A GENERAL ITERATIVE ALGORITHM FOR MONOTONE OPERATORS AND FIXED POINT PROBLEMS IN HILBERT SPACES - Medghalchi, A.R.; Mirzaee, H. (Journal of Nonlinear Analysis and Optimization: Theory & Applications (JNAO), 2012)
Let VI(A,H) be the set of all solutions of the following variational inequality problem: .6cm find .125cmuH .125cm such that v-u,Au≥0,.4cmfor all vH. Where H is a Hilbert space, A:H→ H is a Lipschitz continuous and monotone operator. Assume that F:H→ H is a Lipschitz continuous and strongly monotone operator. Let f:H→ H be a Lipschitz continuous mapping. In this paper, we consider a demiclosed, demicontractive mapping T on H such that Fix(T)VI(A,H) eq. For finding an element x^* which solves the following variational inequality problem: find an x^*Fix(T)VI(A,H) such that v-x^*,μ Fx^*-γ fx^*≥0,.5cmfor all vFix(T)VI(A,H), when μ and γ are positive real numbers which satisfy appropriate conditions, we introduce a new general iterative algorithm and obtain strong convergence results.