Convergence Theorems for a Pair of Nonexpansive Mappings
Wataru Takahashi, Takayuki Tamura · Journal of convex analysis · 1998
Let E E be a real Banach space and let C C be a nonempty closed convex subset of E E . Then a mapping T T of C C into itself is called nonexpansive if \Vert Tx-Ty\Vert \leq \Vert x-y\Vert ∥ T x − T y ∥ ≤ ∥ x − y ∥ for all x,y\in C x , y ∈ C , and quasi-nonexpansive if the set F(T) F ( T ) of all fixed points of T T is nonempty and \Vert Tx-y\Vert \leq \Vert x-y\Vert ∥ T x − y ∥ ≤ ∥ x − y ∥ for all x\in C x ∈ C and y\in F(T) y ∈ F ( T ) . For two mappings S,T S , T of C C into itself G. Das and J. P. Debata ["Fixed points of quasi-nonexpansive mappings", Indian J. Pure Appl. Math. 17 (1986) 1263–1269] considered the following iteration scheme: x_1\in C\ \ \text{and}\ \ x_{n+1} = \alpha_n S [\beta_n Tx_n + (1-\beta_n)x_n] + (1-\alpha_n)x_n\ \ \forall n\geq 1,