Realization of the hybrid method for Mann iterations

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∩Qnx0PCn∩Qnx0 cannot be got, in general. In the case where C = H  , we obtain the specific expression of PCn∩Qnx0PCn∩Qnx0 and thus the hybrid method for Mann’s iteration process can be realized easily. Numerical results show advantages of our result.