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Let E be a real uniformly convex Banach space which admits a weakly sequentially continuous duality mapping from E to E*, C a nonempty closed convex subset of E which is also a sunny nonexpansive retract of E, and T:CÃ¢Â†Â’E a non-expansive nonself-mapping with F(T)Ã¢Â‰Â Ã¢ÂˆÂ…. In this paper, we study the strong convergence of two sequences generated by  xn+1=ÃŽÂ±nx+(1Ã¢ÂˆÂ’ÃŽÂ±n)(1/n+1)Ã¢ÂˆÂ‘j=0n(PT)jxn and  yn+1=(1/n+1)Ã¢ÂˆÂ‘j=0nP(ÃŽÂ±ny+(1Ã¢ÂˆÂ’ÃŽÂ±n)(TP)jyn) for all nÃ¢Â‰Â¥0, where x,x0,y,y0Ã¢ÂˆÂˆC, {ÃŽÂ±n} is a real sequence in an interval [0,1], and P is a sunny non-expansive retraction of E onto C. We prove that {xn} and {yn} converge strongly to Qx and Qy, respectively, as nÃ¢Â†Â’Ã¢ÂˆÂž, where Q is a sunny non-expansive retraction of C onto F(T). The results presented in this paper generalize, extend, and improve the corresponding results of Matsushita and Kuroiwa and many others

Strong Convergence of Cesàro Mean Iterations for Nonexpansive Nonself-Mappings in Banach Spaces