exponential stability of positive almost periodic solutions of the proposed biological model. Our results are new even when the time scale is almost periodic, in particular, for periodic time scales on double-struckT=double-struckR or double-struckT=double-struckZ. An example is given to illustrate the theory. Finally, we present some phenomena which are triggered by almost periodic time scales. Copyright © 2016 John Wiley & Sons, Ltd.

δ-Almost Periodic Functions and Applications to Dynamic Equations

“…Because g ∈ H δ ( f ), there is a sequence α ∈Π such that T δ α f (t, x) = g(t, x) holds uniformly on T * × S. According to Corollary 3.4 from [30], there exists α ⊂ α such that T δ α ϕ(t) = ψ(t) holds uniformly on T * . Therfore, ψ(t) is a solution to (13). Moreover, it follows from |ϕ(t)| ≤ λ that |ψ(t)| ≤ λ.…”

“…Lemma 4. Assume that ϕ(t) is a minimum norm solution for (12) and there is a sequence α ⊆Π such that T δ α f (t, x) = g(t, x) exists uniformly on T * × S. Furthermore, if there is a subsequence α ⊂ α such that T δ α ϕ(t) = ψ(t) holds uniformly on T * , then ψ(t) is a minimum norm solution for (13).…”

“…Lemma 5. Assume that f ∈ C(T × E n , E n ) is uniformly δ-almost periodic and for each g ∈ H δ ( f ), (13) has a unique minimum norm solution. Then these minimum norm solutions are δ-almost periodic.…”

“…Proof. For any given g ∈ H δ ( f ), there is a unique minimum norm solution ψ(t) for (13). Because g(t, x) is uniformly δ-almost periodic, for any sequences α , β ⊆Π, there are common subsequences…”

“…Almost periodic phenomena are common in real world phenomena and the concept of almost periodic functions was first introduced by H. Bohr and later studied by H. Weyl and V. Stepanov, etc. (see [1][2][3][4]). Time scale calculus was first presented by S. Hilger [5] and there were many results concerning almost periodic dynamic equations on time scales (see [6][7][8][9][10][11][12][13][14][15][16][17][18][19]). However, in these papers, the results are only limited to periodic time scales with translation invariance.…”