Best proximity point theory on vector metric spaces

Abstract

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.

Best proximity point theorems for proximal b-cyclic contractions on b-metric spaces

“…Taking into account this situation, a new concept of the cyclic contraction mapping which is a generalization of inequality (1) has been introduced by Eldred and Veeremani [8], and hence obtained the existence of a point such that ( ; H ) = (}; <) which is a best proximity point of H and is also an optimal solution of the minimization problem min 2} ( ; H ). Since a best proximity point result is a natural generalization of …xed point result, many authors have studied to obtain best proximity point results [1,2,16,17]. Now, we recall this notion related result.…”

Some fixed point results via γ-contraction in non-Archimedean fuzzy metric spaces

“…Later, the author defined a new space he obtained with this form as a non-Archimedean fuzzy metric space. Afterward, Banach type contraction theorem was proved and generalized in fuzzy metric spaces just like in metric spaces [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].…”

Ordered Vectorial Quasi and Almost Contractions on Ordered Vector Metric Spaces

“…According to this work, any metric is a vector metric, but the converse is not true in general. In the last two decades, many extensions of the results in [21] have been completed, such as [22][23][24][25]. In [23], the authors united the concept of partial ordering with vector metric and provided some fixed point theorems on ordered vector metric space; hence, they generalized the results of [11,12,21].…”