a b s t r a c tThis paper concentrates on iterative methods for obtaining the multiple roots of nonlinear equations. Using the computer algebra system Mathematica, we construct an iterative scheme and discuss the conditions to obtain fourth-order methods from it. All the presented fourth-order methods require one-function and two-derivative evaluation per iteration, and are optimal higher-order iterative methods for obtaining multiple roots. We present some special methods from the iterative scheme, including some known already. Numerical examples are also given to show their performance.

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Let E a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E * , and K be a closed convex subset of E which is also a sunny nonexpansive retract of E, and T : K → E be nonexpansive mappings satisfying the weakly inward condition and F (T ) = ∅, and f : K → K be a fixed contractive mapping. The implicit iterative sequence {x t } is defined by for t ∈ (0, 1)The explicit iterative sequence {x n } is given bywhere α n ∈ (0, 1) and P is sunny nonexpansive retraction of E onto K. We prove that {x t } strongly converges to a fixed point of T as t → 0, and {x n } strongly converges to a fixed point of T as α n satisfying appropriate conditions. The results presented extend and improve the corresponding results of [H.K. Xu, Viscosity approximation methods for nonexpansive mappings,

In this paper, a necessary and sufficient conditions for the strong convergence to a common fixed point of a finite family of continuous pseudocontractive mappings are proved in an arbitrary real Banach space using an implicit iteration scheme recently introduced by Xu and Ori [H.K. Xu, R.G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Fuct. Anal. Optim. 22 (2001) 767-773] in condition α n ∈ (0, 1], and also strong and weak convergence theorem of a finite family of strictly pseudocontractive mappings of Browder-Petryshyn type is obtained. The results presented extend and improve the corresponding results of M.O. Osilike [M.O. Osilike, Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, J. Math.