To begin with, I am recalling the Banach Fixed Point Theorem.
Let $(X,d)$ be a non-empty complete metric space with a contraction mapping $T:X\to X$. Then $T$ admits a unique fixed-point $x^*$ in $X$ i.e. $T(x^*) = x^*$.
In order to prove that the combinantion of the best response strategies of $N \geqslant 2$ agents consitutes a Nash Equilibrium, we need to use a fixed point theorem argument. Is this necessary only in cases where the agents employ mixed strategies or this holds in the case where they also follow pure strategies? I am searcing for some guidance to use the Banch fixed point theorem in guadratic utility functions when agents act by submitting pure strategies. How can I build this argument? What confuses me the most is that the mapping $T:X\to X$ is about the strategies isn't it?
I can provide more details of my problem if you wish so. I would also be glad if you could answer with great details, since I do not have any idea about the topic