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Viscosity approximation methods for nonexpansive mappings are studied. Consider a nonexpansive self-mapping T of a closed convex subset C of a Banach space X. Suppose that the set Fix(T ) of fixed points of T is nonempty. For a contraction f on C and t ∈ (0, 1), let x t ∈ C be the unique fixed point of the contraction x → tf (x) + (1 − t)T x. Consider also the iteration process {x n }, where x 0 ∈ C is arbitrary and x n+1 = α n f (x n ) + (1 − α n )T x n for n 1, where {α n } ⊂ (0, 1). If X is either Hilbert or uniformly smooth, then it is shown that {x t } and, under certain appropriate conditions on {α n }, {x n } converge strongly to a fixed point of T which solves some variational inequality.  2004 Published by Elsevier Inc.

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Let H be a real Hilbert space. Consider on H a nonexpansive mapping T with a fixed point, a contraction f with coefficient 0 < α < 1, and a strongly positive linear bounded operator A with coefficientγ > 0. Let 0 < γ <γ /α. It is proved that the sequence {x n } generated by the iterative method x n+1 = (I − α n A)T x n + α n γf (x n ) converges strongly to a fixed pointx ∈ Fix(T ) which solves the variational inequality (γf − A)x, x −x 0 for x ∈ Fix(T ).

The split feasibility problem (SFP) (Censor and Elfving 1994 Numer. Algorithms 8 221-39) is to find a point x * with the property that x * ∈ C and Ax * ∈ Q, where C and Q are the nonempty closed convex subsets of the real Hilbert spaces H 1 and H 2 , respectively, and A is a bounded linear operator from H 1 to H 2 . The SFP models inverse problems arising from phase retrieval problems (Censor and Elfving 1994 Numer. Algorithms 8 221-39) and the intensity-modulated radiation therapy (Censor et al 2005 Inverse Problems 21 2071-84). In this paper we discuss iterative methods for solving the SFP in the setting of infinite-dimensional Hilbert spaces. The CQ algorithm of Byrne (2002 Inverse Problems 18 2004 Inverse Problems 20 103-20) is indeed a special case of the gradient-projection algorithm in convex minimization and has weak convergence in general in infinite-dimensional setting. We will mainly use fixed point algorithms to study the SFP. A relaxed CQ algorithm is introduced which only involves projections onto half-spaces so that the algorithm is implementable. Both regularization and iterative algorithms are also introduced to find the minimum-norm solution of the SFP.