On an Iteration Method for Solving a Class of Nonlinear Matrix Equations

A fixed point theorem in partially ordered sets and some applications to matrix equations

“…In particular, this set is mapped into itself. As in [3] one checks that the sequence {G 2j (Q)} ∞ j=0 increases to a limit X −∞ , while the sequence {G 2j+1 } ∞ j=0 decreases to a limit X +∞ . These two matrices form a periodic orbit of period two, and any periodic orbit (including fixed points of G, of which there is at least one), is contained in [X −∞ , X +∞ ].…”

“…For the case m = 1, see [3]. The present case is motivated by Chapter 7 of [10], where the following matrix equation appears:…”

The symmetric linear matrix equation

“…Although in [1] m is equal to 1, the results in that paper can be easily generalized to the case that m ∈ N. In particular, Theorem 5.1 in [1] also holds for the map G + . Combining this theorem and Theorem 2.2 gives us the following result.…”

“…In [1] and [8] solutions of a matrix equation are considered as fixed points of some map G. Also, in the case of equation (1.1) we are interested in fixed points of the map…”

Positive Definite Solutions of the Matrix EquationXr-∑i=1mA

“…During the last few years, many researchers worked to develop the theory and numerical approaches for positive definite solutions to the nonlinear matrix equations of the form (1) [6][7][8][9]. Recently, Duan et al [10] showed that the equation − ∑ =1 * = (0 < | | < 1) always has a unique positive definite solution by using the fixed point theorem of mixed monotone operators.…”