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MR2684111 Kadelburg, Zoran; Radenović, Stojan; Rakočević, Vladimir Topological vector space-valued cone metric spaces and fixed point theorems. Fixed Point Theory Appl. 2010, Art. ID 170253, 17 pp. (Reviewer: Pasquale Vetro)

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Recently, Huang and Zhang [\emph{Cone metric spaces and fixed point theorems of contractive mappings}, J. Math. Anal. Appl., \textbf{332} (2007), 1468 -1476] defined cone metric spaces by substituing an order normed space for the real numbers and proved some fixed point theorems. Let $E$ be a real Hausdorff topological vector space and $P$ a cone in $E$ with int\,$P\neq \emptyset$, where int\,$P$ denotes the interior of $P$. Let $X$ be a nonempty set. A function $d : X \times X\to E$ is called a \emph{tvs}-cone metric and $(X, d)$ is called a \emph{tvs}-cone metric space, if the following conditions hold: (1) $\theta \leq d(x, y)$ for all $x, y \in X$ and $d(x, y)= \theta$ if and only if $x = y$; (2) $d(x, y)=d(y, x)$ for all $x, y \in X$; (3) $d(x, z)\leq d(x, y)+d(y, z)$ for all $x, y, z \in X$. The authors consider a class of convergent sequences in $X$, the same of Huang and Zhang. Then, the authors by using this class of convergent sequences proved several interesting results of common fixed points for three or two mappings satisfying some contractive conditions. The following theorem is one of the main results: \noindent \textbf{Theorem 3.1.} \emph{Let $(X, d)$ be a \emph{tvs}-cone metric space and the mappings $f, g, h : X \to X$ satisfy $$d(fx, gy)\preceq pd(hx, hy) + qd(hx, fx)+ rd(hy, gy)+ sd(hx, gy)+td(hy, fx),$$ for all $x, y \in X$, where $p, q, r, s, t \geq 0$, $p + q + r + s +t < 1$, and $q = r$ or $s = t$. If $f(X) \cup g(X) \subset h(X)$ and $h(X)$ is a complete subspace of $X$, then $f, g$, and $h$ have a unique point of coincidence. Moreover, if $(f, h)$ and $(g, h)$ are weakly compatible, then $f, g$, and $h$ have a unique common fixed point.} For fixed point results in the framework of cone metric space see, also, Arshad, Azam and Vetro [\emph{Some Common Fixed Point Results in Cone Metric Spaces}, Fixed Point Theory Appl., \textbf{2009}, Article ID 493965, 11 pages] 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].