Iterative solution of nonlinear equations of accretive and pseudocontractive types

Let E be a real uniformly smooth Banach space. Let A:D(A)= E→2E be an accretive operator that satisfies the range condition and A−1(0) =∅. Let {λn} and {θn} be two real sequences satisfying appropriate conditions, and for z ∈ E arbitrary, let the sequence {xn} be generated from arbitrary x0 ∈ E by xn+1 = xn − λn(un + θn(xn − z)), un ∈ Axn, n 0. Assume that {un} is bounded. It is proved that {xn} converges strongly to some x∗ ∈ A−1(0). Furthermore, if K is a nonempty closed convex subset of E and T :K → K is a bounded continuous pseudocontractive map with F(T ) := {T x = x} = ∅, it is proved that for arbitrary z ∈ K, the sequence {xn} generated from x0 ∈ K by xn+1 = xn −λn((I −T )xn +θn(xn −z)), n 0, where {λn} and {θn} are real sequences satisfying appropriate conditions, converges strongly to a fixed point of T .  2003 Elsevier Inc. All rights reserved.