Mathematical background

Let a system $F(X)=0$ with $X=\{x_1,\ldots,x_n\}$. Let us consider a ball ${\cal B}=\{[\underline{x_1},\overline{x_1}],\ldots,[\underline{x_n},\overline{x_n}]$ for $X$ and define

$$\begin{eqnarray*} &&[X]_j^+ = \{X \in {\cal B}~{\rm such~that}~x_j=\overline{x_j... ..._j^- = \{X \in {\cal B}~{\rm such~that}~x_j=\underline{x_j}\}\\ \end{eqnarray*}$$

for $j$ in $[1,n]$. If

$$F_j(X)\ge 0 {\rm and}~ F_j(Y)\le 0~~\forall~X\in [X]_j^+, \forall~Y\in [X]_j^-, j=1,\ldots,n$$

or if

$$F_j(X)\le 0 {\rm and}~ F_j(Y)\ge 0~~\forall~X\in [X]_j^+, \forall~Y\in [X]_j^-, j=1,\ldots,n$$

then $F$ has at least one zero in ${\cal B}$ [15].