On the convergence of maximal monotone operators

Penot, Jean-Paul; Zălinescu, Constantin

doi:10.1090/s0002-9939-05-08275-4

Cited by 9 publications

(4 citation statements)

References 20 publications

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“…Remark 2.1. The main ingredient in the previous result is the inequality G M ≥ P M , which already appears in a paper by Penot and Zalinescu; see [33,Lemma 2.3]. By using the same inequality, it is shown in [33,Proposition 3.1]…”

Section: Graph Convergence and Convergence Of The Brézis-haraux Funct...mentioning

confidence: 74%

“…Under this condition, Theorem 3.2(ii) shows that lim t→+∞ (x(t), y(t)) = (x ∞ , 0), for some x ∞ ∈ [−b, b]. For every (x, y) ∈] − a, a[×R, we have By combining (32), (33) and (34), we find for every (x, y) ∈] − a, a[×R,…”

Section: Proposition 42 Assume Hypotheses (H ψ )-(H φ ) (A)mentioning

confidence: 96%

“…This property makes it an effective tool to address the problems governed by maximal monotone operators, using methods of convex analysis. It is the subject of active research; see for example [14,21,29,30,33,37,38]. There are exact calculus formulas for the Fitzpatrick function of sums or compositions of maximal monotone operators, provided that suitable qualification conditions are verified.…”

Section: Graph Convergence and Convergence Of The Brézis-haraux Funct...mentioning

confidence: 99%

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Untitled

2018

Trans. Amer. Math. Soc.

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In a Hilbert setting H, we study the asymptotic behavior of the trajectories of nonautonomous evolution equations ẋ(t) + A t (x(t)) 0, where for each t ≥ 0, A t : H ⇒ H denotes a maximal monotone operator. We provide general conditions guaranteeing the weak ergodic convergence of each trajectory x(•) to a zero of a limit maximal monotone operator A ∞ as the time variable t tends to +∞. The crucial point is to use the Brézis-Haraux function, or equivalently the Fitzpatrick function, to express at which rate the excess of gphA ∞ over gphA t tends to zero. This approach gives a sharp and unifying view of this subject. In the case of operators A t = ∂ϕ t which are subdifferentials of proper closed convex functions ϕ t , we show convergence results for the trajectories. Then, we specialize our results to multiscale evolution equations and obtain asymptotic properties of hierarchical minimization and selection of viscosity solutions. Illustrations are given in the field of coupled systems and partial differential equations.

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Section: Graph Convergence and Convergence Of The Brézis-haraux Funct...mentioning

confidence: 74%

Section: Proposition 42 Assume Hypotheses (H ψ )-(H φ ) (A)mentioning

confidence: 96%

Section: Graph Convergence and Convergence Of The Brézis-haraux Funct...mentioning

confidence: 99%

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Untitled

2018

Trans. Amer. Math. Soc.

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Section: Graph Convergence and Convergence Of The Brézis-haraux Funct...mentioning

confidence: 99%

Asymptotic behavior of nonautonomous monotone and subgradient evolution equations

Attouch

Cabot

Czarnecki

2017

Trans. Amer. Math. Soc.

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Abstract. In a Hilbert setting H, we study the asymptotic behavior of the trajectories of nonautonomous evolution equationsẋ(t) + At(x(t)) ∋ 0, where for each t ≥ 0, At : H ⇒ H denotes a maximal monotone operator. We provide general conditions guaranteeing the weak ergodic convergence of each trajectory x(·) to a zero of a limit maximal monotone operator A∞, as the time variable t tends to +∞. The crucial point is to use the Brézis-Haraux function, or equivalently the Fitzpatrick function, to express at which rate the excess of gph A∞ over gph At tends to zero. This approach gives a sharp and unifying view on this subject. In the case of operators At = ∂ϕt which are subdifferentials of closed convex functions ϕt, we show convergence results for the trajectories. Then, we specialize our results to multiscale evolution equations, and obtain asymptotic properties of hierarchical minimization, and selection of viscosity solutions. Illustrations are given in the field of coupled systems, and partial differential equations.

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