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Long-Time Behavior of a Gradient System Governed by a Quasiconvex Function

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We consider a second-order differential equation governed by a quasiconvex function with a nonempty set of minimizers. Assuming that the gradient of this function is Lipschitz continuous, the existence of solutions to the gradient system is guaranteed. We study the asymptotic behavior of these solutions in continuous and discrete times. More precisely, we show that, if a solution is bounded, then it converges weakly to a critical point of the function; otherwise, it goes to infinity (in norm). We also provide several sufficient conditions for obtaining strong convergence in both continuous and discrete cases. Our work is motivated by an open problem proposed by Khatibzadeh and Moroşanu (J Convex Anal 26:1175–1186, 2019), and we solve this problem in the case, where the gradient of the function is Lipschitz continuous on bounded sets.

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The authors are grateful to anonymous reviewers for their careful reading of the manuscript and for their useful and constructive comments and questions leading to the improvement of the paper.

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Correspondence to Mohsen Rahimi Piranfar.

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Communicated by Nikolai Pavlovich Osmolovskii.

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Rahimi Piranfar, M., Khatibzadeh, H. Long-Time Behavior of a Gradient System Governed by a Quasiconvex Function. J Optim Theory Appl 188, 169–191 (2021).

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