A General Iterative Algorithm for Nonexpansive Mappings in Banach Spaces

Let E be a real q-uniformly smooth Banach space whose duality map is weakly sequentially continuous. Let T : E → E be a nonexpansive mapping with F (T ) 6= ∅. Let A : E → E be an η-strongly accretive map which is also κ-Lipschitzian. Let f : E → E be a contraction map with coefficient 0 < α < 1. Let a sequence {yn} be defined iteratively by y0 ∈ E, yn+1 = αnγf(yn) + (I − αnμA)Tyn, n ≥ 0, where {αn}, γ and μ satisfy some appropriate conditions. Then, we prove that {yn} converges strongly to the unique solution x∗ ∈ F (T ) of the variational inequality 〈(γf−μA)x∗, j(y− x∗)〉 ≤ 0, ∀ y ∈ F (T ). Convergence of the correspondent implicit scheme is also proved without the assumption that E has weakly sequentially continuous duality map. Our results are applicable in lp spaces, 1 < p <∞.