Abstract

Let EE be a real Banach space and let CC be a nonempty closed convex subset of EE. Then a mapping TT of CC into itself is called nonexpansive if ∥Tx−Ty∥≤∥x−y∥\Vert Tx-Ty\Vert \leq \Vert x-y\Vert for all x,y∈Cx,y\in C, and quasi-nonexpansive if the set F(T)F(T) of all fixed points of TT is nonempty and ∥Tx−y∥≤∥x−y∥\Vert Tx-y\Vert \leq \Vert x-y\Vert for all x∈Cx\in C and y∈F(T)y\in F(T). For two mappings S,TS,T of CC 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: x1∈C  and  xn+1=αnS[βnTxn+(1−βn)xn]+(1−αn)xn  ∀n≥1,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, where {αn}\{\alpha_n\} and {βn}\{\beta_n\} are sequences in [0,1][0,1].
We first consider the weak convergence of the iterates {xn}\{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 {xn}\{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].

Author information

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Wataru Takahashi

Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Ohokayama, Meguro-Ku, Tokyo 152, Japan.

Takayuki Tamura

Department of Information Sciences, Tokyo Institute of Technology, Ohokayama, Meguro-Ku, Tokyo 152, Japan

References

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Suggested citation

W. Takahashi, T. Tamura. “Convergence Theorems for a Pair of Nonexpansive Mappings.” Journal of Convex Analysis 5 (1998), No. 1, 45–56. https://doi.org/10.68381/jca05-4

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Published by Heldermann Verlag, 1998. Rights now held by Banach Press LLC.