Weyl double-measure pseudo almost periodic functions and Weyl double-measure pseudo almost periodic solutions to semilinear evolution equations

Abstract

This paper aims to address significant gaps in the theory of almost periodic functions and their applications in differential equations. Although Weyl almost periodic functions generalize Bohr almost periodic functions and Stepanov almost periodic functions, the incompleteness of the space formed by Weyl almost periodic functions and the underdeveloped theory of Weyl pseudo almost periodic functions have limited related research. Furthermore, a theoretical foundation for Weyl measure pseudo almost periodic functions has been lacking in the existing literature. Based on these research gaps, this study makes two main contributions: first, it introduces a new concept of Weyl bi-measure pseudo almost periodic functions via the Weyl semi-norm and systematically investigates their fundamental properties; second, as a key application, it establishes the existence and global exponential stability of such solutions for a class of semilinear differential equations. The theoretical results are supported by illustrative examples, demonstrating that the conditions required by the main theorems of this paper are easily verifiable. This research provides a new framework for the qualitative analysis of differential equations.