一个新非线性方程组解的存在性定理

An New Existence Theorem for Systems of Nonlinear Equations

作为微积分中连续函数介值定理在n维欧氏空间Rn中的推广,G.Aumann建立了一个定义于n维矩形上非线性方程组解的存在性定理,J.Rohn利用Brouwer不动点定理,把定理推广到定义域为紧凸集的情形。利用Brouwer拓扑度理论,证明定理中凸性条件进一步减弱为星形区域时,其结论仍然成立。

As a generalization of the intermediate value theorem in R^n, G. Aumann got an existence theorem of solution for systems of nonlinear equations defined on a n dimensional rectangle. By using Brouwer fixed point theorem,J. Rohn generalized the theorem under the hypothesis that the domain is a compact convex set. In this paper,using Brouwer topological degree theory,it is proved that the theorem still holds if we substitute star shape region for convex set.