Some fixed point results in TVS-cone metric spaces (Q2863544)

A cone metric is such a function that just replacing the set of real numbers with a Banach space \(E\) with a partial order with respect to a cone in \(E\) in the metric function. Investigation of cone metric spaces goes back to 1934 when a Ph.\,D.\ student of the famous French mathematician Maurice Fréchet, the Serbian mathematician Đuro Kurepa, introduced more abstract metric spaces, in which the metric takes values in an ordered vector space.NEWLINENEWLINE\textit{J. S. Vandergraft} [SIAM J. Numer. Anal. 4, 406--432 (1967; Zbl 0161.35302)], \textit{B. Rzepecki} [Publ. Inst. Math., Nouv. Sér. 28(42), 179--186 (1980; Zbl 0482.47029)] and \textit{S.-D. Lin} [Indian J. Pure Appl. Math. 18, 685--690 (1987; Zbl 0622.47057)] reintroduced such spaces calling them \(K\)-metric spaces. \textit{I. Beg}, \textit{M. Abbas} and \textit{T. Nazir} [J. Nonlinear Sci. Appl. 3, No. 1, 21--31 (2010; Zbl 1203.54035)] and \textit{A. Azam}, \textit{I. Beg} and \textit{M. Arshad} [Fixed Point Theory Appl. 2010, Article ID 604084 (2010; Zbl 1197.54057)] replaced the set of an ordered Banach space by a locally convex Hausdorff topological vector space in the definition of a cone metric and a generalized cone metric space. \textit{W.-S. Du} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 5, A, 2259--2261 (2010; Zbl 1205.54040)] obtained a metric \(d_{ p} =\xi_{e}op\) on a topological vector space valued cone metric space \((X,p)\), where \(\xi_{e}\) is a non linear scalarization function defined as \(\xi_{e} (y)=\inf\{r\in{\mathbb{R}}:y\in{re-K}\}\), \(y\in{E}\), and \(K\) is the pointed convex cone, \(\mathbb{R}\) is the set of real numbers, and proved a theorem which is equivalent to the Banach contraction principle.NEWLINENEWLINEIn the paper under review, the authors extend some fixed point theorems from metric spaces to TVS cone metric spaces. NEWLINENEWLINEReviewer's remark. Unfortunately, the authors neglect some references in which fixed point theorems were studied in TVS cone metric spaces, for example, the above mentioned reference [Zbl 1203.54035], and they fail to mention that \textit{H. Çakallı}, \textit{A. Sönmez} and \textit{C. Genç} [Appl. Math. Lett. 25, No. 3, 429--433 (2012; Zbl 1245.54038)] proved that the topology induced by a TVS cone metric coincides with the topology induced by an appropriate metric.