Strong asymptotic convergence of evolution equations governed by maximal monotone operators with Tikhonov regularization
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.
scientific article; zbMATH DE number 5382298
scientific article