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Edelstein-Type Fixed Point Theorems in Compact Tvs-Cone Metric Spaces
(Hacettepe University, 2014) Abdeljawad, Thabet
In this paper we prove two fixed point theorems in compact cone metricspaces over normal cones. The first theorem generalizes Edelstein theorem [8] and is different from the generalization obtained in [11]. Thesecond theorem generalizes the main result in [10] and the first theorem.However, the two theorems fail in different categories. Moreover, different versions of the two theorems are proved in TVS-cone metric spacesby making use of the nonlinear scalarization function used very recentlyby Wei-Shih Du in [A note on cone metric fixed point theory and itsequivalence, Nonlinear Analysis,72(5),2259-2261 (2010).] to prove theequivalence of the Banach contraction principle in cone metric spacesand usual metric spaces.
Citation - WoS: 6
Citation - Scopus: 6
Order Norm Completions of Cone Metric Spaces
(Taylor & Francis inc, 2011) Abdeljawad, Thabet
In this article, a completion theorem for cone metric spaces and a completion theorem for cone normed spaces are proved. The completion spaces are constructed by means of an equivalence relation defined via an ordered cone norm on the Banach space E whose cone is strongly minihedral and ordered closed. This order norm has to satisfy the generalized absolute value property. In particular, if E is a Dedekind complete Banach lattice, then, together with its absolute value norm, satisfy the desired properties.

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