Realization of the hybrid method for Mann iterations

Abstract

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: x 0 ∈ C chosen arbitrarily, y n = α n x n + ( 1 - α n ) Tx n , C n = z ∈ C : ‖ y n - z ‖ ⩽ ‖ x n - z ‖ , Q n = z ∈ C : 〈 x 0 - x n , z - x n 〉 ⩽ 0 , x n + 1 = P C n ∩ Q n x 0 , n = 0 , 1 , 2 , … , where C is a nonempty closed convex subset of H, T : C → C is a nonexpansive mapping and P K 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 P C n ∩ Q n x 0 cannot be got, in general. In the case where C = H, we obtain the specific expression of P C n ∩ Q n x 0 and thus the hybrid method for Mann’s iteration process can be realized easily. Numerical results show advantages of our result.