Asymptotic behavior for a quasi-autonomous gradient system of expansive type governed by a quasiconvex function
Date
2021-03-18
Authors
Djafari Rouhani, Behzad
Rahimi Piranfar, Mohsen
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
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.
Description
Keywords
First-order evolution equation, Expansive type gradient system, Asymptotic behavior, Quasiconvex function, Minimization
Citation
Djafari Rouhani, B., & Rahimi Piranfar, M. (2021). Asymptotic behavior for a quasi-autonomous gradient system of expansive type governed by a quasiconvex function. <i>Electronic Journal of Differential Equations, 2021</i>(15), pp. 1-13.
Rights
Attribution 4.0 International