Abstract
In recent years, fixed-point theorems have attracted increasing attention and have been widely investigated by many authors. Moreover, determining a fixed point has become an interesting topic. In this paper, we provide a constructive proof of the general Brouwer fixed-point theorem and then obtain the existence of a smooth path which connects a given point to the fixed point. We also present a non-interior point homotopy algorithm for solving fixed-point problems on a class of nonconvex sets by numerically tricking this homotopy path.
Highlights
In recent years, fixed-point theorems have attracted increasing attention and have been widely investigated by many authors (e.g., [1,2,3,4] and the references therein) because these theorems play important roles in mechanics, physics, differential equations, and so on
We provide a constructive proof of the general Brouwer fixed-point theorem and obtain the existence of a smooth path which connects a given point to the fixed point
We present a non-interior point homotopy algorithm for solving fixed-point problems on a class of nonconvex sets by numerically tricking this homotopy path
Summary
In recent years, fixed-point theorems have attracted increasing attention and have been widely investigated by many authors (e.g., [1,2,3,4] and the references therein) because these theorems play important roles in mechanics, physics, differential equations, and so on. We provide a constructive proof of the general Brouwer fixed-point theorem and obtain the existence of a smooth path which connects a given point to the fixed point. The abovementioned results generally require certain convexity assumptions; the traditional homotopy method cannot be used to handle the general Brouwer fixed-point theorem.
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