Abstract
We suggest and analyze two new iterative algorithms for a nonexpansive mapping T in Banach spaces. We prove that the proposed iterative algorithms converge strongly to some fixed point of T.
Highlights
Let E be a real Banach space, C a nonempty closed convex subset of E, and T : C → C a nonexpansive mapping; namely, Tx − Ty ≤ x − y, 1.1 for all x, y ∈ C
Received 13 September 2007; Accepted 25 November 2007 Recommended by Massimo Furi We suggest and analyze two new iterative algorithms for a nonexpansive mapping T in Banach spaces
We prove that the proposed iterative algorithms converge strongly to some fixed point of T
Summary
Let E be a real Banach space, C a nonempty closed convex subset of E, and T : C → C a nonexpansive mapping; namely, Tx − Ty ≤ x − y , 1.1 for all x, y ∈ C. The interest and importance of Halpern iterative method lie in the fact that strong convergence of the sequence {xn} is achieved under certain mild conditions on parameter {αn} in a general Banach space. Su and Li 21 introduced the following two new iterative algorithms for a nonexpansive mapping T : for fixed u ∈ C, let the sequences {xn} and {yn} be generated by xn 1 αnu 1 − αn T βnu 1 − βn xn , 1.4 yn 1 αn βnu 1 − βn T yn 1 − αn n n. Motivated and inspired by the above works, in this paper we construct two new iterative algorithms for approximating fixed points of a nonexpansive mapping T. We prove that the proposed iterative algorithms converge strongly to a fixed point of T under some mild conditions
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