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Regularization of nonlinear ill-posed variational inequalities and convergence rates. (English) Zbl 0924.49009
Summary: Let $$H$$ be a Hilbert space and $$K$$ be a nonempty closed convex subset of $$H$$. For $$f\in H$$, we consider the (ill-posed) problem of finding $$u\in K$$ for which $\langle Au- f,v-u\rangle\geq 0\quad\text{for all }v\in K,$ where $$A:H\to H$$ is a monotone (not necessarily linear) operator. We study the approximation of the solutions of the variational inequality by using the following perturbed variational inequality: for $$f_\delta\in H$$, $$\| f_\delta- f\|\leq\delta$$, find $$u^{\delta,\eta}_\varepsilon\in K_\eta$$ for which $\langle Au^{\delta,\eta}_\varepsilon+\varepsilon u^{\delta,\eta}_\varepsilon- f_\delta, v- u^{\delta,\eta}_\varepsilon\rangle\geq 0\quad\text{for all }K_\eta,$ where $$\varepsilon$$, $$\delta$$, and $$\eta$$ are positive parameters, and $$K_\eta$$, a perturbation of the set $$K$$, is a nonempty closed convex set in $$H$$. We establish convergence and a rate $$O(\varepsilon^{1/3})$$ of convergence of the solutions of the regularized variational inequalities to a solution of the original variational inequality using the Mosco approximation of closed convex sets, where $$A$$ is a weakly differentiable inverse-strongly-monotone operator.

##### MSC:
 49J40 Variational inequalities 65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization 65J15 Numerical solutions to equations with nonlinear operators 47H05 Monotone operators and generalizations
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