Approximation of the Nearest Common Fixed Point of Asymptotically Nonexpansive Mappings in Banach Spaces (Communications in Mathematical Research, 2021)
In this paper, the iteration $x_{n+1} = α_ny + (1 − α_n)T^{k(n)}_{i(n)} x_n$ for a familyof asymptotically nonexpansive mappings $T_1, T_2, · · · , T_N$ is originally introduced ina uniformly convex Banach space. Motivated by recent papers, we prove that undersuitable conditions the iteration scheme converges strongly to the nearest commonfixed point of the family of asymptotically nonexpansive mappings. The results presented in this paper expand and improve corresponding ones from Hilbert spaces touniformly convex Banach spaces, or from nonexpansive mappings to asymptoticallynonexpansive mappings.
Keywords: asymptotically nonexpansive mapping; sunny nonexpansive retraction; uniformly Gâteaux differentiable; weakly sequentially continuous duality.