We study the asymptotic behavior of trajectories of the continuous dynamical system(CDS) associat... more We study the asymptotic behavior of trajectories of the continuous dynamical system(CDS) associated with the discrete viscosity approximation method for fixed point problem of nonexpansive mapping (DDS) which was introduced by Moudafi (J Math Anal Appl 241: [46][47][48][49][50][51][52][53][54][55] 2000). We establish that the trajectories x(t) of the continuous dynamical system (CDS) has an asymptotic behavior similar to the behavior of the sequences (x n ) generated by the discrete viscosity approximation (DDS)
In a Hilbert setting H, we study the weak and the strong convergence properties of the trajectori... more In a Hilbert setting H, we study the weak and the strong convergence properties of the trajectories x(t) of the continuous gradient projection dynamical system where Q is a closed, non-empty and convex subset of H, the function f : H → R is regular and convex, the operator P Q : H → Q is the projection onto Q, and λ : [0, +∞[→]0, +∞[ is a an absolutely continuous function. We prove the weak convergence of the trajectories x(t) to a minimizer of f over Q, if one exists, under some mild hypothesis on the function λ(.). We also study the strong convergence and the decay rate to equilibrium of the trajectories under a global Holderian error bound assumption on the objective function f of the form: where κ > 0 and θ ∈]0, 1 2 ] are absolute constants, f * Q = min Q f , and arg min Q f is the set of the minimizers of f over Q.
Let Q be a nonempty closed and convex subset of a real Hilbert space H, S : Q → Q a nonexpansive ... more Let Q be a nonempty closed and convex subset of a real Hilbert space H, S : Q → Q a nonexpansive mapping, A : Q → Q an inverse strongly monotone operator, and f : Q → Q a contraction mapping. We prove, under appropriate conditions on the real sequences {αn} and {λn}, that for any starting point x1 in Q, the sequence {xn} generated by the iterative process converges strongly to a particular element of the set Fix(S) ∩ S V I(A,Q) which we suppose that it is nonempty, where Fix(S) is the set of fixed point of the mapping S and S V I(A,Q) is the set of q ∈ Q such that Aq, x -q ≥ 0 for every x ∈ 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 {αn} on the convergence rate of this algorithm.
Recently, A. Cabot and P. Frankel studied the long time behavior of solutions to the following se... more Recently, A. Cabot and P. Frankel studied the long time behavior of solutions to the following semilinear hyperbolic equation: where γ : R + → R + , the damping term, is a decreasing function, f is the gradient of a given convex function defined on an a real Hilbert space V, and A : V → V ′ is a linear and continuous operator assumed to be symmetric, monotone and semi-coercive. They proved that if the damping term γ(t) behaves like K t α as t → +∞, for some K > 0 and α ∈]0, 1[, then every bounded solution u to the equation (E) (i.e. u ∈ L ∞ (0, +∞; V )) converges weakly in V as t → +∞ toward a solution to the stationary equation Av + f (v) = 0. They left open the question: Does convergence still hold without assuming the boundedness of the solution? In this paper, we give a positive answer to this question. Our approach relies on precise estimates on the decay rates for the energy function along trajectories of (E).
Let Q be a nonempty closed and convex subset of a real Hilbert space H . T : Q → Q is a nonexpans... more Let Q be a nonempty closed and convex subset of a real Hilbert space H . T : Q → Q is a nonexpansive mapping which has a least one fixed point. f : Q → H is a Lipschitzian function, and F : Q → H is a Lipschitzian and strongly monotone mapping. We prove, under appropriate conditions on the functions f and F , the control real sequences {α n } and {β 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 converges strongly to the unique solution of the variational inequality problem Find q ∈ C such that F (q)f (q), xq ≥ 0 for all x ∈ 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.
We study the asymptotic behavior of trajectories of the continuous dynamical system (CDS) associa... more 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 (xn) generated by the the discrete process (DDS) have a very similar asymptotic behaviors.
