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

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 ...

Strong Convergence of a Modified Halpern’s Iteration for Nonexpansive Mappings

A mapping T : C → C is said to be nonexpansive if ‖Tx − Ty‖ ≤ ‖x − y‖, for all x, y ∈ C. We denote by Fix T {x ∈ C : Tx x} the set of fixed points of T . In the last ten years, many papers have been written on the approximation of fixed point for nonlinear mappings by using some iterative processes see, e.g., 1–18 . An explicit iterative process was initially introduced, in 1967, by Halpern 3 i...

Strong Convergence Theorems of Common Fixed Points for a Family of Quasi--Nonexpansive Mappings

Strong Convergence to Common Fixed Points of a Finite Family of Nonexpansive Mappings

Let C be a closed convex subset of a Banach space E. A mapping T of C into itself is called nonexpansive if ‖Tx−Ty‖ ≤ ‖x− y‖ for all x, y ∈ C. We denote by F(T) the set of fixed points of T . Let T1,T2, . . . ,Tr be a finite family of nonexpansive mappings satisfying that the set F =⋂i=1F(Ti) of common fixed points of T1,T2, . . . ,Tr is nonempty. The problem of finding a common fixed point has...

Strong Convergence to Common Fixed Points of Nonexpansive Mappings without Commutativity Assumption

for each x, y ∈ C. For a mapping T of C into itself, we denote by F(T) the set of fixed points of T . We also denote by N and R+ the set of positive integers and nonnegative real numbers, respectively. Baillon [1] proved the first nonlinear ergodic theorem. Let C be a nonempty bounded convex closed subset of a Hilbert spaceH and let T be a nonexpansive mapping of C into itself. Then, for an arb...