Electron. J. Differential Equations, Vol. 2021 (2021), No. 15, pp. 1-13.

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

Behzad Djafari Rouhani, Mohsen Rahimi Piranfar

We consider the quasi-autonomous first-order gradient system

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.

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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
email: behzad@utep.edu
Mohsen Rahimi Piranfar
Isfahan Mathematics House
Isfahan, Iran
email: m.piranfar@gmail.com

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