On completeness of the space of weighted Stepanov-like pseudo almost automorphic (periodic) functions

In this paper, we prove the completeness of the space of weighted Stepanov-like pseudo almost automorphic (periodic) functions under weak conditions. That is, for every ρ∈U∞, the space of weighted Stepanov-like pseudo almost automorphic (periodic) functions is complete under the norm ‖⋅‖Sp. The main method used is to skillfully construct the decomposition of a function, then a norm inequality is established and to get the related convergent subsequences of Cauchy sequences in the space.