Homothetic rectangular metric space and contraction principles

In this article, we introduce an extension of rectangular metric space, called a homothetic rectangular metric space, namely, the rectangular inequality in this space will have the following form: for all \(x,y \in X\) $x,y\in X$ and for all distinct points \(u,v \in X\) $u,v\in X$ each of them are different from x and y, \(\begin{aligned} d(x,y) \le d(x,u) + \nu (x,y)d(u,v)+d(v,y), \end{aligned}\) $$\begin{array}{c}\hfill d(x,y)\le d(x,u)+\nu (x,y)d(u,v)+d(v,y),\end{array}$$ where \(\nu\) $\nu$ is a control function defined from \(X\times X \rightarrow (-\infty ,+\infty )\) $X\times X\to (-\infty ,+\infty )$ . We next show the main properties of sequences in a homothetic rectangular metric space with a negative control function \(\nu\) $\nu$ and also where \(\nu\) $\nu$ takes values in (0, 1). Moreover, we give sufficient conditions for this new extension to becomes a metric space. Analogues of the Banach contraction principle and Kannan’s fixed point theorem are proved in this space under conditions on \(\nu\) $\nu$ .