In this paper, we introduce the notion of a modular $ p $-metric space (an extended modular $ b $-metric space) and establish some fixed point results for $ \alpha $-$ \widehat{\nu} $-Meir-Keeler contractions in this new space. Using these results, we deduce some new fixed point theorems in extended modular metric spaces endowed with a graph and in partially ordered extended modular metric spaces. Also, we develop an important relation between fuzzy-Meir-Keeler and extended fuzzy $ p $-metric with modular $ p $-metric and get certain new fixed point results in triangular fuzzy $ p $-metric spaces. We provide an example and an application to support our results which generalize several well known results in the literature.

In this paper, the notion of sequential ς p -metric spaces has been introduced as a generalization of usual S -metric spaces, S b -metric spaces, S J S metric spaces, and specially of S p -metric spaces. In view of this notion, we prove some fixed point theorems for some classes of ς p -rational Geraghty JS-contractions over such spaces. A supporting example and an application have been given in order to examine the validity of the obtained results.

Ciric and Presic developed the concept of Presic contraction to Ciric-Presic type contractive mappings in the background of a metric space. On the other hand, Altun and Olgun introduced Perov type F-contractions. In this paper, we extend the concept of Ciric-Presic contractions to Perov-Ciric-Presic type F-contractions. Our results modify some known ones in the literature. To support our main result, an example and an application to nonlinear operator systems are presented.

As a natural extension of the metric and the bipolar metric, this article introduces the new abstract bipolar $b-$ metric. The bipolar $b-$metric is a novel technique addressed in this article; it is explained by combining the well-known $b-$metric in the theory of metric spaces, as defined by Mutlu and G\"{u}rdal (2016) \cite{mg1}, with the description of the bipolar metric. In this new definition, well-known mathematical terms such as Cauchy and convergent sequences are utilized. In the bipolar $b-$metric, fundamental topological concepts are also defined to investigate the existence of fixed points implicated in such mappings under different contraction conditions. An example is provided to demonstrate the presented results.

In this study, we obtain some coincidence point theorems for weakly O - α -admissible contractive mappings in an orthogonal extended S -metric space. An example and an application are provided to illustrate the usability of the obtained results. Our results generalize the results of several studies from metric and S -metric frameworks to the setting of orthogonal extended S -metric spaces.