New approach for weighted pseudo-almost periodic functions under the light of measure theory, basic results and applications

Abstract

International audienceThe aim of this work is to present new approach to study weighted pseudo almost periodic functions using the measure theory. We present a new concept of weighted ergodic functions which is more general than the classical one. Then we establish many interesting results on the functional space of such functions like completeness and composition theorems. The theory of this work generalizes the classical results on weighted pseudo almost periodic functions. For illustration, we provide some applications for evolution equations which include reaction diffusion system and partial functional differential equations

Translation invariance and related properties of μ-pseudo almost automorphic (periodic) functions with application

“…Furthermore, by Theorem 3.2 and similar arguments of the proofs of the corresponding results in [19,20], we can obtain the following properties of μ-paa (μ-pap) functions. Here we omit the details of the proofs.…”

“…By the same arguments of the proofs in [19,Theorem 3.9] and [20,Theorem 3.8], we see that f * G ∈ AA(X) (AP(X)) if f ∈ AA(X) (AP(X)), and we omit the details here. As a result, we get the conclusion.…”

“…Remark 2.4 From [19,20], we have the following facts: (i) Let ρ be a nonnegative B-measurable function, and μ the positive measure defined by (ii) One can observe that a ρ-paa (ρ-pap) function is μ-paa (μ-pap) with ρ(t) the Radon-Nikodym derivative of μ:…”

“…(i) In [19,20], the following assumption was used: (H) For all τ ∈ R, there exist β > 0 and a bounded interval I such that…”

“…Recently, by using the measure theory, Blot, Cieutat and Ezzinbi [19,20] introduced the concepts of μ-pseudo almost periodicity and μ-pseudo almost automorphy which are generalization of weighted pseudo almost periodicity and weighted pseudo almost automorphy, respectively. Some basic properties with applications are presented.…”