Behzad Djafari Rouhani, Mohsen Rahimi Piranfar
We consider the quasi-autonomous first-order gradient system
where is a differentiable quasiconvex function such that 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.
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| Behzad Djafari Rouhani |
Department of Mathematical Sciences
University of Texas at El Paso
500 W. University Ave., El Paso, TX 79968, USA
| Mohsen Rahimi Piranfar |
Isfahan Mathematics House
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