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Journal of Convex Analysis 26 (2019), No. 4, 1175--1186
Copyright Heldermann Verlag 2019

Asymptotic Behavior of Solutions to a Second-Order Gradient Equation of Pseudo-Convex Type

Hadi Khatibzadeh
Department of Mathematics, University of Zanjan, P. O. Box 45195-313, Zanjan, Iran

Gheorghe Morosanu
Faculty of Mathematics and Computer Science, Babes-Bolyai University, 1 M. Kogalniceanu Street, 400084 Cluj-Napoca, Romania


Consider in a real Hilbert space $H$ the second order gradient equation $$ u''(t) = \nabla \phi(u(t)), \ \ \ t\geq0 . $$ We state and prove several results on the weak or strong convergence of bounded solutions of this equation to minimizers of $\phi$, where $\phi\colon H\to \mathbb{R}$ is a continuously differentiable, pseudo-convex function with ${\rm Argmin}\,\phi\neq\varnothing$. Our results extend previous results in the literature that are related to the case when $\phi$ is convex.

Keywords: Convex function, pseudo-convex function, minimum point, critical point, second order gradient system, asymptotic behavior.

MSC: 34D05, 34D23, 34D20, 34G20

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