Convergence Analysis of Iterative Methods for Some Variational Inequalities with Pseudomonotone Operators

Abstract

In the present paper, we consider variational inequalities of the second kind with a pseudomonotone operator and a convex nondifferentiable functional in Banach spaces. For example, such inequalities arise in the description of steady-state filtration processes and equilibrium problems for soft shells. For solving variational inequalities, we suggest a two-layer iterative method, which permits one to reduce the original variational inequality to a variational inequality with a duality operator that has better properties than the original operator. Similar iterative processes were considered earlier (e.g., see [1–6]) only for the cases of monotone operators or differentiable functionals (some regularization was used to make the functional differentiable). We analyze the convergence of the iterative process. We show that the iterative sequence is bounded and each weak limit point of this sequence is a solution of the original inequality. In the case of a Hilbert space, we prove the weak convergence of the entire iterative sequence under some additional requirement. We apply the general results to the stationary problem on the filtration of an incompressible fluid governed by a discontinuous filtration law with limit gradient [1, 7, 8] and to problems of finding an equilibrium position of a soft infinitely long cylindrical shell subjected to mass forces and a follower surface load and bounded by a plane obstacle [9, 10]. We also obtain some results about properties of solutions of filtration problems.