  This paper considers the relativistic motion of charged particles coupled
with electromagnetic fields in the higher-order theory proposed by Bopp,
Land\'e--Thomas, and Podolsky. We rigorously derive a world-line integral
expression for the self-force of the charged particle from a distributional
equation for the conservation of four-momentum only. This naturally leads to an
equation of motion for charged particles that incorporates a history-dependent
self-interaction. We show additionally that the same equation of motion follows
from a variational principle for retarded fields. The self-force coincides with
an expression proposed by Zayats and Gratus--Perlick--Tucker on the basis of an
averaging procedure.
