On an equivalence of topological vector space valued cone metric spaces and metric spaces |
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Institution: | a Maltepe University, Department of Mathematics, Faculty of Science and Letters, Marmara E?i?ti?m Köyü, TR 34857, Maltepe, ?stanbul, Turkeyb Department of Mathematics, Gebze Institute of Technology, Cayirova Campus 41400 Gebze-Kocaeli, Turkeyc Department of Mathematics, Maltepe University, Marmara E?i?ti?m Köyü, TR 34857, Maltepe, ?stanbul, Turkey |
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Abstract: | Scalarization method is an important tool in the study of vector optimization as corresponding solutions of vector optimization problems can be found by solving scalar optimization problems. Recently this has been applied by Du (2010) 14] to investigate the equivalence of vectorial versions of fixed point theorems of contractive mappings in generalized cone metric spaces and scalar versions of fixed point theorems in general metric spaces in usual sense. In this paper, we find out that the topology induced by topological vector space valued cone metric coincides with the topology induced by the metric obtained via a nonlinear scalarization function, i.e any topological vector space valued cone metric space is metrizable, prove a completion theorem, and also obtain some more results in topological vector space valued cone normed spaces. |
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Keywords: | TVS-cone metric space Generalized TVS-cone metric space Cone Banach spaces Set-valued operators |
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