MR2670689 Rezapour, Shahram; Khandani, Hassan; Vaezpour, Seyyed M. Efficacy of cones on topological vector spaces and application to common fixed points of multifunctions. Rend. Circ. Mat. Palermo (2) 59 (2010), no. 2, 185–197. (Reviewer: Pasquale Vetro)

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Abstract

Recently, Huang and Zhang defined cone metric spaces by substituting an order normed space for the real numbers and proved some fixed point theorems. For fixed point results in the framework of conemetric space see, also, Di Bari and Vetro [\textit{$\varphi$-pairs and common fixed points in cone metric spaces}, Rend. Circ. Mat.Palermo \textbf{57} (2008), 279--285 and \textit{Weakly $\varphi$-pairs and common fixed points in cone metric spaces}, Rend. Circ. Mat. Palermo \textbf{58} (2009), 125--132].Let $(E,\tau)$ be a topological vector space and $P$ a cone in $E$with int\,$P\neq \emptyset$, where int\,$P$ denotes the interior of$P$. The authors define a topology $\tau_p$ on $E$ so that$(E,\tau_p)$ is a normable topological space and $P$ is a normalcone with constant $M=1$. The topology $\tau_p$ has as basis thefamily $\mathcal{B}=\{N(x,c): x \in E, c \in \textrm{int} P\}$,where $N(x,c)= \{y \in E: -c\ll y-x \ll c\}$ and $z \ll w$ willstand for $w-z \in \textrm{int} P$. $(E,\tau_p)$ is a normabletopological space and the norm is $\mu_V$ the Minkowski functionalof $V=N(0,c)$, $c \in \textrm{int}\, P$.Then, the authors by using this norm proved some interesting resultsof common fixed points for two multifunctions satisfyingcontractive conditions.
Original languageEnglish
Number of pages0
JournalMATHEMATICAL REVIEWS
Volume2011
Publication statusPublished - 2011

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