In this manuscript we discuss, consider, generalize, improve and unify several recent results for so-called F-contraction-type mappings in the framework of complete metric spaces. We also introduce $(\phi ,F)$ -weak contraction and establish the corresponding fixed point result. Using our new approach for the proof that a Picard sequence is a Cauchy in metric space, our obtained results complement and enrich several methods in the existing literature. At the end we give one open question for F-contraction of Ćirić-type mapping.

In this paper we consider $(s-q)$ -graphic contraction mapping in b-metric like spaces. By using our new approach for the proof that a Picard sequence is Cauchy in the context of b-metric-like space, our results generalize, improve and complement several approaches in the existing literature. Moreover, some examples are presented here to illustrate the usability of the obtained theoretical results.

We concern this manuscript with Geraghty type contraction mappings via simulation functions and pull down some sufficient conditions for the existence and uniqueness of point of coincidence for several classes of mappings involving Geraghty functions in the setting of metric spaces. These findings touch up many of the existing results in the literature. Additionally, we elicit one of our main result by a non-trivial example and pose an interesting open problem for the enthusiastic readers.

The purpose of this paper is to consider various contractive conditions in b-metric spaces which have been recently published. Our results improve and complement many recent results from this field. Using the recently obtained result by R. Miculescu and A. Mihail (Miculescu & Mihail, 2017, pp.1-11) the authors of this article show that the proofs of the majority of known results in the context of b-metric spaces can be shortened.