title: On an open question of V. Colao and G. Marino presented in the paper "Krasnoselskii-Mann method for non-self mappings".
abstract: Let H be a Hilbert space and let C be a closed convex nonempty subset of H and [Formula: see text] a non-self nonexpansive mapping. A map [Formula: see text] defined by [Formula: see text]. Then, for a fixed [Formula: see text] and for [Formula: see text], Krasnoselskii-Mann algorithm is defined by [Formula: see text] where [Formula: see text]. Recently, Colao and Marino (Fixed Point Theory Appl 2015:39, 2015) have proved both weak and strong convergence theorems when C is a strictly convex set and T is an inward mapping. Meanwhile, they proposed a open question for a countable family of non-self nonexpansive mappings. In this article, authors will give an answer and will prove the further generalized results with the examples to support them.
authors: Department of Mathematics and Sciences, Hebei GEO University, Shijiazhuang, 050031 China.
authors: Meifang
authors: M
authors: Guo
authors: Department of Mathematics and Sciences, Hebei GEO University, Shijiazhuang, 050031 China.
authors: Xia
authors: X
authors: Li
authors: Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387 China.
authors: Yongfu
authors: 0000-0001-8656-471X
authors: Y
authors: Su
keywords: Hilbert space; Inward condition; Mann iterative scheme; Non-self mapping; Nonexpansive mapping; Strong convergence; Weak convergence