AbstractIn this article, we present a sufficient condition for the existence of a unique positive definite solution of the non-linear matrix equation , where , (the set of all Hermitian positive definite matrices), are non-singular matrices and are order-preserving mappings. We give an example of a non-linear matrix equation of the above form (, A is a non-singular matrix, F is an order-preserving mapping), which is not solvable by previously known methods but solvable by using our new results.