On the iterative method for variational inequalities with nonsmooth unbounded operators in Banach space
The authors continue their investigations concerning approximate methods for solving variational inequalities with unbounded and nonsmooth operators in Banach spaces. Their previous works were concerned with regularization and penality methods, and their convergence and stability. The first author [Sov. Math., Dokl. 20, 871-876 (1979; Zbl 0452.47079)] considered the iterative method \[ x^{n+1}= P_ \Omega (x^ n- \alpha_{n+\sigma} (z^ n- f)),\quad z^ n\in Ax^ n,\quad n=0,1,\dots, \quad x_ 0\in M, \tag{1} \] for the variational inequality \[ (Ax-f, y-x)\geq 0, \qquad \forall y\in \Omega, \tag{2} \] where \(A: H\to H\). (Here \((\cdot, \cdot)\) is the inner product in the Hilbert space \(H\), \(\alpha_{n+\sigma}\) are positive number parameters, \(C\geq \sigma\geq 0\), and \(P_ \Omega\) is a projection operator on the convex closed set \(\Omega \subset H\) in the sense of best approximation.) Here the authors consider the variational inequality (2) with operator \(A\) acting from uniformly convex Banach space \(B\) to \(B^*\), \((z,x)\) where \(z\in B^*\) and \(x\in B\) denotes a dual product pairing in \(B\) and \(\Omega\) is a closed convex set, \(\Omega \subset \text{int } D(A)\), \(\text{int } D(A) \neq \emptyset\). They study the iterative method \[ Ix^{n+1}= Ix^ n- \alpha_ n | z^ n |^{-1} z^ n, \quad n=0,1,\dots, x^ 0\in B, \quad x^ n= I^* Ix^ n, \tag{3} \] \(\forall z^ n\in Ax^ n -f\), prove convergence theorems for iterative processes and establish asymptotic estimates of their convergence rate.
scientific article; zbMATH DE number 720410