Some fixed point theorems for contractive mapping in ordered vector metric spaces

Abstract

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.

Best proximity point theory on vector metric spaces

“…We also introduce a new notion so called -vector proximal contraction mapping including the concept of -proximal contraction de…ned Samet et al [14]. Then, for the …rst time, we obtain some best proximity point results on vector metric spaces ( ; ; E) where (E; ) is -Dedekind complete Riesz space and hence we extend some …xed point results proved on both vector metric spaces and partially ordered vector metric spaces such as [7,8,12].…”

“…If we take (&; ) = 1 in Theorem 14, we deduce the following best proximity point result for the vector proximal contraction mappings. Taking M = N = in Corollary 12 and Corollary 16, we can obtain the following …xed point result in vector metric spaces which is the main result of [12].…”

“…On the other hand, from (12), (13) and the condition (H) there exists a subsequence f& n k g of f& n g such that (& n k ; & ) 1 for all k 2 N. Because of (11) and 14, we get…”

“…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. Then, many authors have studied to obtain various …xed point results in context of vector metric spaces [2,12,16,17,18,21]. However, 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.…”

Unbounded Vectorial Cauchy Completion of Vector Metric Spaces

“…Vector metric spaces equipped with a Riesz space E valued distance map are defined in [8,9]. A vector metric d on a non-empty set X is a map : × → satisfying the followings: for all , , ∈ .…”