Asymptotic Behavior of Solutions to a Second-Order Gradient Equation of Pseudo-Convex Type

Consider in a real Hilbert space

the second order gradient equation

u''(t) = \nabla \phi(u(t)), \ \ \ t\geq0. u ′ ′ ( t ) = ∇ ϕ ( u ( t ) ) ,     t ≥ 0.

We state and prove several results on the weak or strong convergence of bounded solutions of this equation to minimizers of

\phi ϕ

, where

\phi\colon H\to \mathbb{R} ϕ  ⁣ : H → R

is a continuously differentiable, pseudo-convex function with

{\rm Argmin}\,\phi\neq\varnothing A r g m i n   ϕ ≠ ∅

. Our results extend previous results in the literature that are related to the case when

\phi ϕ

is convex.