TY - JOUR

T1 - Strong asymptotic convergence of evolution equations governed by maximal monotone operators with Tikhonov regularization

AU - Cominetti, R.

AU - Peypouquet, J.

AU - Sorin, S.

N1 - Funding Information:
E-mail addresses: rcominet@dim.uchile.cl (R. Cominetti), juan.peypouquet@usm.cl (J. Peypouquet), sorin@math.jussieu.fr (S. Sorin). 1 Supported by FONDAP grant in Applied Mathematics, CONICYT-Chile. 2 Supported by MECESUP grant UCH0009 and FONDAP grant in Applied Mathematics, CONICYT-Chile. 3 Supported by grant ANR-05-BLAN-0248-01.

PY - 2008/12/15

Y1 - 2008/12/15

N2 - We consider the Tikhonov-like dynamics - over(u, ̇) (t) ∈ A (u (t)) + ε (t) u (t) where A is a maximal monotone operator on a Hilbert space and the parameter function ε (t) tends to 0 as t → ∞ with ∫0∞ ε (t) d t = ∞. When A is the subdifferential of a closed proper convex function f, we establish strong convergence of u (t) towards the least-norm minimizer of f. In the general case we prove strong convergence towards the least-norm point in A-1 (0) provided that the function ε (t) has bounded variation, and provide a counterexample when this property fails.

AB - We consider the Tikhonov-like dynamics - over(u, ̇) (t) ∈ A (u (t)) + ε (t) u (t) where A is a maximal monotone operator on a Hilbert space and the parameter function ε (t) tends to 0 as t → ∞ with ∫0∞ ε (t) d t = ∞. When A is the subdifferential of a closed proper convex function f, we establish strong convergence of u (t) towards the least-norm minimizer of f. In the general case we prove strong convergence towards the least-norm point in A-1 (0) provided that the function ε (t) has bounded variation, and provide a counterexample when this property fails.

KW - Maximal monotone operators

KW - Tikhonov regularization

UR - http://www.scopus.com/inward/record.url?scp=54149100315&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2008.08.007

DO - 10.1016/j.jde.2008.08.007

M3 - Article

AN - SCOPUS:54149100315

VL - 245

SP - 3753

EP - 3763

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 12

ER -