Compact almost automorphic dynamics of non-autonomous differential equations with exponential dichotomy and applications to biological models with delay
arxiv(2024)
摘要
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 ℝ^2,
this implies that G(·, ·) is Δ_2 - like uniformly continuous,
where Δ_2 is the principal diagonal of ℝ^2, 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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