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

摘要

Let E be a real q-uniformly smooth Banach space with q≥1+d{sub}q. Let K be a closed, convex and nonempty subset of E. Let ({T{sub}i}{sub}(i=1)){sup}∞ be a family of nonexpansive self-mappings of K. For arbitrary fixed δ∈(0, 1) define a family of nonexpansive maps ({S{sub}i}{sub}(i=1)){sup}∞ by S{sub}i:=(1-δ)I+δT{sub}i where I is the identity map of K. Let F :=(∩{sub}(i=1)){sup}∞)(F(T{sub}i)≠0 Assume either at least one of the T{sub}i's is demicompact or E admits weakly sequentially continuous duality map. It is proven that the fixed point sequence {z{sub}(t{sub}n)} g converges strongly to a common fixed point of the family ({T{sub}i}{sub}(i=1)){sup}∞, where Z{sub}(t{sub}n)=t{sub}n u+∑((σ{sub}(i,n)S{sub}iz{sub}(t{sub}n))(i≥1), and {t{sub}n} is a sequence in (0, 1), satisfying appropriate conditions. As an application, it is proven that the iterative sequence {x{sub}n} defined by: x{sub}0∈K, x{sub}(n+1)=α{sub}n u+∑((σ{sub}(i,n)S{sub}i x{sub}n)(i≥1), n≥0 converges strongly to a common fixed point of the family ({T{sub}i}{sub}(i=1)){sup}∞ where {α{sub}n} and {σ{sub}(i,n)} are sequences in (0, 1) satisfying appropriate conditions.