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Ramzi May

Publications and source records attributed to Ramzi May.

18 recordsLinked to original sources
arXiv2025

On the convergence of a perturbed one dimensional Mann's process

We consider the perturbed Mann's iterative process \begin{equation} x_{n+1}=(1-\theta_n)x_n+\theta_n f(x_n)+r_n, \end{equation} where $f:[0,1]\rightarrow[0,1]$ is a continuous function, $\{\theta_n\}\in [0,1]$ is a given sequence, and $\{r_n\}$ is the error term. We establish that if the sequence $\{\theta_n\}$ converges relatively slowly to $0$ and the error term $r_n$ becomes enough small at infinity, any sequences $\{x_n\}\in [0,1]$ satisfying the process converges to a fixed point of the function $f$. We also study the asymptotic behavior of the trajectories $x(t)$ as $t\rightarrow\infty$ of a continuous version of the the considered. We investigate the similarities between the asymptotic behaviours of the sequences generated by the considered discrete process and the trajectories $x(t)$ of its corresponding continuous version.

math.GM↗
arXiv2022

On the convergence of the continuous version of the Moudafi's viscosity approximation method

We study the asymptotic behavior of trajectories of the continuous dynamical system (CDS) associated to the the discrete viscosity approximation method for fixed point problem of nonexpansive mapping (DDS) which was introduced by Moudafi in 2000 [A. Moudafi, Viscosity approximation methods for fixed points problems, J. Math. Anal. Appl. 241 (2000), 46-55]. We establish that the trajectories $x(t)$ of the system (CDS) and the sequences $(x_{n})$ generated by the the discrete process (DDS) have a very similar asymptotic behaviors.

math.CA↗
arXiv2022

Viscosity approximation method for a variational problem

Let $Q$ be a nonempty closed and convex subset of a real Hilbert space $% \mathcal{H}$, $S:Q\rightarrow Q$ a nonexpansive mapping, $A:Q\rightarrow Q$ an inverse strongly monotone operator, and $f:Q\rightarrow Q$ a contraction mapping. We prove, under appropriate conditions on the real sequences $% \{\alpha_{n}\}$ and $\{\lambda_{n}\},$ that for any starting point $x_{1}$ in $Q,$ the sequence $\{x_{n}\}$ generated by the iterative process \begin{equation} x_{n+1}=\alpha_{n}f(x_{n})+(1-\alpha_{n})SP_{Q}(x_{n}-\lambda_{n}Ax_{n}) \label{Alg} \end{equation} converges strongly to a particular element of the set $F_{ix}(S)\cap S_{VI(A,Q)}$ which we suppose that it is nonempty, where $F_{ix}(S)$ is the set of fixed point of the mapping $% S$ and $S_{VI(A,Q)}$ is the set of $q\in Q$ such that $\langle Aq,x-q\rangle\geq0$ for every $x\in Q.$ Moreover, we study the strong convergence of a perturbed version of the algorithm generated by the above process. Finally, we apply the main result to construct an algorithm associated to a constrained convex optimization problem and we provide a numerical experiment to emphasize the effect of the parameter $\{\alpha_{n}\}$ on the convergence rate of this algorithm.

math.DS↗
arXiv2021

On the strong convergence of a perturbed algorithm to the unique solution of a variational inequality problem

Let $Q$ be a nonempty closed and convex subset of a real Hilbert space $% \mathcal{H}$. $T:Q\rightarrow Q$ is a nonexpansive mapping which has a least one fixed point. $f:Q\rightarrow \mathcal{H}$ is a Lipschitzian function, and $% F:Q\rightarrow \mathcal{H}$ is a Lipschitzian and strongly monotone mapping. We prove, under appropriate conditions on the functions $f$ and $F$, the control real sequences $\{\alpha _{n}\}$ and $\{\beta _{n}\},$ and the error term $\{e_{n}\},$ that for any starting point $x_{0}$ in $Q,$ the sequence $% \{x_{n}\}$ generated by the perturbed iterative process \[ x_{n+1}=\beta _{n}x_{n}+(1-\beta _{n})P_{Q}\left( \alpha _{n}f(x_{n})+(I-\alpha _{n}F)Tx_{n}+e_{n}\right) \] converges strongly to the unique solution of the variational inequality problem \[ \text{Find }q\in C\text{ such that }\langle F(q)-f(q),x-q\rangle \geq 0\text{ for all }x\in C \] where $C=F_{ix}(T)$ is the set of fixed points of $T.$ Our main result unifies and extends many well-known previous results.

