Strong convergence of an inertial Halpern type algorithm in Banach spaces

In this article, we obtain the strong convergence of the new modified Halpern iteration process xn+1=αnu+(1-αn)TnP(xn+θn(xn-xn-1)),n=1,2,3,…,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} x_{n+1} = \alpha _{n}u + (1-\alpha _{n})T_{n}P(x_{n} + \theta _{n}(x_{n} - x_{n-1})), \ \ \ \ \ \ n=1,2,3,\ldots , \end{aligned}$$\end{document}to a common fixed point of {Tn}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ T_{n}\}$$\end{document}, where {Tn}n=1∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ T_{n}\}_{n=1}^{\infty }$$\end{document} is a family of nonexpansive mappings on the closed and convex subset C of a Banach space X, P:X⟶C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P: X \longrightarrow C$$\end{document} is a nonexpansive retraction, {αn}⊂[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\alpha _n\} \subset [0, 1]$$\end{document} and {θn}⊂R+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\theta _n\}\subset R^+$$\end{document}. Some applications of this result are also presented.