  The electronic response of a two-dimensional (2D) electron system represents
a key quantity in discussing one-electron properties of electrons in
semiconductor heterojunctions, on the surface of liquid helium and in
copper-oxide planes of high-temperature superconductors. We here report an
evaluation of the wave-vector and frequency dependent dynamical quadratic
density-response function of a 2D electron gas (2DEG), within a self-consistent
field approximation. We use this result to find the $Z_1^3$ correction to the
stopping power of a 2DEG for charged particles moving at a fixed distance from
the plane of the 2D sheet, $Z_1$ being the projectile charge. We reproduce, in
the high-density limit, previous full nonlinear calculations of the stopping
power of a 2DEG for slow antiprotons, and we go further to calculate the
$Z_1^3$ correction to the stopping power of a 2DEG for a wide range of
projectile velocities. Our results indicate that linear response calculations
are, for all projectile velocities, less reliable in two dimensions than in
three dimensions.
