Ishikawa Iteration Process for Nonlinear Lipschitz Strongly Accretive Mappings

Let E = Lp, p ≥ 2 and let T:E → E be a Lipschitzian and strongly accretive mapping. Let S:E → E be defined by Sx = ƒ − Tx + x. It is proved that under suitable conditions on the real sequences {αn}∞n=0 and {βn}∞n=0, the iteration process, x0 ∈ E, xn+1 = (1 − αn)xn + αnS[(1 − βn)xn + βnSxn], n ≥ 0, converges strongly to the unique solution of Tx = ƒ. A related result deals with the iterative approximation of fixed points for Lipschitz strongly pseudocontractive mappings in E. A consequence of our result gives an affirmative answer to a problem posed by C. E. Chidume (J. Math. Anal. Appl. 151, No. 2 (1990), 453-461).