Positive almost periodic solutions of nonautonomous evolution equations
and application to Lotka--Volterra systems
Abstract
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.