Electronic Journal of Differential Equations, 
Vol. 2021 (2021), No. 15, pp. 1-13.

Title: Asymptotic behavior for a quasi-autonomous gradient system of expansive
       type governed by a quasiconvex function
       
Authors: Behzad Djafari Rouhani (University of Texas at El Paso, TX, USA)
         Mohsen Rahimi Piranfar (Isfahan Mathematics House, Iran)

Abstract:
 We consider the quasi-autonomous first-order gradient system
 $$\displaylines{ 
 \dot{u}(t)=\nabla\phi(u(t))+f(t),\quad t\in [0,+\infty)\cr
 u(0)=x_0\in H,
 }$$
 where $\phi:H\to\mathbb{R}$ is a differentiable quasiconvex function such that $\nabla\phi$
 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.

Submitted January 25, 2021. Published March 18, 2021.
Math Subject Classifications: 34G20, 47J35, 39A30, 37N40.
Key Words: First-order evolution equation; expansive type gradient system;
           asymptotic behavior; quasiconvex function; minimization.