Iterative Algorithm for Common Fixed Points of Infinite Family of Nonexpansive Mappings in Banach Spaces

Songnian He and Jun Guo

Journal of Applied Mathematics, 2012, vol. 2012, 1-13

Abstract:

Let C be a nonempty closed convex subset of a real uniformly smooth Banach space X , { T k } k = 1 ∞ : C → C an infinite family of nonexpansive mappings with the nonempty set of common fixed points ⋂ k = 1 ∞ Fix ⠡ ( T k ) , and f : C → C a contraction. We introduce an explicit iterative algorithm x n + 1 = α n f ( x n ) + ( 1 - α n ) L n x n , where L n = ∑ k = 1 n ( ω k / s n ) T k , S n = ∑ k = 1 n ω k ,    and w k > 0 with ∑ k = 1 ∞ ω k = 1 . Under certain appropriate conditions on { α n } , we prove that { x n } converges strongly to a common fixed point x * of { T k } k = 1 ∞ , which solves the following variational inequality: 〈 x * - f ( x * ) , J ( x * - p ) 〉 ≤ 0 ,       p ∈ ⋂ k = 1 ∞ Fix ( T k ) , where J is the (normalized) duality mapping of X . This algorithm is brief and needs less computational work, since it does not involve W -mapping.

Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnljam:787419

DOI: 10.1155/2012/787419

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