Strong convergence of an iterative algorithm for nonself multimaps in Banach spaces

✍ Scribed by Lu-Chuan Ceng; Jen-Chih Yao

• Publisher: Elsevier Science
• Year: 2009
• Tongue: English
• Weight: 596 KB
• Volume: 71
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2009.03.007

✦ Synopsis

Let E be a uniformly convex Banach space having a uniformly Gâteaux differentiable norm, D a nonempty closed convex subset of E, and T : D → K (E) a nonself multimap such that F (T ) = ∅ and P T is nonexpansive, where F (T ) is the fixed point set of T , K (E) is the family of nonempty compact subsets of E and P T (x) = {u x ∈ Tx : xu x = d(x, Tx)}. Suppose that D is a nonexpansive retract of E and that for each v ∈ D and t ∈ (0, 1), the contraction S t defined by S t x = tP T x + (1 -t)v has a fixed point x t ∈ D. Let {α n }, {β n } and {γ n } be three real sequences in (0, 1) satisfying approximate conditions. Then for fixed u ∈ D and arbitrary x 0 ∈ D, the sequence {x n } generated by

x n ∈ α n u + β n x n-1 + γ n P T (x n ), ∀n ≥ 0, converges strongly to a fixed point of T .