Strong Convergence of Modified Implicit Iteration Processes for Common Fixed Points of Nonexpansive Mappings

Throughout this paper, let H be a real Hilbert space with inner product 〈·,·〉 and norm ‖ · ‖. Let C be a nonempty closed convex subset of H , we denote by PC(·) the metric projection from H onto C. It is known that z = PC(x) is equivalent to 〈z− y,x− z〉 ≥ 0 for every y ∈ C. Recall that T : C → C is nonexpansive if ‖Tx− Ty‖ ≤ ‖x− y‖ for all x, y ∈ C. A point x ∈ C is a fixed point of T provided that Tx = x. Denote by F(T) the set of fixed points of T , that is, F(T)= {x ∈ C : Tx = x}. It is known that F(T) is closed and convex. Construction of fixed points of nonexpansive mappings (and asymptotically nonexpansive mappings) is an important subject in the theory of nonexpansive mappings and finds application in a number of applied areas, in particular, in image recovery and signal processing (see, e.g., [1–5]). However, the sequence {Tnx}n=0 of iterates of the mapping T at a point x ∈ C may not converge even in the weak topology. Thus averaged iterations prevail. Indeed, Mann’s iterations do have weak convergence. More precisely, Mann’s iteration procedure is a sequence {xn} which is generated in the following recursive way: