## Properties of the solutions of rational matrix difference equations.(English)Zbl 1055.39033

Consider the matrix difference equation $\begin{split} X_t=A^*X_{t+1}A+Q+\Pi_1(X_{t+1})-[S+A^*X_{t+1}B+ \Pi_{12}(X_{t+1})]\times\\ \times[R+B^*X_{t+1}B+\Pi_2(X_{t+1})]^+[S+A^*X_{t+1} B+\Pi_{12}(X_{t+1})]^*\end{split}\tag{A}$ and the corresponding algebraic equations $\begin{split} A^*XA-X+Q+\Pi_1(X)-[S+A^*XB+\Pi_{12}(X)]\times\\ \times [R+B^*XB+\Pi_2(X)]^+[S+A^*XB+\Pi_{12}(X)]^*= 0.\end{split}\tag{B}$ Sufficient conditions for the existence and uniqueness of stabilizing solutions for (A) are established. Finally, it is shown that under certain conditions on the coefficients the solutions of (A) converge for any positive value to the stabilizing solutions of (B).

### MSC:

 39A20 Multiplicative and other generalized difference equations 39A70 Difference operators
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### References:

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