  This paper introduces a second-order differential inclusion for unconstrained
convex optimization. In continuous level, solution existence in proper sense is
obtained and exponential decay of a novel Lyapunov function along with the
solution trajectory is derived as well. Then in discrete level, based on
numerical discretizations of the continuous model, two inexact proximal point
algorithms are proposed, and some new convergence rates are established via a
discrete Lyapunov function.
