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.
