  This paper presents a variational approach to doubly-nonlinear (gradient)
flows (P) of nonconvex energies along with nonpotential perturbations (i.e.,
perturbation terms without any potential structures). An elliptic-in-time
regularization of the original equation ${\rm (P)}_\varepsilon$ is introduced,
and then, a variational approach and a fixed-point argument are employed to
prove existence of strong solutions to regularized equations. More precisely,
we introduce a functional (defined for each entire trajectory and including a
small approximation parameter $\varepsilon$) whose Euler-Lagrange equation
corresponds to the elliptic-in-time regularization of an unperturbed (i.e.
without nonpotential perturbations) doubly-nonlinear flow. Secondly, due to the
presence of nonpotential perturbation, a fixed-point argument is performed to
construct strong solutions $u_\varepsilon$ to the elliptic-in-time regularized
equations ${\rm (P)}_\varepsilon$. Here, the minimization problem mentioned
above defines an operator $S$ whose fixed point corresponds to a solution
$u_\varepsilon$ of ${\rm (P)}_\varepsilon$. Finally, a strong solution to the
original equation (P) is obtained by passing to the limit of $u_\varepsilon$ as
$\varepsilon \to 0$. Applications of the abstract theory developed in the
present paper to concrete PDEs are also exhibited.
