Stepanov-Like Pseudo-Almost Periodic Functions in Lebesgue Spaces with Variable Exponents L p(x)

Almost periodic and asymptotically almost periodic type functions in Lebesgue spaces with variable exponents Lp(x)

“…For additional details upon Lebesgue spaces with variable exponents L p(x) , we refer the reader to the following sources: [6], [7], [8], [10] and [23].…”

“…S (I : X). This is not the case with the notion introduced by Diagana and Zitane [6]- [7], since there the space L p(x) S (I : X) may or may not be translation invariant depending on p(x). Furthermore, let us note that the notion introduced in these papers is meaningful even in the case that p ∈ P(R).…”

“…The concept of almost periodicity (respectively, almost automorphy, pseudo-almost periodicity, and pseudo-almost automorphy) in the Lebesgue space with variable exponent L p(x) (I, X) was first introduced and studied by Diagana and Zitane [6,7]. However, the translation-invariance of these newly introduced spaces depends heavily upon the function p ∈ C(R + ).…”

Ergodicity in Stepanov-Orlicz spaces

“…Zhang's ergodicity has undergone various important generalizations, such as weighted pseudo almost periodicity and μ-pseudo almost periodicity introduced by Blot et al [2,3] for which the other previous definitions become just a particular case. In [5], Diagana and Zitane introduce and study a new class of weighted Stepanov-like pseudo-almost periodic functions with variable exponents, which include Stepanov pseudo almost periodicity [4] as a special case. These notions of pseudo almost periodicity and their generalizations, including Stepanov ergodicity [4] defined in L p spaces, have been successfully researched in abstract differential equations, evolution equations, and integro-differential equations because of their theoretical applications in control theory, mathematical biology, etc.…”

“…The direct impetus of this work comes from Diagana and Zitane's paper [5] where a new notion called Stepanov-like pseudo-almost periodic functions in Lebesgue spaces with variable exponents L p(.) is explored.…”