Asymptotically Weyl almost periodic functions in Lebesgue spaces with variable exponents

Abstract

In this paper, we introduce and analyze several different notions of Weyl almost periodic functions and Weyl ergodic components in Lebesgue spaces with variable exponent L p(x) . We investigate the invariance of (asymptotical) Weyl almost periodicity with variable exponent under the actions of convolution products, providing also some illustrative applications to abstract fractional differential inclusions in Banach spaces. The introduced classes of generalized (asymptotically) Weyl almost periodic functions are new even in the case that the function p(x) has a constant value p ≥ 1, provided that the functions φ(x) and F (l, t) under our consideration satisfy φ(x) = x or F (l, t) = l (−1)/p .

Generalized Almost Periodicity in Lebesgue Spaces with Variable Exponents, Part II

“…If X does not contain an isomorphic copy of the sequence space c 0 , φ(x) = x and F(l, t) ≡ F(t), where lim t→+∞ F(t) = +∞, then there does not exist a non-periodic trigonometric polynomial f (•) and function p ∈ P (R) such that f ∈ e − W (p,x,F) ur (R : X). This can be verified based on the argumentation contained in Reference [12] (example 2.4 (iii)).…”

“…This statement continues to hold for generalized uniformly recurrent functions introduced above. For example, the use of same arguments as in Reference [12] shows that any continuous Stepanov p-almost periodic function f (•) which is not periodic cannot be Weyl-(p, x, 1)-uniformly recurrent, while it is always equi-Weyl-(p, x, 1)-almost periodic.…”

“…Let us recall that any equi-Weyl-p-almost periodic function needs to be Weyl p-almost periodic, while the converse statement does not hold in general. In Reference [12], we showed that an equi-Weyl-(p, φ, ψ)-almost periodic function, resp. equi-Weyl-(p, φ, ψ) i -almost periodic function, does not need to be Weyl-(p, φ, ψ)-almost periodic, resp.…”

“…The spaces introduced in these papers are not translation invariant, in contrast to the spaces of Stepanov p(x)-almost periodic functions and Stepanov p(x)-almost automorphic functions considered by Diagana and Kostić in Reference [10,11]. The class of (asymptotically) Weyl almost periodic functions in Lebesgue spaces with variable exponents has been analyzed in Reference [12], while the classes of Stepanov uniformly recurrent functions, Doss uniformly recurrent functions, and Doss almost periodic functions in Lebesgue spaces with variable exponents were analyzed in the first part of this research study by Kostić and Du [13].…”

“…The spaces introduced in Definitions 3-5 may not be translation invariant, in general, which is not the case with the spaces introduced in Definitions 6-8. The main aim of Section 2 was to explain without proofs how the structural results and characterizations established for generalized (equi-)Weyl almost periodic functions in Reference [12] can be straightforwardly extended for the corresponding classes of generalized (equi-)Weyl uniformly recurrent functions. In Definition 9, we introduce the class of quasi-asymptotically uniformly recurrent functions.…”