Weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in Banach spaces

By Charles E. Chidume, Bashir Ali. 2007

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Abstract Let E be a real uniformly convex Banach space whose dual space E ∗ satisfies the Kadec–Klee property, K be a closed convex nonempty subset of E. Let T 1 , T 2 , … , T m : K → K be asymptotically nonexpansive mappings of K into E with sequences (respectively) { k i n } n = 1 ∞ satisfying k i n → 1 as n → ∞ , i = 1 , 2 , … , m , and ∑ n = 1 ∞ ( k i n − 1 ) ∞ . For arbitrary ϵ ∈ ( 0 , 1 ) , let { α i n } n = 1 ∞ be a sequence in [ ϵ , 1 − ϵ ] , for each i ∈ { 1 , 2 , … , m } (respectively). Let { x n } be a sequence generated for m ⩾ 2 by { x 1 ∈ K , x n + 1 = ( 1 − α 1 n ) x n + α 1 n T 1 n y n + m − 2 , y n + m − 2 = ( 1 − α 2 n ) x n + α 2 n T 2 n y n + m − 3 , ⋮ y n = ( 1 − α m n ) x n + α m n T m n x n , n ⩾ 1 . Let ⋂ i = 1 m F ( T i ) ≠ ∅ . Then, { x n } converges weakly to a common fixed point of the family { T i } i = 1 m . Under some appropriate condition on the family { T i } i = 1 m , a strong convergence theorem is also proved.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/b1bb1f7e29e6a978f2287307270b0de319fef68a