Modified Ishikawa Iteration of Cesàro Means for Asymptotically Non-expansive Mappings
Pei Yonggan · Mathematica Applicata · 2014
Let Cbe a nonempty closed convex subset of a uniformly convex Banach space Ewith a uniformly Gteauxdifferentiablenorm.Suppose that T∶C →Cis an asymptotically nonexpansive mapping.For an arbitrary initial value x ∈C,we introduce the modified Ishikawa iteration of its Cesaro means:x0 =x∈C arbitrarily chosen,yn=γnun+δnxn+(1-γn-δ)n+11∑j=0nTjxn,xn+1=μnvn+αnγf(xn)+βnxn+[(1-μn-βn)I-αnA]n+11∑j=0nTjyn,n≥0and prove its strong and weak convergence under some mild conditions.