We investigate the strong and the weak convergence properties of the following gradient projectio... more We investigate the strong and the weak convergence properties of the following gradient projection algorithm with Tikhonov regularizing term where P Q is the projection operator from a Hilbert space H onto a given nonempty, closed and convex subset Q, f : H → R a regular convex function, φ : H → 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. ( ); 245:3753-3763], we establish the strong convergence of (x n ) n to a particular minimizer x * 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 .
This paper is essentially a translation from French of my article M1 published in 2003. Let u∈ C(... more This paper is essentially a translation from French of my article M1 published in 2003. Let u∈ C([0,T^∗[;L^3(R maximal solution of the Navier-Stokes equations. We prove that u is C^∞ on ]0,T^∗[×R^3 and there exists a constant ε_∗>0 independent of u such that if T^∗ is finite then, for all ω∈S(R ^-1,∞, we have _t→ T^∗ u(t)-ω_B_∞^-1,∞≥ε_∗.
In this article, we prove that if the initial data θ 0 and its Riesz transforms (R 1 (θ 0) and R ... more In this article, we prove that if the initial data θ 0 and its Riesz transforms (R 1 (θ 0) and R 2 (θ 0)) belong to the space (S(R 2)) B 1−2α,∞ ∞ , 1/2 < α < 1, then the 2-D Quasi-Geostrophic equation with dissipation α has a unique global in time solution θ. Moreover, we show that if in addition θ 0 ∈ X for some functional space X such as Lebesgue, Sobolev and Besov's spaces then the solution θ belongs to the space C([0, +∞[, X).
This is a translation from French of my paper [R. May, Extension d'une classe d'unicite p... more 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. Poincar\'{e}-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 $r\in ]-1,-\frac{1}{2}].$
We study the strong convergence properties of the discreet projected gradient method and the asym... more We study the strong convergence properties of the discreet projected gradient method and the asymptotic behavior of the trajectories of its continuous version under a general growth condition on the convex objective function.
Let Q be a nonempty closed and convex subset of a real Hilbert space H , T : Q → Q a nonexpansive... more Let Q be a nonempty closed and convex subset of a real Hilbert space H , T : Q → Q a nonexpansive mapping which has a least one fixed point f : Q → H a Lipschitzian function, and F : Q → H a Lipschitzian and strongly monotone mapping. We prove, under appropriate conditions on the functions f and F , the control real sequences {αn} and {βn}, and the error term {en}, that for any starting point x0 in Q, the sequence {xn} generated by the iterative process xn+1 = βnxn + (1− βn)PQ (αnf(xn) + (I − αnF )Txn + en) converges strongly to the unique solution of the variational inequality problem Find q ∈ C such that 〈F (q)− f(q), x− q〉 ≥ 0 for all x ∈ C where C = Fix(T ) is the set of fixed points of T. Our main result unifies and extends many well-known previous results.
In a Hilbert setting H, we study the weak and the strong convergence properties of the trajectori... more In a Hilbert setting H, we study the weak and the strong convergence properties of the trajectories x(t) of the continuous gradient projection dynamical system where Q is a closed, non-empty and convex subset of H, the function f : H → R is regular and convex, the operator P Q : H → Q is the projection onto Q, and λ : [0, +∞[→]0, +∞[ is a an absolutely continuous function. We prove the weak convergence of the trajectories x(t) to a minimizer of f over Q, if one exists, under some mild hypothesis on the function λ(.). We also study the strong convergence and the decay rate to equilibrium of the trajectories under a global Holderian error bound assumption on the objective function f of the form: where κ > 0 and θ ∈]0, 1 2 ] are absolute constants, f * Q = min Q f , and arg min Q f is the set of the minimizers of f over Q.
In this short note, we recover by a different method the new result due to Attouch, Chbani, Peyro... more In this short note, we recover by a different method the new result due to Attouch, Chbani, Peyrouqet and Redont concerning the weak convergence as t → +∞ of solutions x(t) to the second order differential equation where K > 3 and Φ is a smooth convex function defined on an Hilbert Space H. Moreover, we improve their result on the rate of convergence of Φ(x(t))min Φ.
