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

2024/04/25 by Chávez, Alan, Aragonés, Nelson, Pinto, Manuel +1
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2404.16758

Abstract

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.

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