}+(0-\gamma)\beta\bigl\langle A_{0}\bigl(x_{0}^{k}-x_{0}^{k+0} \bigr),A_{0}x_{0}^{k}+A_{0}x_{0}^{k}-b \bigr\rangle +\bigl\Vert x_{0}^{k+0}-x_{0}^{k} \bigr\Vert _{G_{0}}^{0} \\ &\qquad{}+\bigl\langle x_{0}^{k}-x_{0}^{k+0}, G_{0}\bigl(x_{0}^{k}-x_{0}^{k-0} \bigr)\bigr\rangle \\ &\qquad{}+\frac{\beta}{0}\bigl(\bigl\Vert A_{0}x_{0}^{k+0}-A_{0}x_{0} \bigr\Vert ^{0}-\bigl\Vert A_{0}x_{0}^{k}-A_{0}x_{0} \bigr\Vert ^{0}+\bigl\Vert A_{0}x_{0}^{k+0}-A_{0}x_{0}^{k} \bigr\Vert ^{0}\bigr). \end{aligned}$$ \end{document}$$  Proof {#FPar00} -----  Setting $\documentclass[00pt]{minimal}                 \usepackage{amsmath}                 \usepackage{wasysym}                  \usepackage{amsfonts}                  \usepackage{amssymb}                  \usepackage{amsbsy}                 \usepackage{mathrsfs}                 \usepackage{upgreek}                 \setlength{\oddsidemargin}{-00pt}                 \begin{document}$x_{0}=x_{0}^{k