d-Young distance function and existence of fixed points in complete metric spaces

We introduce the concept of d-Young distance function with respect to the 5-uplet \((p,q,\tau ,\kappa ,\xi )\), where d is a metric on a certain set \(\Lambda \), \(1<p,q<\infty \) with \(\frac{1}{p}+\frac{1}{q}=1\), \(\tau : \Lambda \times \Lambda \rightarrow [0,\infty )\), \(\kappa >0\), and \(\xi : [0,\infty )\rightarrow [0,\infty )\) satisfies the condition \(\inf _{t>0} \frac{\xi (t)}{t^\kappa }>0\). We establish some properties of the introduced distance function. Next, we study the existence and uniqueness of fixed points for some classes of mappings \(F: \Lambda \rightarrow \Lambda \) satisfying contractions involving the d-Young distance function. In particular, for a special choice of the 5-uplet \((p,q,\tau ,\kappa ,\xi )\), we recover the Banach fixed point theorem. We also provide an example, where our approach can be used, but the Banach fixed point theorem is inapplicable.