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.

Highlights • Unbounded vectorial convergence in vector metric spaces is studied. • Relations between uo-convergence and ,-convergence are examined. • An unbounded Cauchy completion of any vector metric space is constructed. • A more general completion of vector metric spaces is obtained.

In this paper, we define ordered vectorial quasi contractions. We show that ordered quasi contractions are ordered vectorial quasi contractions, but the reverse is not true. We also define ordered vectorial almost contractions and present fixed point theorems for this type of contractions. Hence, we disclose many results in the literature. With the help of examples, we illustrate the relationship between these two types of contractions and some others in the literature.

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In this paper we present a fixed point theorem for order-preserving Prešić type operators on ordered vector metric spaces. This result extends many results in the literature obtained for Prešić type operators both on metric spaces and partially ordered metric spaces. We also emphasize the relationships between our work and the previous ones in the literature. Finally we give an example showing the fact that neither results for Prešić type contractions on metric spaces nor the results for ordered Prešić type contractions on ordered metric space is applicable to it.