First-order continuous and iterative methods with a generalized projection operator for monotone variational inequalities in a Banach space

For nonlinear variational inequalities with monotone operators in a Banach space, two variants of the first-order continuous method are constructed. These variants are structurally based on a generalized projection operator on a convex closed set in a Banach space. Discrete analogues of continuous methods generating sequences having the properties of trajectories of continuous methods are studied. For the methods considered, sufficient conditions for strong convergence are obtained and their convergence rate is investigated.