Convergence Theorems for a Pair of Nonexpansive Mappings

Let

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, x 1 ∈ C   and   x n + 1 = α n S [ β n T x n + ( 1 − β n ) x n ] + ( 1 − α n ) x n    ∀ n ≥ 1 ,

where

\{\alpha_n\} { α n }

and

\{\beta_n\} { β n }

are sequences in

[0,1] [ 0 , 1 ]

. We first consider the weak convergence of the iterates

\{x_n\} { x n }

in such iteration schemes in a uniformly convex Banach space which satisfies Opial's condition or whose norm is Fréchet differentiable. Further, we discuss the strong convergence of the iterates

\{x_n\} { x n }

in a strictly convex Banach space. The theorems generalize results of W. Takahashi and G.-E. Kim ["Approximating fixed points of nonexpansive mappings in Banach spaces", Math. Jap. 48/1 (1998) 1–9].