Strong convergence of the composite Halpern iteration

作者：Qin Xiaolong^*; Su Yongfu; Shang Meijuan

来源：Journal of Mathematical Analysis and Applications, 2008, 339(2): 996-1002.

DOI：10.1016/j.jmaa.2007.07.062

• iteration scheme
• sunny and nonexpansive retraction
• nonexpansive mapping
• Banach space

Let C be a closed convex subset of a uniformly smooth Banach space E and let T : C -> C be a nonexpansive mapping with a nonempty fixed points set. Given a point u epsilon C, the initial guess x(0) epsilon C is chosen arbitrarily and given sequences {alpha(n)}(n=0)(infinity), {beta(n))(n=0)(infinity) and {gamma(n))(n=0)(infinity) in (0, 1), the following conditions are satisfied:
(i) Sigma(infinity)(n=0) alpha(n) = infinity;
(ii) alpha(n) -> 0, beta(n) -> 0 and 0 < a <= gamma(n), for some a epsilon (0, 1)
(iii) Sigma(infinity)(n=0) vertical bar alpha(n+1)-alpha(n)vertical bar < infinity, Sigma(infinity)(n=0) vertical bar beta(n+1)-beta(n)vertical bar < infinity and Sigma(infinity)(n=0) vertical bar gamma(n+1)-gamma(n)vertical bar < infinity. Let {x(n)}(n=1)(infinity) be a composite iteration process defined by
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then {x(n)}(n=1)(infinity) converges strongly to a fixed point of T.

• 出版日期2008-3-15
• 单位石家庄学院; 天津工业大学