Representation functions in the set of natural numbers

Let $\mathbb{N}$ be the set of all nonnegative integers. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_S(n)$ denote the number of solutions of the equation $n=s+s^{\mathrm{\prime }}$, $s,s^{\mathrm{\prime }}\in S$, $s<s^{\mathrm{\prime }}$. In this paper, we determine the structure of all sets $A$ and $B$ such that $A\cup B=\mathbb{N}\setminus \{r+mk:k\in \mathbb{N}\}$, $A\cap B=\mathrm{\varnothing }$ and $R_A(n)=R_B(n)$ for every positive integer $n$, where $m$ and $r$ are two integers with $m\ge 2$ and $r\ge 0$.