Journal of Mathematical Analysis and Applications, 2015
Recently, A. Cabot and P. Frankel studied the long time behavior of solutions to the following se... more Recently, A. Cabot and P. Frankel studied the long time behavior of solutions to the following semilinear hyperbolic equation: where γ : R + → R + , the damping term, is a decreasing function, f is the gradient of a given convex function defined on an a real Hilbert space V, and A : V → V ′ is a linear and continuous operator assumed to be symmetric, monotone and semi-coercive. They proved that if the damping term γ(t) behaves like K t α as t → +∞, for some K > 0 and α ∈]0, 1[, then every bounded solution u to the equation (E) (i.e. u ∈ L ∞ (0, +∞; V )) converges weakly in V as t → +∞ toward a solution to the stationary equation Av + f (v) = 0. They left open the question: Does convergence still hold without assuming the boundedness of the solution? In this paper, we give a positive answer to this question. Our approach relies on precise estimates on the decay rates for the energy function along trajectories of (E).
This paper is essentially a translation from French of my article )) be a maximal solution of the... more This paper is essentially a translation from French of my article )) be a maximal solution of the Navier-Stokes equations. We prove that u is C ∞ on ]0, T * [×R 3 and there exists a constant ε * > 0 independent of u such that if T * is finite then, for all ω ∈ S(R 3 )
In this paper, we prove that if the initial data $\theta_0$ and its Riesz transforms ($\mathcal{R... more In this paper, we prove that if the initial data $\theta_0$ and its Riesz transforms ($\mathcal{R}_1(\theta_0)$ and $\mathcal{R}_2(\theta_0)$) belong to the space $(\overline{S(\mathbb{R}^2))}^{B_{\infty}^{1-2\alpha ,\infty}}$, where $\alpha \in ]1/2,1[$, then the 2D Quasi-Geostrophic equation with dissipation $\alpha$ has a unique global in time solution $\theta$. Moreover, we show that if in addition $\theta_0 \in X$ for some functional space $X$ such as Lebesgue, Sobolev and Besov's spaces then the solution $\theta$ belongs to the space $C([0,+\infty [,X).$
We study the asymptotic behavior of trajectories of the continuous dynamical system(CDS) associat... more We study the asymptotic behavior of trajectories of the continuous dynamical system(CDS) associated with the discrete viscosity approximation method for fixed point problem of nonexpansive mapping (DDS) which was introduced by Moudafi (J Math Anal Appl 241: [46][47][48][49][50][51][52][53][54][55] 2000). We establish that the trajectories x(t) of the continuous dynamical system (CDS) has an asymptotic behavior similar to the behavior of the sequences (x n ) generated by the discrete viscosity approximation (DDS)
In a Hilbert setting H, we study the weak and the strong convergence properties of the trajectori... more In a Hilbert setting H, we study the weak and the strong convergence properties of the trajectories x(t) of the continuous gradient projection dynamical system where Q is a closed, non-empty and convex subset of H, the function f : H → R is regular and convex, the operator P Q : H → Q is the projection onto Q, and λ : [0, +∞[→]0, +∞[ is a an absolutely continuous function. We prove the weak convergence of the trajectories x(t) to a minimizer of f over Q, if one exists, under some mild hypothesis on the function λ(.). We also study the strong convergence and the decay rate to equilibrium of the trajectories under a global Holderian error bound assumption on the objective function f of the form: where κ > 0 and θ ∈]0, 1 2 ] are absolute constants, f * Q = min Q f , and arg min Q f is the set of the minimizers of f over Q.
Let Q be a nonempty closed and convex subset of a real Hilbert space H, S : Q → Q a nonexpansive ... more Let Q be a nonempty closed and convex subset of a real Hilbert space H, S : Q → Q a nonexpansive mapping, A : Q → Q an inverse strongly monotone operator, and f : Q → Q a contraction mapping. We prove, under appropriate conditions on the real sequences {αn} and {λn}, that for any starting point x1 in Q, the sequence {xn} generated by the iterative process converges strongly to a particular element of the set Fix(S) ∩ S V I(A,Q) which we suppose that it is nonempty, where Fix(S) is the set of fixed point of the mapping S and S V I(A,Q) is the set of q ∈ Q such that Aq, x -q ≥ 0 for every x ∈ 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 {αn} on the convergence rate of this algorithm.
