  In the paper, we consider a path-dependent Hamilton-Jacobi equation with
coinvariant derivatives over the space of continuous functions. Such equations
arise from optimal control problems and differential games for time-delay
systems. We study generalized solutions of the considered Hamilton-Jacobi
equation both in the minimax and in the viscosity sense. A minimax solution is
defined as a functional which epigraph and subgraph satisfy certain conditions
of weak invariance, while a viscosity solution is defined in terms of a pair of
inequalities for coinvariant sub- and super-gradients. We prove that these two
notions are equivalent, which is the main result of the paper. As a corollary,
we obtain comparison and uniqueness results for viscosity solutions of a Cauchy
problem for the considered Hamilton-Jacobi equation and a right-end boundary
condition. The proof is based on a certain property of the coinvariant
subdifferential. To establish this property, we develop a technique going back
to the proofs of multidirectional mean-value inequalities. In particular, the
absence of the local compactness property of the underlying continuous function
space is overcome by using Borwein-Preiss variational principle with an
appropriate guage-type functional.
