Realization of the hybrid method for Mann iterations
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摘要
The Mann iterations for nonexpansive mappings have only weak convergence even in a Hilbert space H. In order to overcome this weakness, Nakajo and Takahashi proposed the hybrid method for Mann’s iteration process:x0∈Cchosen arbitrarily,yn=αnxn+(1-αn)Txn,Cn=z∈C:‖yn-z‖⩽‖xn-z‖,Qn=z∈C:〈x0-xn,z-xn〉⩽0,xn+1=PCn∩Qnx0,n=0,1,2,…,where C is a nonempty closed convex subset of H, T : C → C is a nonexpansive mapping and PK is the metric projection from H onto a closed convex subset K of H. However, it is difficult to realize this iteration process in actual computing programs because the specific expression of PCn∩Qnx0 cannot be got, in general. In the case where C = H, we obtain the specific expression of PCn∩Qnx0 and thus the hybrid method for Mann’s iteration process can be realized easily. Numerical results show advantages of our result.
论文关键词:Nonexpansive mapping,Strong convergence,Weak convergence,Hybrid method,Metric projection,Fixed point,Mann’s iteration
论文评审过程:Available online 26 October 2010.
论文官网地址:https://doi.org/10.1016/j.amc.2010.10.039