Abstract

Consider in a real Hilbert space HH the second order gradient equation u′′(t)=∇ϕ(u(t)),   t≥0.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 ϕ\phi, where ϕ ⁣:H→R\phi\colon H\to \mathbb{R} is a continuously differentiable, pseudo-convex function with Argmin ϕ≠∅{\rm Argmin}\,\phi\neq\varnothing. Our results extend previous results in the literature that are related to the case when ϕ\phi is convex.

Author information

Contact details are reproduced from the original publication and may be historical.

Hadi Khatibzadeh

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

hkhatibzadeh@znu.ac.ir

Gheorghe Moroşanu

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

morosanu@math.ubbcluj.ro

Keywords

  • Convex function
  • pseudo-convex function
  • minimum point
  • critical point
  • second order gradient system
  • asymptotic behavior

Mathematics Subject Classification

34D05, 34D23, 34D20, 34G20

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Suggested citation

H. Khatibzadeh, G. Moroşanu. “Asymptotic Behavior of Solutions to a Second-Order Gradient Equation of Pseudo-Convex Type.” Journal of Convex Analysis 26 (2019), No. 4, 1175–1186. https://doi.org/10.68381/jca26064

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.