A General Iterative Method for Nonexpansive Mappings

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 < ?? < 1, A a strongly positive boundedlinear operator with coefficient ?????? > 0, and 0 < ?? < ?????? /??. It is proved that both sequences{ } n x and { } n w generated by the iterative method xn = ??n?? f (xn) + (I ??? ( ??n + ??n) A)Txn + ??nun ,and wn+1 = ??n?? f (wn) + (I ??? ( ??n + ??n) A)Twn + ??nun converge strongly to a fixed pointx???? ??? F(T ) which solves the variational inequality ???(A ??? ?? 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.