Best proximity point theory on vector metric spaces
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.