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In this paper, we introduce the concept of cyclic p-contraction pair for single-valued mappings. Then we present some best proximity point results for such mappings defined on proximally complete pair of subsets of a metric space. Also, we provide some illustrative examples that compared our results with some earliest. Finally, by taking into account a fixed point consequence of our main result we give an existence and uniqueness result for a common solution of a system of second order boundary value problems.

In this paper, considering an order relation on a vector metric space which is introduced by Ç evik and Altun in 2009, we present some fundamental fixed point results. Then, we provide some nontrivial examples show that the investigation of this work is significant.

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In this paper, we …rst give a new de…nition of-Dedekind complete Riesz space (E;) in the frame of vector metric space (; ; E) and we investigate the relation between Dedekind complete Riesz space and our new concept. Moreover, we introduce a new contraction so called-vector proximal contraction mapping. Then, we prove certain best proximity point theorems for such mappings on vector metric spaces (; ; E) where (E;) is-Dedekind complete Riesz space. Thus, for the …rst time, we acquire best proximity point results on vector metric spaces. As a result, we generalize some …xed point results proved on both vector metric spaces and partially ordered vector metric spaces. Further, we provide nontrivial and comparative examples to show the e¤ectiveness of our main results.