General iteration algorithm and convergence rate optimal model for common fixed points of nonexpansive mappings

The purpose of this paper is to introduce a new general composite implicit iteration scheme for approximating the common fixed points of a finite family of nonexpansive mappings {Ti}i=1N in Banach spaces and to prove some weak and strong convergence theorems, this new iteration scheme is defined as follows:xn=αnxn-1+(1-αn)Tnyn,yn=rnxn+snxn-1+tnTnxn+wnTnxn-1,rn+sn+tn+wn=1,{αn},{rn},{sn},{tn},{wn}∈[0,1],where Tn = Tn mod N. The general composite implicit iteration scheme presented in this paper included various concrete iteration schemes. Hence, the results presented in this paper extend, generalize and improve the results of Mann, Ishikawa, Xu and Ori, and other authors. Meanwhile, this paper establish the convergence rate optimal model and give the convergence theorems with the estimation of convergence rate.