Geodesic
Asymptotic Behavior of Solutions to a Second-Order Gradient Equation of Pseudo-Convex Type
Journal of convex analysis, Tome 26 (2019) no. 4, pp. 1175-1186
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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.
Classification : 34D05, 34D23, 34D20, 34G20
Mots-clés : Convex function, pseudo-convex function, minimum point, critical point, second order gradient system, asymptotic behavior 
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     author = {H. Khatibzadeh and G. Morosanu},
     title = {Asymptotic {Behavior} of {Solutions} to a {Second-Order} {Gradient} {Equation} of {Pseudo-Convex} {Type}},
     journal = {Journal of convex analysis},
     pages = {1175--1186},
     year = {2019},
     volume = {26},
     number = {4},
     url = {http://geodesic.mathdoc.fr/item/JCA_2019_26_4_JCA_2019_26_4_a8/}
}
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H. Khatibzadeh; G. Morosanu. Asymptotic Behavior of Solutions to a Second-Order Gradient Equation of Pseudo-Convex Type. Journal of convex analysis, Tome 26 (2019) no. 4, pp. 1175-1186. http://geodesic.mathdoc.fr/item/JCA_2019_26_4_JCA_2019_26_4_a8/