Iterative solution of nonlinear equations of accretive and pseudocontractive types

AbstractLet 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):={Tx=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