math.DS↗
arXiv2019

On The Strong Convergence of The Gradient Projection Algorithm with Tikhonov regularizing term

We investigate the strong and the weak convergence properties of the following gradient projection algorithm with Tikhonov regularizing term \[ x_{n+1}=P_{Q}(x_{n}-γ_{n}\nabla f(x_{n})-γ_{n}α_{n}\nabla ϕ(x_{n})), \] where $P_{Q}$ is the projection operator from a Hilbert space $\mathcal{H}$ onto a given nonempty, closed and convex subset $Q,$ $f:\mathcal{H}% \rightarrow \mathbb{R}$ a regular convex function, $ϕ:\mathcal{H}% \rightarrow \mathbb{R}$ a regular strongly convex function, and $γ_{n}$ and $α_{n}$ are positive real numbers. Following a Lyuapunov approach inspired essentially from the paper [Comminetti R, Peypouquet J Sorin S. Strong asymptotic convergence of evolution equations governed by maximal monotone operators with Tikhonov regularization. J. Differential Equations. (2001); 245:3753-3763], we establish the strong convergence of $(x_{n})_{n}$ to a particular minimizer $x^{\ast }$ of $f$ on $Q$ under some simple and natural conditions on the objective function $f$\ and the sequences $(γ_{n})_{n}$ and $(α_{n})_{n}$

math.OC↗
arXiv2018

On the convergence of the continuous gradient projection method

We prove the weak and the strong convergence of the trajectories of the continuous gradient projection method under some mild assumptions on the objective function and the step size function. Moreover, we estimate the decay rate to equilibrium when the objective function satisfies a global Holderian error bound inequality.

math.OC↗
arXiv2018

An extension of an unicity class for Navier-Stokes equations

This is a translation from French of my paper [R. May, Extension d'une classe d'unicite pour les equations de Navier-Stokes, Ann. I. H. Poincaré-AN 27 (2010) 705-718. doi:10.1016/j.anihp.2009.11.007]. Q. Chen, C. Miao, and Z. Zhang \cite{CMZ} have proved that weak Leray solutions of the Navier-Stokes are unique in the class $L^{\frac{2}{1+r}% }([0,T].B_{\infty}^{r,\infty}(\mathbb{R}^{3})$ with $r\in]-\frac{1}{2},1].$ In this paper, we establish that this criterion remains true for $r\in ]-1,-\frac{1}{2}].$

math.AP↗
arXiv2016

Asymptotic for the perturbed heavy ball system with vanishing damping term

We investigate the long time behavior of solutions to the differential equation $\ddot{x}(t)+\frac{c}{\left( t+1\right) ^α}\dot{x}(t)+\nabla Φ\left( x(t)\right) =g(t),~t\geq0, $ where $c$ is nonnegative constant, $α\in\lbrack0,1[,$ $Φ$ is a $C^{1}$ convex function on a Hilbert space $\mathcal{H}$ and $g\in L^{1} (0,+\infty;\mathcal{H}).$ We obtain sufficient conditions on the source term $g(t)$ ensuring the weak or the strong convergence of any trajectory $x(t)$ as $t\rightarrow+\infty$ to a minimizer of the function $Φ$ if one exists.

math.OC↗
arXiv2016

Asymptotic for a second order evolution equation with convex potential and vanishing damping term

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 \[ x^{\prime\prime}(t)+\frac{K}{t}x^{\prime}(t)+\nablaΦ(x(t))=0, \] where $K>3$ and $Φ$ is a smooth convex function defined on an Hilbert Space $\mathcal{H}.$ Moreover, we improve slightly their result on the rate of convergence of $Φ(x(t))-\minΦ.$

math.OC↗
arXiv2016

Asymptotic for a semilinear hyperbolic equation with asymptotically vanishing damping term, convex potential, and integrable source

We investigate the long time behavior of solutions to semilinear hyperbolic equation (E$_{\alpha}$): $ u^{\prime\prime}(t)+\gamma(t)u^{\prime}(t)+Au(t)+f(u(t))=g(t),~t\geq0, $ where $A$ is a self-adjoint nonnegative operator, $f$ a function which derives from a convex function, and $\gamma$ a nonnegative function which behaviors, for $t$ large enough, as $\frac{K}{t^{\alpha}}$ with $K>0$ and $\alpha \in\lbrack0,1[.$ We obtain sufficient conditions on the source term $g(t),$ ensuring the weak or the strong convergence of any solution $u(t)$ of (E$_{\alpha}$) as $t\rightarrow+\infty$ to a solution of the stationary equation $Av+f(v)=0$ if one exists.

