Abstract
In this paper, we first study some basic properties of Stepanov-like asymptotical almost periodic functions including the completeness of the space of Stepanov-like asymptotical almost periodic functions. Then, as an application, based on these and the contraction mapping principle, we obtain sufficient conditions for the existence and uniqueness of Stepanov-like asymptotical almost periodic solutions for a class of semilinear delay differential equations.
Highlights
Almost periodic functions, which are an important generalization of periodic functions, were introduced into the field of mathematics by Bohr [1, 2]
Stepanov proposed a weaker concept of almost periodic functions in the sense of Bohr
Due to the fact that almost periodic phenomena exist in the real world, more and more scholars are interested in the almost periodicity and its various generalizations
Summary
Almost periodic functions, which are an important generalization of periodic functions, were introduced into the field of mathematics by Bohr [1, 2]. The concept of Stepanovlike weighted pseudo almost periodicity was introduced by Diagana et al [14] This notion is more extensive than Stepanov-like pseudo almost periodicity. The concept of the asymptotically almost periodicity was introduced into the research field by French mathematician Frechet [17, 18] Such a notion is a natural generalization of the concept of the almost periodicity in the sense of Bohr. Motivated by the above discussions, in this paper, we first study some basic properties of Stepanov-like asymptotical almost periodic functions Based on these properties and by using the contraction mapping principle, we investigate the existence and uniqueness of Stepanovlike asymptotical almost periodic solutions for a class of semilinear delay differential equations
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