Weak and strong convergence theorems of modified Ishikawa iteration for an infinitely countable family of pointwise asymptotically nonexpansive mappings in Hilbert spaces

Abstract

In this paper, we first verify that the sequence generated by the Ishikawa iterative scheme is weakly convergent to a fixed point of a uniformly Lipschitzian and pointwise asymptotically nonexpansive mapping T in a Hilbert space. Then, we introduce a new kind of monotone hybrid method which is a modification of the Ishikawa iterative scheme for finding a common fixed point of an infinitely countable family of uniformly Lipschitzian and pointwise asymptotically nonexpansive mappings in a Hilbert space. We also prove the strongly convergent of the sequence generated by the proposed monotone hybrid method, for an infinitely countable family of uniformly Lipschitzian and pointwise asymptotically nonexpansive mappings in a Hilbert space. The results presented in this paper extend and improve some known results in the literature.