Composition Theorems of Stepanov Almost Periodic Functions and Stepanov-Like Pseudo-Almost Periodic Functions

Abstract

We establish a composition theorem of Stepanov almost periodic functions, and, with its help, a composition theorem of Stepanov-like pseudo almost periodic functions is obtained. In addition, we apply our composition theorem to study the existence and uniqueness of pseudo-almost periodic solutions to a class of abstract semilinear evolution equation in a Banach space. Our results complement a recent work due to Diagana 2008 .

Weyl almost periodic solutions to abstract linear and semilinear equations with Weyl almost periodic coefficients

“…Summing up, we have shown that, for every ε > 0, there corresponds a positive number l = l ( ε ) and there exists a relatively dense set such that which implies the Weyl almost periodicity of the solution u . Uniqueness of the solution follows from the same arguments as those in previous studies . We omit the details. □…”

Almost periodic and asymptotically almost periodic functions: part I

“…For the purpose of research of (asymptotically) almost periodic properties of solutions to semilinear Cauchy inclusions, we need to remind ourselves of the following well-known definitions and results (see, e.g., Zhang [34], Long and Ding [35] and Proposition 2.6 below). The following composition principles are well known in the existing literature (see, e.g., [34]).…”

The existence and uniqueness of almost periodic and asymptotically almost periodic solutions of semilinear Cauchy inclusions

“…For this purpose, we introduce the class of asymptotically Stepanov almost periodic functions depending on two parameters and prove some new composition principles in this direction (see e.g. [4], [23] and references therein). It seems that our main results, Theorem 2.8-Theorem 2.11, are new even for abstract semilinear non-degenerate differential equations with almost sectorial operators ( [25]- [26]).…”

“…We open the second section of paper by proving some new composition principles for Stepanov almost periodic two-parameter functions and asymptotically Stepanov almost periodic two-parameter functions. The main aim of Theorem 2.1 is to clarify that the composition principle [23,Theorem 2.2], proved by W. Long and S.-H. Ding, continues to hold for the functions defined on the real semi-axis I = [0, ∞). The use of usual Lipschitz assumption has some advantages compared to the condition f ∈ L r (R × X : X) used in the formulation of the above-mentioned theorem since, in this case, we can include the order of (asymptotic) Stepanov almost periodicity p = 1 in our analyses (cf.…”

“…For the purpose of research of (asymptotically) almost periodic properties of solutions to semilinear Cauchy inclusions, we need to remind ourselves of the following well-known definitions and results (see e.g. C. Zhang [29], W. Long, S.-H. Ding [23], and Proposition 1.4 below):…”