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Let  H  be a Hilbert space and let  C  be a closed convex nonempty subset of  H  and    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$T : C\rightarrow H$$\end{document}    T  :  C  →  Ha non-self nonexpansive mapping. A map    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$h : C\rightarrow R$$\end{document}    h  :  C  →  Rdefined by    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$h(x) := \inf \{\lambda \ge 0 : \lambda x+(1-\lambda )Tx \in C \}$$\end{document}    h  (  x  )  :  =  inf  {  λ  ≥  0  :  λ  x  +  (  1  -  λ  )  T  x  ∈  C  }. Then, for a fixed    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$x_0 \in C$$\end{document}x  0   ∈  Cand for    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\begin{aligned}{\alpha _0} = \max \left\{ {\frac{1}{2},h({x_0})} \right\}\end{aligned}$$\end{document}α  0   =  max1  2   ,  h   (   x  0   ), Krasnoselskii–Mann algorithm is defined by    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$x_{n+1}=\alpha _n+(1-\alpha _n)Tx_n,$$\end{document}x   n  +  1    =   α  n   +   (  1  -   α  n   )   T   x  n   ,where    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\alpha _{n+1}=\max \{\alpha _n, h(x_{x_{n+1}})\}$$\end{document}α   n  +  1    =  max   {   α  n   ,  h   (   x   x   n  +  1)   }. 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.

On an open question of V. Colao and G. Marino presented in the paper “Krasnoselskii–Mann method for non-self mappings”