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RESEARCH PRODUCT

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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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 cone metric 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 normal cone with constant $M=1$. The topology $\tau_p$ has as basis the family $\mathcal{B}=\{N(x,c): x \in E, c \in \textrm{int} P\}$, where $N(x,c)= \{y \in E: -c\ll y-x \ll c\}$ and $z \ll w$ will stand for $w-z \in \textrm{int} P$. $(E,\tau_p)$ is a normable topological space and the norm is $\mu_V$ the Minkowski functional of $V=N(0,c)$, $c \in \textrm{int}\, P$. Then, the authors by using this norm proved some interesting results of common fixed points for two multifunctions satisfying contractive conditions.