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 ) ) 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 ) ‖ α x ( t ) = f ( t ) ( α ⩾ 0 ) .