Fixed Point Theorem: Variants, Affine Context and Some Consequences

In this work, we will present variants Fixed Point Theorem for the affine and classical contexts, as a consequence of general Brouwer's Fixed Point Theorem. For instance, the affine results will allow working on affine balls, which are defined through the affine $L^{p}$ functional $\mathcal{E}_{p,\Omega}^p$ introduced by Lutwak, Yang and Zhang in the work \textit{Sharp affine L_p Sobolev inequalities}, J. Differential Geom. 62 (2002), 17-38 for $p > 1$ that is non convex and does not represent a norm in $\mathbb{R}^m$. Moreover, we address results for discontinuous functional at a point. As an application, we study critical points of the sequence of affine functionals $\Phi_m$ on a subspace $W_m$ of dimension $m$ given by $$\Phi_m(u)=\frac{1}{p}\mathcal{E}_{p, \Omega}^{p}(u) - \frac{1}{\alpha}\|u\|^{\alpha}_{L^\alpha(\Omega)}- \int_{\Omega}f(x)u dx,$$ where $1<\alpha<p$, $[W_m]_{m \in \mathbb{N}}$ is dense in $W^{1,p}_0(\Omega)$ and $f\in L^{p'}(\Omega)$, with $\frac{1}{p}+\frac{1}{p'}=1$.