Compact almost automorphic dynamics of non-autonomous differential equations with exponential dichotomy and applications to biological models with delay

TL;DR

This paper establishes the properties of compact almost automorphic solutions and Green functions for non-autonomous differential equations with exponential dichotomy, with applications to biological models involving delays and harvesting terms.

Contribution

It proves the invariance of the compact almost automorphic function space under convolution and studies positive solutions for biological delay models.

Findings

Green function is compact Bi-almost automorphic and $ ext{Delta}_2$-like.

Invariance of function space under convolution with Green function.

Existence of positive almost automorphic solutions in biological delay models.

Abstract

In the present work, we prove that, if $A(\cdot)$ is a compact almost automorphic matrix and the system $$x'(t) = A(t)x(t)\, ,$$ possesses an exponential dichotomy with Green function $G(\cdot, \cdot)$, then its associated system $$y'(t) = B(t)y(t)\, ,$$ where $B(\cdot) \in H(A)$ (the hull of $A(\cdot)$) also possesses an exponential dichotomy. Moreover, the Green function $G(\cdot, \cdot)$ is compact Bi-almost automorphic in $\mathbb{R}^2$, this implies that $G(\cdot, \cdot)$ is $\Delta_2$ - like uniformly continuous, where $\Delta_2$ is the principal diagonal of $\mathbb{R}^2$, an important ingredient in the proof of invariance of the compact almost automorphic function space under convolution product with kernel $G(\cdot, \cdot)$. Finally, we study the existence of a positive compact almost automorphic solution of non-autonomous differential equations of biological interest having non-linear harvesting terms and mixed delays.