Convergence theorems for common fixed points for finite families of nonexpansive mappings in reflexive Banach spaces

• Published in: Nonlinear analysis 2008-06, Vol.68 (11), p.3410-3418
• Main Authors: Chidume, C.E., Ali, Bashir
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Functional analysis; Global analysis, analysis on manifolds; Mathematical analysis; Mathematics; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Uniformly Gâteaux differentiable norm; Uniformly smooth real Banach spaces
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2007.03.032

Let E be a real reflexive Banach space with a uniformly Gâteaux differentiable norm. Let K be a nonempty closed convex subset of E . Suppose that every nonempty closed convex bounded subset of K has the fixed point property for nonexpansive mappings. Let T 1 , T 2 , … , T N be a family of nonexpansive self-mappings of K , with F ≔ ⋂ i = 1 N Fix ( T i ) ≠ 0̸ , F = Fix ( T N T N − 1 … T 1 ) = Fix ( T 1 T N … T 2 ) = … = Fix ( T N − 1 T N − 2 … T 1 T N ) . Let { λ n } be a sequence in ( 0 , 1 ) satisfying the following conditions: C 1 : lim λ n = 0 ; C 2 : ∑ λ n = ∞ . For a fixed δ ∈ ( 0 , 1 ) , define S n : K → K by S n x ≔ ( 1 − δ ) x + δ T n x ∀ x ∈ K where T n = T n mod N . For an arbitrary fixed u , x 0 ∈ K , let B ≔ { x ∈ K : T N T N − 1 … T 1 x = γ x + ( 1 − γ ) u ,  for some  γ > 1 } be bounded, and let the sequence { x n } be defined iteratively by x n + 1 = λ n + 1 u + ( 1 − λ n + 1 ) S n + 1 x n , for  n ≥ 0 . Assume that lim n → ∞ ‖ T n x n − T n + 1 x n ‖ = 0 . Then, { x n } converges strongly to a common fixed point of the family T 1 , T 2 , … , T N . This convergence theorem is also proved for non-self maps.