The existence of pseudo-almost periodic solutions for some nonlinear differential equations in Banach spaces

Abstract

Let ( E , ‖ . ‖ ) be a real Banach, where [ x , h ] − denotes the limit of the quotient ( ‖ x ‖ − ‖ x − t h ‖ ) / t as 0 < t → 0 . Consider the differential equation d x d t = A ( t , x ) + f ( t ) where we assume, amongst other properties, that A is strongly dissipative in the following sense: [ x − y , A ( t , x ) − A ( t , y ) ] − ≤ p ( t ) ‖ x − y ‖ θ ( ‖ x − y ‖ ) where θ is a function of the type u α ( α ≥ 0 ) and p is a function majorized by an almost periodic function with negative mean value. First, we show that this equation has a unique pseudo-almost periodic solution. Then we apply this result to the class of the following equations: d x d t + q ( t ) ‖ x ‖ α x = f ( t ) ( α ≥ 0 ) , where q is a pseudo-almost periodic function with positive mean value.