Monotone system of functions of many variables (Q1896894)

In this paper the authors introduce the notion of Markov monotone system and establish a sufficient condition for a system to be monotone. More precisely, they prove the following main result: If \(f_1(x_1,\dots, x_m),\dots, f_n(x_1,\dots, x_m)\), \(n\leq m\), \(x_i\in\mathbb{R}\), \(i= 1,2,\dots, m\), is a system of continuously differentiable functions, \(V\) is a subset of \(\mathbb{R}^m\) with one of the properties \hskip 17mm a) \(V= \{{\mathbf x}= (x_1,\dots, x_m);\;-\infty\leq a_i< x_i< b_i\leq +\infty,\;i= 1,2,\dots, m\}\) or \hskip 17mm b) \(V= \{{\mathbf x}= (x_1,\dots, x_m);\;-\infty\leq a< x_1<\cdots< x_m< a\leq+\infty\}\) or \hskip 17mm c) \(V= \{{\mathbf x}= (x_1,\dots, x_m);\;+\infty\geq b> x_1>\cdots >x_m> a\geq -\infty\}\) and for all \({\mathbf x}\in V\), \(\left({\partial f_i({\mathbf x})\over\partial x_j}\right)_{\substack{ i=1,2,\dots,n\\ j=1,2,\dots,m}}\) is an \(M\)-matrix, then the given system is monotone.