Recently, A. Cabot and P. Frankel studied the long time behavior of solutions to the following se... more Recently, A. Cabot and P. Frankel studied the long time behavior of solutions to the following semilinear hyperbolic equation: where γ : R + → R + , the damping term, is a decreasing function, f is the gradient of a given convex function defined on an a real Hilbert space V, and A : V → V ′ is a linear and continuous operator assumed to be symmetric, monotone and semi-coercive. They proved that if the damping term γ(t) behaves like K t α as t → +∞, for some K > 0 and α ∈]0, 1[, then every bounded solution u to the equation (E) (i.e. u ∈ L ∞ (0, +∞; V )) converges weakly in V as t → +∞ toward a solution to the stationary equation Av + f (v) = 0. They left open the question: Does convergence still hold without assuming the boundedness of the solution? In this paper, we give a positive answer to this question. Our approach relies on precise estimates on the decay rates for the energy function along trajectories of (E).
Let Q be a nonempty closed and convex subset of a real Hilbert space H . T : Q → Q is a nonexpans... more Let Q be a nonempty closed and convex subset of a real Hilbert space H . T : Q → Q is a nonexpansive mapping which has a least one fixed point. f : Q → H is a Lipschitzian function, and F : Q → H is a Lipschitzian and strongly monotone mapping. We prove, under appropriate conditions on the functions f and F , the control real sequences {α n } and {β 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 converges strongly to the unique solution of the variational inequality problem Find q ∈ C such that F (q)f (q), xq ≥ 0 for all x ∈ 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.
We study the asymptotic behavior of trajectories of the continuous dynamical system (CDS) associa... more 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 (xn) generated by the the discrete process (DDS) have a very similar asymptotic behaviors.
We investigate the strong and the weak convergence properties of the following gradient projectio... more We investigate the strong and the weak convergence properties of the following gradient projection algorithm with Tikhonov regularizing term where P Q is the projection operator from a Hilbert space H onto a given nonempty, closed and convex subset Q, f : H → R a regular convex function, φ : H → 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. ( ); 245:3753-3763], we establish the strong convergence of (x n ) n to a particular minimizer x * 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 .
This paper is essentially a translation from French of my article M1 published in 2003. Let u∈ C(... more This paper is essentially a translation from French of my article M1 published in 2003. Let u∈ C([0,T^∗[;L^3(R maximal solution of the Navier-Stokes equations. We prove that u is C^∞ on ]0,T^∗[×R^3 and there exists a constant ε_∗>0 independent of u such that if T^∗ is finite then, for all ω∈S(R ^-1,∞, we have _t→ T^∗ u(t)-ω_B_∞^-1,∞≥ε_∗.
In this article, we prove that if the initial data θ 0 and its Riesz transforms (R 1 (θ 0) and R ... more In this article, we prove that if the initial data θ 0 and its Riesz transforms (R 1 (θ 0) and R 2 (θ 0)) belong to the space (S(R 2)) B 1−2α,∞ ∞ , 1/2 < α < 1, then the 2-D Quasi-Geostrophic equation with dissipation α has a unique global in time solution θ. Moreover, we show that if in addition θ 0 ∈ X for some functional space X such as Lebesgue, Sobolev and Besov's spaces then the solution θ belongs to the space C([0, +∞[, X).
This is a translation from French of my paper [R. May, Extension d'une classe d'unicite p... more 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. Poincar\'{e}-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 $r\in ]-1,-\frac{1}{2}].$
We study the strong convergence properties of the discreet projected gradient method and the asym... more We study the strong convergence properties of the discreet projected gradient method and the asymptotic behavior of the trajectories of its continuous version under a general growth condition on the convex objective function.
