*Mathematical Methods in the Applied Sciences*Positive almost periodic solutions of nonautonomous evolution equations
and application to Lotka--Volterra systems

Consider the nonautonomous semilinear evolution equation of type:
$(\star) \; u’(t)=A(t)u(t)+f(t,u(t)),
\; t \in \mathbb{R},$
where $ A(t), \ t\in
\mathbb{R} $ is a family of closed linear operators on
a Banach space $X$, the nonlinear term $f$, acting on some real
interpolation spaces, is assumed to be almost periodic only in a weak
sense (i.e. in Stepanov sense) with respect to $t$ and Lipschitzian in
bounded sets with respect to the second variable. We prove the existence
and uniqueness of positive almost periodic solutions in the strong sense
(Bohr sense) for equation $ (\star) $ using the
exponential dichotomy approach. Then, we establish a new composition
result of Stepanov almost periodic functions by assuming only the
continuity of $f$ in the second variable. Moreover, we provide an
application to a nonautonomous system of reaction–diffusion equations
describing a Lotka–Volterra predator–prey model with diffusion and
time–dependent parameters in a generalized almost periodic environment.

26 Feb 2022

01 Mar 2022

01 Mar 2022

10 Mar 2022

21 Aug 2022

27 Aug 2022

25 Sep 2022

26 Sep 2022

26 Sep 2022

17 Oct 2022

24 Jan 2023

25 Jan 2023