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

Marko Kostić,Wei-Shih Du

doi:10.3390/math8071052

Abstract

In this paper, we introduce and analyze several different notions of almost periodic type functions and uniformly recurrent type functions in Lebesgue spaces with variable exponent L p ( x ) . We primarily consider the Stepanov and Weyl classes of generalized almost periodic type functions and generalized uniformly recurrent type functions. We also investigate the invariance of generalized almost periodicity and generalized uniform recurrence with variable exponents under the actions of convolution products, providing also some illustrative applications to the abstract fractional differential inclusions in Banach spaces.

Highlights

Introduction and PreliminariesThe notion of almost periodicity was first studied by Bohr around 1925 and later generalized by Stepanov, Weyl, and Besicovitch, amongst many others
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 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]

Summary

Introduction and Preliminaries

The notion of almost periodicity was first studied by Bohr around 1925 and later generalized by Stepanov, Weyl, and Besicovitch, amongst many others. As mentioned in the abstract, the main aim of this paper was to analyze several different notions of almost periodic type functions and uniformly recurrent type functions in Lebesgue spaces with. The paper is conceptualized as a certain continuation of our recent research studies [13,14], where we recently analyzed quasi-asymptotically almost periodic functions and their Stepanov generalizations ( see Reference [15]). If φ : [0, ∞) → R is a concave function, the above inequalities reverse

Almost Periodic and Uniformly Recurrent Type Functions

Stepanov Classes of Quasi-Asymptotically Uniformly Recurrent Type Functions

Applications to the Abstract Volterra Integro-Differential Equations

Conclusions