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

Abstract

In the present work, we prove that, if $A(\cdot )$ is a compact almost automorphic matrix and the system $$x^{\prime }(t)=A(t)x(t),$$ possesses an exponential dichotomy with Green function $G(\cdot ,\cdot )$, then its associated system $$y^{\prime }(t)=B(t)y(t),$$ 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.