Implicit iteration methods for variational inequalities in Banach spaces

In this paper, we introduce three new iteration methods, which are implicit and converge strongly, based on the steepest descent method with a strongly accretive and strictly pseudocontractive mapping and the modified Halpern’s iterative scheme, for finding a solution of variational inequalities over the set of common fixed points of a nonexpansive semigroup on a real Banach space which has a uniformly Gˆateaux differentiable norm.