Fixed Point Theorem: Variants, Affine Context and Some Consequences

Abstract

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 $\mathcal{E}_{p,Ω}^p$ 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 $Φ_m$ on a subspace $W_m$ of dimension $m$ given by \[ Φ_m(u)=\frac{1}{p}\mathcal{E}_{p, Ω}^{p}(u) - \frac{1}α\|u\|^α_{L^α(Ω)}- \int_Ωf(x)u dx, \] where $1<α<p$, $[W_m]_{m \in \mathbb{N}}$ is dense in $W^{1,p}_0(Ω)$ and $f\in L^{p'}(Ω)$, with $\frac{1}{p}+\frac{1}{p'}=1$.