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
hkhatibzadeh@znu.ac.ir
Gheorghe Morosanu
Faculty of Mathematics and Computer Science, Babes-Bolyai University, 1 M. Kogalniceanu Street, 400084 Cluj-Napoca, Romania
morosanu@math.ubbcluj.ro
[Abstract-pdf]
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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