  In this paper the nonlinear matrix equation X-A^{*}X^{-p}A=Q with p>0 is
investigated. We consider two cases of this equation: the case p>1 and the case
0<p<1. In the case p>1, a new sufficient condition for the existence of a
unique positive definite solution for the matrix equation is obtained. A
perturbation estimate for the positive definite solution is derived. Explicit
expressions of the condition number for the positive definite solution are
given. In the case 0<p<1, a new sharper perturbation bound for the unique
positive definite solution is evaluated. A new backward error of an approximate
solution to the unique positive definite solution is obtained. The theoretical
results are illustrated by numerical examples.
