We derive some necessary and sufficient conditions for the existence of an Hermitian positive definite solution. We mainly consider its solvability conditions and the question that under which conditions the nonlinear equation has only two different Hermitian positive definite solutions. Then we give a sufficient condition for the nonlinear equation to have only two different HPD solutions and obtain the formulae for these solutions. In addition, we consider the nonlinear equation with the case . We derive a necessary condition on the spectral radius of for the existence of a Hermitian positive definite solution and present some new properties of the solutions.