\(\Pi\)-semigroup for invariant under translations time scales and abstract weighted pseudo almost periodic functions with applications (Q2825623)

The first part of the paper deals with the so-called \(\Pi\)-semigroups, which generalize the classical concept of semigroups of linear operators in a Banach space. \(\Pi\)-semigroups have the form \(\{T_\tau;\tau\in \Pi^+\}\), where \(T_\tau\) are bounded linear operators on a Banach space \(X\) and \(\Pi^+\) is a certain subset of \([0,\infty)\). To describe \(\Pi^+\), one needs the notion of a time scale invariant under translations. A time scale \(\mathbb T\), which is a nonempty closed subset of \(\mathbb R\), is called invariant under translations if the set NEWLINE\[NEWLINE\Pi=\{\tau\in\mathbb R;t\pm\tau\in\mathbb T\text{ for each }t\in\mathbb T\}NEWLINE\]NEWLINE is nonempty. The set \(\Pi^+\) is then simply \(\Pi^+=\Pi\cap[0,\infty)\), and \(\{T_\tau;\tau\in \Pi^+\}\) is a one-parameter operator semigroup. By requiring a certain kind of continuity of \(T_\tau\) with respect to \(\tau\), one arrives at the notion of a strongly continuous operator semigroup -- a \(\Pi\)-semigroup. Proceeding similarly as in the classical case when \(\Pi^+=[0,\infty)\), the authors introduce the notion of an infinitesimal generator of a \(\Pi\)-semigroup.NEWLINENEWLINEThe second part of the paper begins by recalling the concept of an almost periodic function \(f:\mathbb T\times X\to X\), where \(\mathbb T\) is invariant under translations. Then the authors proceed to their main goal, which is the study of weighted pseudo almost periodic functions. In the classical case when \(\mathbb T=\mathbb R\), such functions were introduced in \textit{T. Diagana}'s paper [C. R., Math., Acad. Sci. Paris 343, No. 10, 643--646 (2006; Zbl 1112.43005)]. Given a suitable weight function \(\rho:\mathbb T\to(0,\infty)\), a bounded continuous function \(f:\mathbb T\times X\to X\) is called weighted pseudo almost periodic if it has the form \(f=g+\phi\), where \(g:\mathbb T\times X\to X\) is almost periodic and \(\phi:\mathbb T\times X\to X\) satisfies NEWLINE\[NEWLINE\lim_{r\to\infty}\frac{\int_{t_0-r}^{t_0+r}\|f(t,x)\|\rho(t)\Delta t}{\int_{t_0-r}^{t_0+r}\rho(t)\Delta t}=0NEWLINE\]NEWLINE uniformly for all \(x\in X\). The integrals appearing here are the \(\Delta\)-integrals, which generalize the usual concept of Newton integral.NEWLINENEWLINEIn the final part, the authors consider a fixed weight function \(\rho\) and study weighted pseudo almost periodic solutions of the neutral dynamic equation NEWLINE\[NEWLINEx^\Delta(t)=Ax(t)+F^\Delta(t,x(t-g(t)))+G(t,x(t),x(t-g(t))),\quad t\in\mathbb T,NEWLINE\]NEWLINE where \(\mathbb T\) is invariant under translations, \(A\) is the infinitesimal generator of a \(\Pi\)-semigroup, \(F\), \(G\) are almost periodic, and \(g\) takes values in \(\Pi\). Using Krasnoselskii's fixed point theorem, they obtain some sufficient conditions (which are too technical to be reproduced here) for the existence and uniqueness of weighted pseudo almost periodic solutions. The result is illustrated on an example with the weight function \(\rho(t)=1+t^2+t\sigma(t)+\sigma^2(t)\).