Fixed point results in incomplete extended $b$-metric space with $t$-property

Abstract

The main objective of this paper is to prove some fixed point theorems in partially ordered, extended $b$-metric spaces which are not  complete. The idea behind the proof is based on the so-called $t$-property. Contractions of various kinds, including Boyd-Wong type contraction mappings defined on incomplete, extended  $b$-metric spaces are studied. Several examples are provided in order to demonstrate the novel concepts and findings and also, to verify the fact that the existence theorems proved in this paper can be used whenever the well-known theorems in the literature are not applicable.