Viscosity approximative methods to Cesà ro mean iterations for nonexpansive nonself-mappings in Banach spaces

Let E be a real uniformly convex Banach space which admits a weakly sequentially continuous duality mapping from E to E∗, C a closed convex subset of E which is also a sunny nonexpansive retract of E. Let T:C→E be a nonexpansive nonself-mapping with F(T)≠∅, and f:C→C be a fixed contractive mapping. The implicit viscosity iterative sequence {zm} is defined by {tm}⊂(0,1), zm=1m+1∑j=0mP(tmf(zm)+(1-tm)(TP)jzm), for m⩾0 and two explicit viscosity iterative sequences {xn} and {yn} are given by x0∈C,xn+1=αnf(xn)+(1-αn)1n+1∑j=0n(PT)jxn and y0∈C,yn+1=1n+1∑j=0nP(αnf(yn)+(1-αn)(TP)jyn) for n⩾0, where {αn} is a sequence in (0,1) and P is a sunny nonexpansive retraction of E onto C. We prove that under appropriate conditions imposed on {tm} and {αn}, the sequences {zm}, {xn} and {yn} converge strongly to some fixed point of T which solves some variational inequalities. The results presented extend and improve the corresponding results of Matsushita and Kuroiwa [S. Matsushita, D. Kuroiwa, Strong convergence of averaging iterations of nonexpansive nonself-mappings, J. Math. Anal. Appl. 294 (2004) 206-214], Song and Chen [Y. Song, R. Chen, Viscosity approximation methods to Cesàro means for nonexpansive mappings, Appl. Math. Comput. 186 (2007) 1120-1128] and many authors.