Strong asymptotic convergence of evolution equations governed by maximal monotone operators with Tikhonov regularization
type: Item
instance of: scholarly article
title: Strong asymptotic convergence of evolution equations governed by maximal monotone operators with Tikhonov regularization
zbMATH Open document ID: 1169.34045
published in: Journal of Differential Equations
publication date: 2008-12-12T00:00:00Z
full work available at URL: strong-asymptotic-convergence-of-evolution-equations-governed-by-maximal-monotone-operators-with-tikhonov-regularization(e5419fd5-1697-45f8-9664-df447a2ddc45).html
review text: The authors study the asymptotic behavior of the strong solution to the differential inclusion \[ \begin{gathered} -\dot u(t)\in A(u(t))+ \varepsilon(t) u(t),\\ u(0)= x_0\end{gathered} \] in a Hilbert space, where \(A\) is a maximal monotone operator, \(\varepsilon\) is nonnegative and measurable, \(\int^\infty_0 \varepsilon(t)\,dt= \infty\) and \(\varepsilon(t)\to 0\) as \(t\to\infty\). Existence and uniqueness of a strong solution is assumed. This problem is a perturbation of \[ \begin{gathered} -\dot u(t)\in A(u(t)),\\ u(0)= x_0\end{gathered} \] and it is assumed that the set \(S\) of rest points to the unperturbed problem is nonempty. In Theorem 2, it is assumed that \(A\) is the subdifferential of a closed, proper, convex function, and strong convergence of \(u(t)\) to \(x^*\) as \(t\to\infty\) is proven, where \(x^*\) is the element of least norm of \(S\). The proof makes use of a Gronwall-type inequality. In Proposition 5, for a general maximal monotone operator \(A\) by adapting the argument in [\textit{H. Attouch} and \textit{R. Cominetti}, J. Differ. Equations 128, No.~2, 519--540 (1996; Zbl 0886.49024)], it is proven that for decreasing \(\varepsilon\) we have \(u(t)\to x^*\) strongly. In Theorem 9, the monotonicity is replaced by bounded variation; this provides an extension of Theorem 1.4 in [\textit{S. Reich}, Nonlinear Anal., Theory, Methods Appl. 1, 319--330 (1977; Zbl 0359.34059)]. Finally, an example is given to demonstrate bounded variation is needed in Theorem 9.
reviewed by: Daniel C. Biles
Mathematics Subject Classification ID: 34G25
Mathematics Subject Classification ID: 47H14
Mathematics Subject Classification ID: 47J35
Mathematics Subject Classification ID: 26A45
Mathematics Subject Classification ID: 34E10
zbMATH DE Number: 5382298
zbMATH Keywords: maximal monotone operator
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: perturbation
zbMATH Keywords: differential inclusion
zbMATH Keywords: bounded variation
zbMATH Keywords: Tikhonov regularization
author: Roberto Cominetti
author: Juan Peypouquet
author: Sylvain Sorin
MaRDI profile type: Publication
OpenAlex ID: W2013921128
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DOI: 10.1016/J.JDE.2008.08.007
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description: scientific article; zbMATH DE number 5382298