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)

Risultato della ricerca: Review article


Recently, Huang and Zhang [\emph{Cone metric spaces and fixed pointtheorems of contractive mappings}, J. Math. Anal. Appl.,\textbf{332} (2007), 1468 -1476] defined cone metric spaces bysubstituing an order normed space for the real numbers and provedsome fixed point theorems.Let $E$ be a real Hausdorff topological vector space and $P$ a conein $E$ with int\,$P\neq \emptyset$, where int\,$P$ denotes theinterior of $P$. Let $X$ be a nonempty set. A function $d : X \times X\to E$ is called a \emph{tvs}-cone metric and $(X, d)$ iscalled a \emph{tvs}-cone metric space, if the following conditionshold: (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 $X$, thesame of Huang and Zhang. Then, the authors by using this class ofconvergent sequences proved several interesting results of commonfixed points for three or two mappings satisfying some contractiveconditions. 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 PointResults 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 metricspaces}, Rend. Circ. Mat. Palermo \textbf{57} (2008), 279--285 and\textit{Weakly $\varphi$-pairs and common fixed points in conemetric spaces}, Rend. Circ. Mat. Palermo \textbf{58} (2009),125--132].
Lingua originaleEnglish
Numero di pagine0
Stato di pubblicazionePublished - 2011


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