The matrix equation is considered, where .
Some necessary and/or sufficient conditions are given for the existence of a Hermitian positive definite solution. For instance, Theorem 2.2 states that such solution exists if and only if all the matrices in the sequence are positive definite and the sequence is uniformly bounded, where , and .
Two iterative algorithms are provided to determine the maximal positive definite solution.
Two examples illustrate the behaviour of the proposed algorithms.