Nonnegative matrix equations having positive solutions

Abstract:

Suppose à is a nonnegative invertible matrix with a positive diagonal Diag $(\bar A) > 0$ and $\bar y > 0$ is a positive vector. Let $A = {D^{ - 1}}\tilde A$ and $y = {D^{ - 1}}\tilde y$. If $0 < 2y - Ay$ , then $2y - Ay \leqq x \leqq y$, where ${A^{ - 1}}y$.