Convergence of hybrid steepest descent method for variational inequalities in Banach spaces

Let E be a q-uniformly smooth real Banach space with constant dq, q > 1. Let Ti : E → E, i = 1, 2, … , r be a finite family of nonexpansive mappings with K≔∩i=1rFix(Ti)≠∅ and K = Fix(TrTr−1 … T1) = Fix(T1Tr … T2) = ⋯ = Fix(Tr−1Tr−2 …  Tr). Let G : E → E be an η-strongly accretive map which is also κ-Lipschitzian. A hybrid steepest descent method introduced by Yamada [25] and studied by various authors is proved to converge strongly to the unique solution of the variational inequality problem VI(G, K) in q-uniformly smooth real Banach space, in particular, in Lp spaces 1 < p < ∞.