Krasnoselskii-Mann method for non-self mappings

Let H be a Hilbert space and let C be a closed, convex and nonempty subset of H. If T:C→H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T:C\to H$\end{document} is a non-self and non-expansive mapping, we can define a map h:C→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$h:C\to\mathbb{R}$\end{document} by h(x):=inf{λ≥0:λx+(1−λ)Tx∈C}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$h(x):=\inf\{\lambda\geq 0:\lambda x+(1-\lambda)Tx\in C\}$\end{document}. Then, for a fixed x0∈C\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} and for α0:=max{1/2,h(x0)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha_{0}:=\max\{1/2, h(x_{0})\}$\end{document}, we define the Krasnoselskii-Mann algorithm xn+1=αnxn+(1−αn)Txn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x_{n+1}=\alpha _{n}x_{n}+(1-\alpha_{n})Tx_{n}$\end{document}, where αn+1=max{αn,h(xn+1)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha_{n+1}=\max\{\alpha_{n},h(x_{n+1})\}$\end{document}. We will prove both weak and strong convergence results when C is a strictly convex set and T is an inward mapping.