| Affiliation: | 1.Nonlinear Analysis Research Group,Ton Duc Thang University,Ho Chi Minh City,Vietnam;2.Faculty of Mathematics and Statistics,Ton Duc Thang University,Ho Chi Minh City,Vietnam;3.Department of Energy, Information Engineering and Mathematical Models (DEIM),University of Palermo,Palermo,Italy |
| Abstract: | Let (left( E,C,tright) ) 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 (left( X,dright) ) 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)). 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. |
