Fixed Point Theorem: Variants, Affine Context and Some Consequences

2023/05/05 by Anderson L. A. de Araújo, de Araujo, Anderson Luis Albuquerque, Edir Ferreira Leite +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2305.03791

openalex publication_date 2023/05/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 E_p,Ω^p introduced by Lutwak, Yang and Zhang in the work 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 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) - \frac1α‖u‖^α_L^α(Ω)- ∫_Ωf(x)u dx, where 1<α

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