Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 91, pp. 1-19.

Title: Compact almost automorphic dynamics of linear non-autonomous differential equations
with exponential dichotomy and of delayed biological models

Authors: Alan Chavez (Univ. Nacional de Trujillo, Peru)
         Nelson Aragones (Univ. Nacional de Trujillo, Peru)
         Ulices Zavaleta (Univ. Nacional de Trujillo, Peru)
         Manuel Pinto (Univ. de Chile, Santiago, Chile)

Abstract:
In this work, we study the dynamics of linear non-autonomous differential equations
with exponential dichotomy and compact almost automorphic perturbations.
First, we prove that if the homogeneous system is exponentially dichotomous and the
coefficient matrix is compact almost automorphic, then the associated Green's function
is compact bi-almost automorphic and uniformly continuous relative to the principal
diagonal of the two-dimensional Euclidean space. Next, we demonstrate the invariance
of the compact almost automorphic function space under convolution products with
Green's function as the kernel. These results ensures that the unique bounded solution
of a linear non-autonomous differential equation, under exponential dichotomy and with
compact almost automorphic perturbation, is itself compact almost automorphic.
Finally, we investigate the existence and the global exponential stability of a
unique positive compact almost automorphic solution for a nonlinear non-autonomous
delayed biological model with nonlinear harvesting or immigration terms and mixed delays.

Submitted May 7, 2025. Published September 02, 2025.
Math Subject Classifications: 34A30, 34A34, 34C27, 34K14, 35B99.
Key Words: Compact almost automorphic functions; bi-almost automorphic functions; non-autonomous differential systems; exponential dichotomy; biological models.