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Asymptotic for the perturbed heavy ball system with vanishing damping term

  • *Corresponding author: Ramzi May

    *Corresponding author: Ramzi May
The authors are grateful to the Deanship of Scientific Research at King Faisal University for financially and morally supporting this work under Project 160052
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  • We investigate the long time behavior of solutions to the differential equation:

    $\ddot{x}(t)+\frac{c}{{{\left( 1+t \right)}^{\alpha }}}\dot{x}(t)+\nabla \Phi \left( x(t) \right)=g(t),~t\ge 0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 1 \right)$

    where $c$ is nonnegative constant, $α∈\lbrack0,1[,Φ \ \ {\rm{is \ \ a}}\ \ C^{1}$ convex function defined on a Hilbert space $\mathcal{H}$ and $g∈ L^{1}(0,+∞;\mathcal{H}).$ We obtain sufficient conditions on the source term $g(t)$ ensuring the weak or the strong convergence of any trajectory $x(t)$ of (1) as $t\to ∞$ to a minimizer of the function $Φ$ if one exists.

    Mathematics Subject Classification: Primary: 34A34, 34A40; Secondary: 34D05, 34E10.


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