We consider the equation \begin{gather*} y(t_1,\dots,t_n)-\sum_{q_1\dots q_n}A_{q_1\dots q_n}y(t_1-m^{(1)}_{q_1\dots q_n}a_1,\dots,t_n-m^{(n)}_{q_1\dots q_n}a_n)=f \\ (m^{(k)}_{q_1\dots q_n} \text{ -- are integers} \geqslant0;\ a_k>0;\ 0\leqslant t_1,\dots,t_n\infty), \end{gather*} where the $A_{q_1\dots q_n}=A_{q_1\dots q_n}(t_1,\dots,t_n)$ are continuous linear operator-functions operating in a complex Banach space. We establish necessary and sufficient tests for the boundedness of the solutions $y(t_1,\dots,t_n)$ of these equations for all bounded right-hand sides $f=f(t_1,\dots,t_n)$