Generalized normed spaces and Fixed point Theorems

Characterization of a generalized triangle inequality in normed spaces

For a normed linear space $(X,|\cdot|)$ and $p>0$ we characterize all $n$-tuples $(μ_1,...,μ_n)\in\mathbb{R}^{n}$ for which the generalized triangle inequality of the second type $$\|x_1+...+x_n\|^p\leq\frac{|x_1|^p}{μ_1}+...+\frac{|x_n|^p}{μ_n}$$ holds for any $x_1,...,x_n\in X$. We also characterize $(μ_1,...,μ_n)\in\mathbb{R}^{n}$ for which the reverse of the inequality above holds.

Fixed points of enriched contraction mappings in 2-normed spaces

Very recently, Berinde and Păcurar obtained in [V. Berinde and M. Păcurar, Approximating fixed points of enriched contractions in Banach spaces. Journal of Fixed Point Theory and Applications. \textbf{22}(2) (2020), 1--10.] an interesting generalization of the Banach contractive mapping principle in the framework of Banach spaces. The aim of this paper is to prove some fixed point theorems which extend the results of Berinde and Păcurar to the case of 2-normed spaces.

赋范向量空间中的镜面微分：拟中值定理与拟费马定理

Specular differentiation in normed vector spaces

本文提出镜面微分概念，推广了赋范向量空间中的Gâteaux和Fréchet微分，建立了弱形式的中值定理和费马定理，并利用镜面微分识别凸函数的Fréchet次微分的特定元素。

Some polynomial inequalities on real normed spaces

We consider various inequalities for polynomials, with an emphasis on the most fundamental inequalities of approximation theory. In the sequel a key role is played by the generalized Minkowski functional α(K,x), already being used by Minkowski and contemporaries and having occurred in approximation theory in the work of Rivlin and Shapiro in the early sixties. We try to compare real, geometric methods and complex, pluripotential theoretical approaches, where possible, and formulate a number of questions to be decided in the future. An extensive bibliography is given to direct the reader even i…

乘积空间中分量形式的 Krasnosel'skii 型不动点定理及其应用

Component-wise Krasnosel'skii type fixed point theorem in product spaces and applications

提出了乘积空间中范数型锥压缩与锥扩张条件下的 Krasnosel'skii 型不动点定理，基于锥中不动点指数理论证明共存不动点的存在性，并应用于证明含奇异非线性和混合项的二阶微分方程组正周期解的存在性。

混合与分量型Leggett-Wilson型固定点定理在积空间中的应用

Hybrid and Component-wise Leggett-Williams type Fixed Point Theorems in Product Spaces with Applications

本文提出了新的算子在二维范数线性空间笛卡尔积上的多重固定点定理，展示了每个分量中Leggett-Wilson型条件保证存在九个不同的固定点，其中四个是共存固定点。此外，结合Leggett-Wilson条件与Krasnosel'skii压缩-扩展条件的混合方法可获得三个固定点。应用方面，建立了非线性二阶方程组在两点边界条件下的多重正解存在性。