集合值化的二次函数方程

Pub Date : 2024-04-21 DOI:10.1007/s00010-024-01067-z

Elham Mohammadi, Abbas Najati, Kazimierz Nikodem

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引用次数: 0

摘要

考虑一个实向量空间，记为 X，让 cc(Y) 表示实 Hausdorff 拓扑向量空间 Y 的所有凸紧凑子集的集合。本文研究了对于未知函数 $f_1,f_2,f_3,f_4:X\rightarrow cc(Y)$ 的 Pexiderized 二次函数方程 $$begin{aligned} f_1(x+y)+f_2(x-y)=f_3(x)+f_4(y), \end{aligned}$$的集合值解。这个函数方程包含了许多函数方程，包括二次方程、柯西方程和德赖加斯方程。本文提出了该函数方程的集值解的特征。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Set valued pexiderized quadratic functional equation

Consider a real vector space denoted as X, and let cc(Y) represent the collection of all convex and compact subsets of a real Hausdorff topological vector space Y. This paper investigates set-valued solutions of the Pexiderized quadratic functional equation

$$\begin{aligned} f_1(x+y)+f_2(x-y)=f_3(x)+f_4(y), \end{aligned}$$

for unknown functions $f_1,f_2,f_3,f_4:X\rightarrow cc(Y)$. This functional equation incorporates many functional equations including the quadratic, Cauchy’s and Drygas’ equations. A characterization for set-valued solutions of this functional equation is presented in this paper.

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