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.