Waring-Goldbach problems for unlike powers with almost equal variables

We prove that almost all positive even integers n can be written as n = p22 + p33 + p44 + p55 with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| {p_k^k - \tfrac{N} {4}} \right| \leqslant N^{\tfrac{{321}} {{325}} + \varepsilon } $$\end{document} for 2 ⩽ k ⩽ 5. Moreover, it is proved that each sufficiently large odd integer N can be represented as N = pp + p22 + p33 + p44 + p55 with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| {p_k^k - \tfrac{N} {5}} \right| \leqslant N^{\tfrac{{321}} {{325}} + \varepsilon }$$\end{document} for 1 ⩽ k ⩽ 5.