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

Abstract

Let E be a real uniformly convex Banach space which admits a weakly sequentially continuous duality mapping from E to E∗E∗, C a closed convex subset of E which is also a sunny nonexpansive retract of E . Let T:C→ET:C→E be a nonexpansive nonself-mapping with F(T)≠∅F(T)≠∅, and f:C→Cf:C→C be a fixed contractive mapping. The implicit viscosity iterative sequence {zm}{zm} is defined by {tm}⊂(0,1){tm}⊂(0,1), zm=1m+1∑j=0mP(tmf(zm)+(1-tm)(TP)jzm), for m⩾0m⩾0 and two explicit viscosity iterative sequences {xn}{xn} and {yn}{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⩾0n⩾0, where {αn}{αn} is a sequence in (0,1)(0,1) and P is a sunny nonexpansive retraction of E onto C . We prove that under appropriate conditions imposed on {tm}{tm} and {αn}{αn}, the sequences {zm}{zm}, {xn}{xn} and {yn}{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 Cesaro means for nonexpansive mappings, Appl. Math. Comput. 186 (2007) 1120–1128] and many authors.