Iterative Solution of Nonlinear Equations with Strongly Accretive Operators

Let E be a real Banach space with a uniformly convex dual. Suppose T: E → E is a strongly accretive map with bounded range such that for each ƒ ∈ E the equation Tx = ƒ has a solution in E. It is proved that each of the two well known fixed point iteration methods (the Mann and Ishikawa iteration methods), under suitable conditions, converges strongly to a solution of the equation Tx = ƒ. Furthermore, our method shows that such a solution is necessarily unique. Explicit error estimates are given. Our results resolve in the affirmative an open problem (J. Math, Anal, Appl. 151, No. 2 (1990), 460) and generalize important known results.