不动点定理:变式、仿射上下文和一些结果

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY Accounts of Chemical Research Pub Date : 2023-10-24 DOI:10.1007/s43034-023-00304-x
Anderson L. A. de Araujo, Edir J. F. Leite
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引用次数: 0

摘要

在这项工作中,作为一般Brouwer不动点定理的结果,我们将提出仿射和经典背景下的变体不动点定理。例如,仿射结果将允许对仿射球进行处理,仿射球是通过仿射\(L^{p}\)泛函\(\mathcal定义的{E}_{p,\Omega}^p\),由Lutwak等人(J Differ Geom 62:17–382002)为非凸且不表示\(\mathbb{R}^m\)中的范数的\(p>;1\)引入。此外,我们讨论了一点上不连续泛函的结果。作为一个应用,我们研究了由$$\beagin{aligned}\Phi _m(u)=\frac{1}{p}\mathcal给出的维数为m的子空间\(W_m)上的仿射泛函序列\(\Phi _m\)的临界点{E}_{p,\Omega}^{p}(u)-\frac{1}{\alpha}\Vert u\Vert^{\aalpha}_{L^\alpha(\Omega)}-\int _{\Omega}f(x)u\textrm{d}x,\end{aligned}$$其中\(1<;\alpha<;p\),\([W_m]_{m\in\mathbb{N}}\)在\(W)中稠密^{1,p}_0(\Omega)\)和\(f\在L^{p'}(\Omega\)中,其中\(\frac{1}{p}+\frac{1}{p'}=1\)。
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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 \(\frac{1}{p}+\frac{1}{p'}=1\).

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来源期刊
Accounts of Chemical Research
Accounts of Chemical Research 化学-化学综合
CiteScore
31.40
自引率
1.10%
发文量
312
审稿时长
2 months
期刊介绍: Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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