On Wiener theorem in studying periodic at infinity functions with respect to subspaces of vanishing at infinity functions

In the article under consideration we study periodic at infinity functions from $C_b(\mathbb{J},X),$ i.e., bounded continuous functions defined on the real axis with their values in a complex Banach space $X.$ On the basis of the well-known Wiener theorem we introduce a concept of a set satisfying Wiener condition. Together with an ordinary subspace $C_0\subset C_b$ we consider various subspaces of continuous functions vanishing at infinity in different senses, not necessarily tending to zero at infinity. For example, integrally vanishing at infinity functions and functions whose convolution with any function from the set satisfying Wiener condition gives a function tending to zero at infinity. Those subspaces we also call vanishing at infinity and denote then as $\mathcal{C}_0$. So, by choosing one of the subspaces $\mathcal{C}_0$ we introduce different types of slowly varying and periodic at infinity functions (with respect to the chosen subspace). A function $x\in C_{b,u}$ is called slowly varying at infinity with respect to the subspace $\mathcal{C}_0$ if $(S(t)x - x)\in \mathcal{C}_0$ for all $t\in\mathbb{J}.$ Respectively, for some $\omega>0$ a function $x\in C_{b,u}$ is called $\omega$-periodic at infinity with respect to the subspace $\mathcal{C}_0$ if $(S(\omega)x - x)\in \mathcal{C}_0.$ Nevertheless, these functions are constructed as extensions of the classes of slowly varying and periodic at infinity functions respectively, we proved them to be congruent with these classes. Ordinary periodic at infinity functions appear naturally as bounded solutions of certain classes of differential and difference equations. So, in our research, we also study the solutions of differential and difference equations of some kind. It is proved that for those equations, where the right hand side of the equation is a function from any of the subspaces $\mathcal{C}_0$ of vanishing at infinity functions, the solutions are periodic at infinity. The results were received with essential use of isometric representations and Banach modules theories.