(Q, T)-affine-periodic solutions and Pseudo (Q, T)-affine-periodic solutions for Dynamic Equations on Time Scales

Abstract

The aim of this paper is to study the existence of $\left(Q,T\right)$ -affine-periodic solutions for affine-periodic systems on time scales of the type $x^{\mathrm{\Delta }}\left(t\right)=A\left(t\right)x\left(t\right)+f\left(t\right)$ and $x^{\mathrm{\Delta }}\left(t\right)=A\left(t\right)x\left(t\right)+g\left(t,x\left(t\right)\right),t\in \mathbb{T}$ , assuming that corresponding homogeneous equation of this system admits exponential dichotomy. The result is also extended to the case of pseudo $\left(Q,T\right)$ -affine-periodic solutions. The main approaches are based on the Banach contraction mapping principle, but certain technical aspects on time scales are more complicated.