The Banach contraction principle is the most important result. This principle has many applications and some authors was interested in this principle in various metric spaces as Brianciari.
The author initiated the notion of the generalized metric space as a generalization of a metric space by replacing the triangle inequality by a more general inequality, $d(x,y)\leq d(x,u)+d(u,v)+d(v,y)$ for all pairwise distinct points $x,y,u,v$ of $X$. As such, any metric space is a generalized metric space but the converse is not true. He proved the Banach fixed point theorem in such a space. Some authors proved different types of fixed point theorems by extending the Banach's result. Wardowski introduced a new contraction, which generalizes the Banach contraction. He using a mapping $F: \mathbb{R}^{+} \rightarrow \mathbb{R}$ introduced a new type of contraction called $F$-contraction and proved a new fixed point theorem concerning $F$-contraction. In this paper, we have dealt with $F$-contraction and $F$-weak contraction in complete generalized metric spaces. We prove some results for $F$-contraction and $F$-weak contraction and we show that the existence and uniqueness of fixed point for satisfying $F$-contraction and $F$-weak contraction in complete generalized metric spaces. Some examples are supplied in order to support the useability of our results. The obtained result is an extension and a generalization of many existing results in the literature.
A characterization of completeness of generalized metric spaces using generalized Banach contraction principle
