Convergence of path and an iterative method for families of nonexpansive mappings

Abstract

Let E be a real q-uniformly smooth Banach space with q ≥ 1 + d _q . Let K be a closed, convex and nonempty subset of E. Let be a family of nonexpansive self-mappings of K. For arbitrary fixed δ ∈ (0, 1) define a family of nonexpansive maps by S _i := (1 − δ)I + δT _i where I is the identity map of K. Let Assume either at least one of the T _i 's is demicompact or E admits weakly sequentially continuous duality map. It is proven that the fixed point sequence {z _{t n} } converges strongly to a common fixed point of the family where

and {t _n } is a sequence in (0, 1), satisfying appropriate conditions. As an application, it is proven that the iterative sequence {x _n } defined by: x ₀ ∈ K,

converges strongly to a common fixed point of the family where {α _n } and {σ _i,n } are sequences in (0, 1) satisfying appropriate conditions.

Keywords:

• Demicompact operator
• Nonexpansive mapings
• q-Uniformly smooth spaces
• Uniformly Gâteaux differentiable norm
• Uniformly smooth real Banach spaces

2000 Mathematics Subject Classifications :

• 47H09
• 47J25

Acknowledgement

Research supported by the Japanese Mori Fellowship of UNESCO at The Abdus Salam International Center for Theoretical Physics, Trieste, Italy.