当前位置: X-MOL 学术Optim. Lett. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Uniform convexity, uniform monotonicity and $$(\phi ,c)$$ ( ϕ , c ) -subdifferentiability with application to nonlinear PDE
Optimization Letters ( IF 1.6 ) Pub Date : 2021-07-16 , DOI: 10.1007/s11590-021-01783-4
Abdessamad Oussarhan 1 , Tijani Amahroq 2 , Ikram Daidai 2 , Aicha Syam 2
Affiliation  

This paper introduces a new subclass of convex functions called uniformly convex functions [this class is a slight extension of the class introduced earlier by Zălinescu (J Math Anal Appl 95:344–374, 1983; Convex analysis in general vector spaces, World Scientific, Singapore)] and studies some of its properties. It also defines a new subdifferentiability called \((\phi ,c)\)-subdifferentiability and investigates some of its properties. As a main result, we prove that a lower semicontinuous function defined on a Banach space is uniformly convex if and only if its Clarke subdifferential is uniformly monotone set-valued map. Then we move to present a result concerning optimization problems for uniformly convex functions. Finally, we apply our theory to conclude the Minty-Browder theorem for a subclass of monotone operators as well as the Lax–Milgram theorem without recourse to use the Riesz–Fréchet theorem. Consequently, we establish the existence and uniqueness of weak solution for the p-Laplacian problem.



中文翻译:

均匀凸性、均匀单调性和 $$(\phi ,c)$$ ( ϕ , c ) - 可微分,应用于非线性 PDE

这篇论文介绍了一个新的凸函数子类,称为一致凸函数[这个类是 Zălinescu 早先介绍的类的一个轻微扩展(J Math Anal Appl 95:344–37​​4, 1983; 一般向量空间中的凸分析,World Scientific,新加坡)]并研究了它的一些特性。它还定义了一个新的次可微性,称为\((\phi ,c)\)- 次可微性并研究它的一些性质。作为主要结果,我们证明了定义在 Banach 空间上的下半连续函数是一致凸的,当且仅当其 Clarke 次微分是一致单调集值映射。然后我们开始介绍关于一致凸函数的优化问题的结果。最后,我们应用我们的理论来总结单调算子子类的 Minty-Browder 定理以及 Lax-Milgram 定理,而无需求助于 Riesz-Fréchet 定理。因此,我们建立了p- Laplacian 问题弱解的存在唯一性。

更新日期:2021-07-18
down
wechat
bug