Hybrid iteration method for common fixed points of an infinite family of nonexpansive mappings in Banach spaces

作者：Deng Wei Qi^*

来源：Fixed Point Theory and Applications, 2012, 1-7.

DOI：10.1186/1687-1812-2012-22

• infinite family of nonexpansive mappings
• strong and weak convergence
• common fixed point
• hybrid iteration

Let E be a real uniformly convex Banach space, and let K be a nonempty closed convex subset of E. Let {T-i}(i=1)(infinity) be a sequence of nonexpansive mappings from K to itself with F :={x is an element of K :T(i)x = x, for all i >= 1} not equal {empty set}. For an arbitrary initial point x(1) is an element of K, the modified hybrid iteration scheme {x(n)} is defined as follows:
x(n+1) = alpha(n)x(n) + (1 - alpha(n)) (T*(n)x(n) - lambda(n+1)mu A(T*(n)x(n))), n = 1, where A: K -> K is an L
Lipschitzian mapping, T*(n) = T-i with i satisfying: n = [(k-i+1)(i+k)/2]+[1+(i-1)(i+2)/2], k >= i-1(i = 1,2,...),{lambda(n)} subset of [0,1), and {alpha(n)} is a sequence in [a, 1 - a] for some a is an element of (0,1). Under some suitable conditions, the strong and weak convergence theorems of {x(n)} to a common fixed point of the nonexpansive mappings {T-i}(i=1)(infinity) are obtained. The results in this article extend those of the authors whose related researches are restricted to the situation of finite families of nonexpansive mappings.

• 出版日期2012
• 单位云南大学