An asymptotic formula for the number of solutions of a Diophantine equation

Suppose that $k$, $s$, $m_1,\dots,m_k$, $m_1',\dots,m_s'$ are fixed positive integers, $m$ is a fixed integer, $p$ is an increasing positive integer, and suppose that a sequence of integers $\{n_k\}$ satisfies the following conditions: 1) $n_{k+1}\geqslant n_k(1+k^{-1/2+\varepsilon})$ , where $\varepsilon>0$ is arbitrarily small; 2) for fixed $m,n,a,B$, the number of solutions of the Diophantine equation
$$ mn_{x+a}-nn_x=B $$
in $x$ in the half-open interval $[0,p)$ does not exceed some constant $q$ which does not depend on $m,n,a,B$.
Under these assumptions, an asymptotic formula with remainder term is derived for the number of solutions of the Diophantine equation
$$ m_1n_{x_1}+\dots+m_kn_{x_k}=m_1'n_{y_1}+\dots+m_s'n_{y_s}+m $$
in integers $0\leqslant x_1,\dots,x_k$; $y_1,\dots,y_s<p$.
The results obtained extend and refine several results obtained by other authors.
Bibliography: 7 titles.