Let Q be a nonempty closed and convex subset of a real Hilbert space H , T : Q → Q a nonexpansive... more Let Q be a nonempty closed and convex subset of a real Hilbert space H , T : Q → Q a nonexpansive mapping which has a least one fixed point f : Q → H a Lipschitzian function, and F : Q → H a Lipschitzian and strongly monotone mapping. We prove, under appropriate conditions on the functions f and F , the control real sequences {αn} and {βn}, and the error term {en}, that for any starting point x0 in Q, the sequence {xn} generated by the iterative process xn+1 = βnxn + (1− βn)PQ (αnf(xn) + (I − αnF )Txn + en) converges strongly to the unique solution of the variational inequality problem Find q ∈ C such that 〈F (q)− f(q), x− q〉 ≥ 0 for all x ∈ C where C = Fix(T ) is the set of fixed points of T. Our main result unifies and extends many well-known previous results.
In a Hilbert setting H, we study the weak and the strong convergence properties of the trajectori... more In a Hilbert setting H, we study the weak and the strong convergence properties of the trajectories x(t) of the continuous gradient projection dynamical system where Q is a closed, non-empty and convex subset of H, the function f : H → R is regular and convex, the operator P Q : H → Q is the projection onto Q, and λ : [0, +∞[→]0, +∞[ is a an absolutely continuous function. We prove the weak convergence of the trajectories x(t) to a minimizer of f over Q, if one exists, under some mild hypothesis on the function λ(.). We also study the strong convergence and the decay rate to equilibrium of the trajectories under a global Holderian error bound assumption on the objective function f of the form: where κ > 0 and θ ∈]0, 1 2 ] are absolute constants, f * Q = min Q f , and arg min Q f is the set of the minimizers of f over Q.
In this short note, we recover by a different method the new result due to Attouch, Chbani, Peyro... more In this short note, we recover by a different method the new result due to Attouch, Chbani, Peyrouqet and Redont concerning the weak convergence as t → +∞ of solutions x(t) to the second order differential equation where K > 3 and Φ is a smooth convex function defined on an Hilbert Space H. Moreover, we improve their result on the rate of convergence of Φ(x(t))min Φ.
Journal of Mathematical Analysis and Applications, 2015
Recently, A. Cabot and P. Frankel studied the long time behavior of solutions to the following se... more Recently, A. Cabot and P. Frankel studied the long time behavior of solutions to the following semilinear hyperbolic equation: where γ : R + → R + , the damping term, is a decreasing function, f is the gradient of a given convex function defined on an a real Hilbert space V, and A : V → V ′ is a linear and continuous operator assumed to be symmetric, monotone and semi-coercive. They proved that if the damping term γ(t) behaves like K t α as t → +∞, for some K > 0 and α ∈]0, 1[, then every bounded solution u to the equation (E) (i.e. u ∈ L ∞ (0, +∞; V )) converges weakly in V as t → +∞ toward a solution to the stationary equation Av + f (v) = 0. They left open the question: Does convergence still hold without assuming the boundedness of the solution? In this paper, we give a positive answer to this question. Our approach relies on precise estimates on the decay rates for the energy function along trajectories of (E).
This paper is essentially a translation from French of my article )) be a maximal solution of the... more This paper is essentially a translation from French of my article )) be a maximal solution of the Navier-Stokes equations. We prove that u is C ∞ on ]0, T * [×R 3 and there exists a constant ε * > 0 independent of u such that if T * is finite then, for all ω ∈ S(R 3 )
In this paper, we prove that if the initial data $\theta_0$ and its Riesz transforms ($\mathcal{R... more In this paper, we prove that if the initial data $\theta_0$ and its Riesz transforms ($\mathcal{R}_1(\theta_0)$ and $\mathcal{R}_2(\theta_0)$) belong to the space $(\overline{S(\mathbb{R}^2))}^{B_{\infty}^{1-2\alpha ,\infty}}$, where $\alpha \in ]1/2,1[$, then the 2D Quasi-Geostrophic equation with dissipation $\alpha$ has a unique global in time solution $\theta$. Moreover, we show that if in addition $\theta_0 \in X$ for some functional space $X$ such as Lebesgue, Sobolev and Besov's spaces then the solution $\theta$ belongs to the space $C([0,+\infty [,X).$
Uploads
Papers by Ramzi May