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 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 \(\frac{1}{p}+\frac{1}{p'}=1\).