Abstract

We consider the equation \(G(x,x)=y\), where \(G\colon X\times X\to Y\) and \(X\) and \(Y\) are metric spaces. This operator equation is compared with the “model” equation \(g(t,t)=0\), where the function \(g\colon \mathbb{R}_+\times \mathbb{R}_+ \to\mathbb{R}\) is continuous, nondecreasing in the first argument, and nonincreasing in the second argument. Conditions are obtained under which the existence of solutions of this operator equation follows from the solvability of the “model” equation. Conditions for the stability of the solutions under small variations in the mapping \(G\) are established. The statements proved in the present paper extend the Kantorovich fixed-point theorem for differentiable mappings of Banach spaces, as well as its generalizations to coincidence points of mappings of metric spaces.

中文翻译：

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研究度量空间中方程的比较方法

摘要

我们考虑方程\（G（x，x）= y \），其中\（G \冒号X \ X到Y \）和\（X \）和\（Y \）是度量空间。将该运算符方程与“模型”方程\（g（t，t）= 0 \）进行比较，其中函数\（g \冒号\ mathbb {R} _ + \ times \ mathbb {R} _ + \ to \ mathbb {R} \）是连续的，第一个参数不减，第二个参数不增。从“模型”方程的可解性中获得了该运算符方程的解的存在条件。映射\（G \）的小变化下解的稳定性的条件建立。本文证明的陈述将Kantorovich不动点定理扩展到Banach空间的可微映射，以及将其推广到度量空间映射的重合点。