A general iterative method for nonexpansive mappings in Hilbert spaces

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Abstract

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 coefficient γ¯>0. Let 0<γ<γ¯/α. It is proved that the sequence {xn} generated by the iterative method xn+1=(IαnA)Txn+αnγf(xn) converges strongly to a fixed point x˜Fix(T) which solves the variational inequality (γfA)x˜,xx˜0 for xFix(T).

Keywords

Nonexpansive mapping
Iterative method
Variational inequality
Fixed point
Projection
Viscosity approximation

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Supported in part by the National Research Foundation of South Africa.