A General Iterative Method for Nonexpansive Mappings

Somyot Plubtieng, Prapairat Junlouchai

Abstract


Let H be a real Hilbert space and T: H → H be a nonexpansive mapping,f : H → H a contraction mapping with coefficient 0 < α < 1, A a strongly positive boundedlinear operator with coefficient γ􀀄 > 0, and 0 < γ < γ􀀄 /α. It is proved that both sequences{ } n x and { } n w generated by the iterative method xn = αnγ f (xn) + (I – ( αn + βn) A)Txn + βnun ,and wn+1 = αnγ f (wn) + (I – ( αn + βn) A)Twn + βnun converge strongly to a fixed pointx􀀄 ∈ F(T ) which solves the variational inequality 〈(A – γ f ) x􀀄 , x􀀄 – x〉 ≤ 0 for x ∈ F(T ).Our results extend and improve the corresponding results of G. Marino and H.K. Xu[A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal.Appl. 318(2006), 43-52], and may others.

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