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 E p , Ω 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 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 ) = 1 p E p , Ω p ( u ) - 1 α ‖ u ‖ L α ( Ω ) α - ∫ Ω f ( x ) u d x , where 1 < α < p , [ W m ] m ∈ N is dense in W 0 1 , p ( Ω ) and f ∈ L p ′ ( Ω ) , with 1 p + 1 p ′ = 1 .