math.OC↗
arXiv2014

Long time behavior for a semilinear hyperbolic equation with asymtotically vanishing damping term and convex potential

We investigate the asymptotic behavior, as t goes to infinity, for a semilinear hyperbolic equation with asymptotically smal dissipation and convex potential. We prove that if the damping term behaves like K/t^αfor t large enough, k>0 and 0</alpha<1 then every global solution converges weakly to an equilibrium point. This result is a positive answer to a question left open in the paper [A. Cabot and P. Frankel, Asymptotics for some semilinear hyperbolic equation with non-autonomous damping. J. Differential Equations 252 (2012) 294-322.]

math.AP↗
arXiv2009

Global well-posedness for a Modified 2D dissipative quasi-geostrophic equation with initial data in the critical Sobolev space $H^1$

In this paper, we consider the following modified quasi-geostrophic equations $\partial_tθ+Λ^αθ+u\vec\nablaθ=0$, $u=Λ^{α-1}\mathcal{R}^\perp(θ)$ where $α\in ]0,1[$ is a fixed parameter. This equation was recently introduced by P. Constantin, G. Iyer and J. Wu in \cite{CIW} as a modification of the classical quasi-geostrophic equation. In this paper, we prove that for any initial data $θ_\ast$ in the Sobolev space $H^1(\mathbb{R}^2),$ the equation (MQG) has a global and smooth solution $θ$ in $C(\mathbb{R}^{+},H^1(\mathbb{R}^2)) .$

math.AP↗
arXiv2009

The role of the Besov space $\mathbf{B}_{\infty}^{-1,\infty}$% in the control of the eventual explosion in finite time of the regular solutions of the Navier-Stokes equations

This paper is essentially a translation from French of my article \cite{M1} published in 2003. Let $u\in C([0,T^{\ast}[;L^{3}(\mathbb{R}% ^{3})) $ be a maximal solution of the Navier-Stokes equations. We prove that $u$ is $C^{\infty}$ on $]0,T^{\ast}[\times \mathbb{R}^{3}$ and there exists a constant $\varepsilon_{\ast}>0$ independent of $u$ such that if $T^{\ast}$ is finite then, for all $ω\in \overline{S(\mathbb{R%}^{3})}^{B_{\infty }^{-1,\infty}},$ we have $\overline{\lim_{t\to T^{\ast}}}\Vert u(t)-ω\Vert_{\mathbf{B}_{\infty}^{-1,\infty}}\geq \varepsilon_{\ast}. $

math.AP↗
arXiv2009

Global Existence of Solutions to the 2D subcritical dissipative Quasi-Geostrophic equation and persistency of the initial regularity

In this paper, we prove that if the initial data $θ_0$ and its Riesz transforms ($\mathcal{R}_1(θ_0)$ and $\mathcal{R}_2(θ_0)$) belong to the space $(\overline{S(\mathbb{R}^2))}^{B_{\infty}^{1-2α,\infty}}$, where $α\in ]1/2,1[$, then the 2D Quasi-Geostrophic equation with dissipation $α$ has a unique global in time solution $θ$. Moreover, we show that if in addition $θ_0 \in X$ for some functional space $X$ such as Lebesgue, Sobolev and Besov's spaces then the solution $θ$ belongs to the space $C([0,+\infty [,X).$

math.AP↗
arXiv2009

Rôle de léspace de Besov $\mathbf{B}_{\infty}^{-1,\infty}$dans le contrôle de léxplosion èventuelle en temps fini des solutions régulières des équations de Navier-Stokes

Let $u\in C([0,T^{\ast}[;L^{n}(\mathbb{R}% ^{n})^{n})$ be a maximal solution of the Navier-Stokes equations. We prove that $u$ is $C^{\infty}$ on $]0,T^{\ast}[\times \mathbb{R}^{n}$ and there exists a constant $\varepsilon _{\ast}>0$, which depends only on $n,$ such that if $T^{\ast}$ is finite then, for all $ω\in S(\mathbb{R}% ^{n})^{n},$ we have $\overline{\lim_{t\to T^{\ast}}}\Vert u(t)-ω\Vert_{\mathbf{B}_{\infty}^{-1,\infty}}\geq \varepsilon_{\ast}.$

math.AP↗