Abstract
In this paper, we first give a new definition of Ω-Dedekind complete Riesz space (E,≤) in the frame of vector metric space (Ω,ρ,E) and we investigate the relation between Dedekind complete Riesz space and our new concept. Moreover, we introduce a new contraction so called α-vector proximal contraction mapping. Then, we prove certain best proximity point theorems for such mappings in vector metric spaces (Ω,ρ,E) where (E,≤) is Ω-Dedekind complete Riesz space. Thus, for the first time, we acquire best proximity point results on vector metric spaces. As a result, we generalize some fixed point results proved in both vector metric spaces and partially ordered vector metric spaces such as main results of V4 . Further, we provide nontrivial and comparative examples to show the effectiveness of our main results.
Highlights
Introduction and PreliminariesCevik et al [11] brought to the literature a notion of vector metric and proved Banach ...xed point theorem [7] which is considered starting of metric ...xed point theory in these spaces
Introducing two new concepts so called -Dedekind complete Riesz space and -vector proximal contraction mapping, we prove some best proximity point theorems for such mappings on vector metric spaces ( ; ; E) where (E; ) is -Dedekind complete Riesz space
We ...rst give a new de...nition of -Dedekind complete Riesz space (E; ) on a vector metric space ( ; ; E), and so we obtain a new family of Riesz space which is a larger than the class of Dedekind complete Riesz space in the frame of vector metric space
Summary
Cevik et al [11] brought to the literature a notion of vector metric and proved Banach ...xed point theorem [7] which is considered starting of metric ...xed point theory in these spaces. Many authors have studied to obtain various ...xed point results in context of vector metric spaces [2, 12, 16, 17, 18, 21]. When consider the topological structure of vector metric spaces, it may not be easy to prove a result existing in the real valued metric spaces. This is an important and interesting point for the authors.
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