Asymptotic behavior for a quasi-autonomous gradient system of expansive type governed by a quasiconvex function

Abstract

We consider the quasi-autonomous first-order gradient system u̇(t) = ∇φ(u(t)) + ƒ(t), t ∈ [0, +∞) u(0) = x0 ∈ H, where φ : H → ℝ is a differentiable quasiconvex function such that ∇φ is Lipschitz continuous. We study the asymptotic behavior of solutions to this system in continuous and discrete time. We show that each solution either approaches infinity in norm or converges weakly to a critical point of φ. This further concludes that the existence of bounded solutions and implies that φ has a nonempty set of critical points. Some strong convergence results, as well as numerical examples, are also given in both continuous and discrete cases.