Some Remarks on Perov Type Mappings in Cone Metric Spaces

Let E,C,t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( E,C,t\right) $$\end{document} be a real ordered topological vector space and let (X, d) be a tvs-cone metric space over cone C. Using Proposition 19.9 of Deimling (Nonlinear functional analysis, Springer, Berlin, 1985), we show that E can be equipped with a norm such that C is a normal monotone solid cone. Hence, a tvs-cone metric space X,d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( X,d\right) $$\end{document} over a solid cone C is a normal cone metric space over the same cone C. This assures that tvs-cone metric spaces are not a genuine generalization of cone metric spaces introduced by Huang and Zhang, recently. Further, if the cone C is solid then we have only cone metric spaces over normal solid cone (with coefficient of normality K=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K=1$$\end{document}). Here, we introduce also the notion of Sehgal–Guseman–Perov type mappings and we establish a result of existence and uniqueness of fixed points for this class of mappings.