Existence, uniqueness and attractiveness of a pseudo almost automorphic solutions for some dissipative differential equations in Banach spaces

We give sufficient conditions ensuring the existence, uniqueness and global attractiveness of a pseudo compact almost automorphic solution of the following differential equation:x′(t)=f(t,x(t))x′(t)=f(t,x(t)) in a Banach space E  , where f:R×E→E is a pseudo almost automorphic function with respect to the first argument. We essentially assume that the function f is dissipative. Then we apply the main results to the following equation:x′(t)+q(t)‖x(t)‖αx(t)=f(t)(α⩾0).