  Motivated by optimal control problems and differential games for functional
differential equations of retarded type, the paper deals with a Cauchy problem
for a path-dependent Hamilton--Jacobi equation with a right-end boundary
condition. Minimax solutions of this problem are studied. The existence and
uniqueness result is obtained under assumptions that are weaker than those
considered earlier. In contrast to previous works, on the one hand, we do not
require any properties concerning positive homogeneity of the Hamiltonian in
the impulse variable, and on the other hand, we suppose that the Hamiltonian
satisfies a Lipshitz continuity condition with respect to the path variable in
the uniform (supremum) norm. The progress is related to the fact that a
suitable Lyapunov--Krasovskii functional is built that allows to prove a
comparison principle. This functional is in some sense equivalent to the square
of the uniform norm of the path variable and, at the same time, it possesses
appropriate smoothness properties. In addition, the paper provides non-local
and infinitesimal criteria of minimax solutions, their stability with respect
to perturbations of the Hamiltonian and the boundary functional, as well as
consistency of the approach with the non-path-dependent case. Connection of the
problem statement under consideration with some other possible statements
(regarding the choice of path spaces and derivatives used) known in the theory
of path-dependent Hamilton--Jacobi equations is discussed. Some remarks
concerning viscosity solutions of the studied Cauchy problem are given.
