Asymptotic for a second-order evolution equation with convex potential andvanishing damping term | Zendy

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Asymptotic for a second-order evolution equation with convex potential andvanishing damping term

Author(s) -

Ramzi May

Publication year - 2017

Publication title -

turkish journal of mathematics

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.454

H-Index - 27

eISSN - 1303-6149

pISSN - 1300-0098

DOI - 10.3906/mat-1512-28

Subject(s) - nabla symbol , mathematics , order (exchange) , prime (order theory) , hilbert space , convex function , regular polygon , term (time) , function (biology) , combinatorics , mathematical analysis , convergence (economics) , physics , geometry , quantum mechanics , finance , economics , evolutionary biology , omega , biology , economic growth

In this short note, we recover by a different method the new result due to Attouch, Peyrouqet and Redont concerning the weak convergence as $t\rightarrow+\infty$ of solutions $x(t)$ to the second order differential equation \[ x^{\prime\prime}(t)+\frac{K}{t}x^{\prime}(t)+\nabla\Phi(x(t))=0, \] where $K>3$ and $\Phi$ is a smooth convex function defined on an Hilbert Space $\mathcal{H}.$ Moreover, we improve slightly their result on the rate of convergence of $\Phi(x(t))-\min\Phi.$

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