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 $\mathcal {E}_{p,\Omega }^p$ introduced by Lutwak et al. (J Differ Geom 62:17–38, 2002) 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

$$\begin{aligned} \Phi _m(u)=\frac{1}{p}\mathcal {E}_{p, \Omega }^{p}(u) - \frac{1}{\alpha }\Vert u\Vert ^{\alpha }_{L^\alpha (\Omega )}- \int _{\Omega }f(x)u \textrm{d}x, \end{aligned}$$

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 $\frac{1}{p}+\frac{1}{p'}=1$.