  The paper studies the quantum action for the five-dimensional real
$\phi^3$-theory in the case of a general formulation using the background field
method. The three-loop renormalization is performed with the usage of a cutoff
regularization in the coordinate representation. The explicit form of the first
three coefficients for the renormalization constants is presented. The absence
of non-local singular contributions and partial results for the fourth
correction are discussed.
