Let
${{a}_{1}},...,{{a}_{9}}$
be nonzero integers and
$n$
any integer. Suppose that
${{a}_{1}}+\cdot \cdot \cdot +{{a}_{9}}\,\equiv \,n\,\left( \bmod \,2 \right)$
and
$\left( {{a}_{i}},\,{{a}_{j}} \right)\,=\,1$
for
$1\,\le \,i\,<\,j\,\le \,9$
. In this paper we prove the following:
(i) If
${{a}_{j}}$
are not all of the same sign, then the cubic equation
${{a}_{1}}p_{1}^{3}\,+\cdot \cdot \cdot +\,{{a}_{9}}\,p_{9}^{3}\,=\,n$
has prime solutions satisfying
${{p}_{j}}\,\ll \,{{\left| n \right|}^{{1}/{3}\;}}\,+\,\max {{\left\{ \left| {{a}_{j}} \right| \right\}}^{14+\varepsilon }}$
.
(ii) If all
${{a}_{j}}$
are positive and
$n\,\gg \,\max {{\left\{ \left| {{a}_{j}} \right| \right\}}^{43+\varepsilon }}$
, then
${{a}_{1}}p_{1}^{3}\,+\cdot \cdot \cdot +\,{{a}_{9}}\,p_{9}^{3}\,=\,n$
is solvable in primes
${{p}_{j}}$
.
These results are an extension of the linear and quadratic relative problems.