The goal of this paper is to find the solution and of the generalized variational inequality such that
where is a nonempty closed convex set in , is a multi-valued mapping from into with nonempty values, and and denote the inner product and norm in , respectively.
The authors assume that the solution set of the problem (1) is nonempty and is continuous on with nonempty compact convex values satisfies the inequality
(Here property (2) holds if is pseudomonotone on C in the sense of Karamardian. In particular, if is monotone, then (2) holds.) Main result:
If is continuous with nonempty compact convex values on and the condition (2) holds, then either the proposed algorithm terminates in a finite number of iterations or generates an infinite sequence converging to a solution of (1).
If is also Lipschitz continuous with modulus and if there exist positive constants such that , , then there is a constant such that for sufficiently large one has: .
The details of the algorithm are presented and several preliminary results for convergence analysis are proved. The authors give a unified framework of a projection-type algorithm for multi-valued variational inequalities. At the end the results of some numerical experiments for the proposed algorithm are presented.