Iterative solution of nonlinear equations of accretive and pseudocontractive types

Abstract

Let E be a real uniformly smooth Banach space. Let A :D(A)=E→2 E be an accretive operator that satisfies the range condition and A −1(0)≠∅. Let { λ n } and { θ n } be two real sequences satisfying appropriate conditions, and for z∈ E arbitrary, let the sequence { x n } be generated from arbitrary x 0∈ E by x n+1 = x n − λ n ( u n + θ n ( x n − z)), u n ∈ Ax n , n⩾0. Assume that { u n } is bounded. It is proved that { x n } 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):={ Tx= x}≠∅, it is proved that for arbitrary z∈ K, the sequence { x n } generated from x 0∈ K by x n+1 = x n − λ n (( I− T) x n + θ n ( x n − z)), n⩾0, where { λ n } and { θ n } are real sequences satisfying appropriate conditions, converges strongly to a fixed point of T.