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

In the present work, we prove that, if A(·) is a compact almost automorphic matrix and the system $$x'(t) = A(t)x(t)\, ,$$ possesses an exponential dichotomy with Green function G(·, ·), then its associated system $$y'(t) = B(t)y(t)\, ,$$ where B(·) ∈ H(A) (the hull of A(·)) also possesses an exponential dichotomy. Moreover, the Green function G(·, ·) is compact Bi-almost automorphic in R², this implies that G(·, ·) is Δ₂ - like uniformly continuous, where Δ₂ is the principal diagonal of R², an important ingredient in the proof of invariance of the compact almost automorphic function space under convolution product with kernel G(·, ·). 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.