Fixed Point Theorem: Variants, Affine Context and Some Consequences

arXiv (Cornell University) Pub Date : 2023-05-05 DOI:10.48550/arxiv.2305.03791

de Araujo, Anderson Luis Albuquerque, Leite, Edir Junior Ferreira

引用次数: 0

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, 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 $\Phi_m$ on a subspace $W_m$ of dimension $m$ given by \[ \Phi_m(u)=\frac{1}{p}\mathcal{E}_{p, \Omega}^{p}(u) - \frac{1}{\alpha}\|u\|^{\alpha}_{L^\alpha(\Omega)}- \int_{\Omega}f(x)u dx, \] where $1<\alpha

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不动点定理:变体、仿射上下文和一些结果

在这项工作中，我们将提出仿射和经典背景下不动点定理的变体，作为一般browwer不动点定理的结果。例如，仿射结果将允许在仿射球上工作，它是通过Lutwak, Yang和Zhang在工作$\textit{Sharp affine $ L＿p $ Sobolev inequalities}$, J. Differential Geom. 62(2002)， 17-38中引入的仿射$L^{p}$泛函$\mathcal{E}_{p,\Omega}^p$定义的，对于$p > 1$是非凸的，不代表$\mathbb{R}^m$中的范数。此外，我们还处理了点上不连续泛函的结果。作为应用，我们研究了\[ \Phi_m(u)=\frac{1}{p}\mathcal{E}_{p, \Omega}^{p}(u) - \frac{1}{\alpha}\|u\|^{\alpha}_{L^\alpha(\Omega)}- \int_{\Omega}f(x)u dx, \]给出的$m$维数的子空间$W_m$上仿射泛函序列$\Phi_m$的临界点，其中$1<\alpha

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来源期刊

arXiv (Cornell University)

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