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Properties and composition of pseudo almost periodic functions and application to semilinear differential equations in Banach spaces. (English) Zbl 1183.34082

The authors firstly generalize some well-known properties of pseudo almost periodic functions in the sense introduced in 1994 by C. Zhang. Secondly they establish some results on the composition of pseudo almost periodic functions that generalize some recent results by C. Cuevas and M. Pinto. Similar results can be found in a 2008 work by Jin Liang et al. for the pseudo almost automorphic case. Finally, an application to semilinear differential equation of the form

x ' (t)=Ax(t)+f(t,x),

where A generates an exponentially stable semigroup and f satisfies some Lipschitz-type condition. The tool used to obtain the existence and uniqueness of a pseudo almost periodic mild solution to the equation is the Banach contraction mapping principle. This extends some recent results by Ezzinbi, Arino and the second author. The paper contains also an example of an ergodic perturbation of a pseudo almost periodic function which is unbounded.

MSC:
34G20Nonlinear ODE in abstract spaces
34C27Almost and pseudo-almost periodic solutions of ODE