Generalization of two theorems of M. I. Kadets concerning the indefinite integral of abstract almost periodic functions

Conditions are obtained for the almost periodicity (or almost automorphy) of an abstract function $f(t)$ on a group $G$ satisfying the difference equations $f(t\gamma)-f(t)=g_\gamma(t)$, where, for each $\gamma\in G$, the function $f(t)$ is almost periodic (or almost automorphic) (the difference problem). The investigation of the almost periodicity of the integral $\int_0^x\varphi(t)dt$ of an almost periodic function $\varphi(t)$ on the real line $R$ is reduced to a study of the difference problem.