A Fixed Point Theorem for Monotone Maps and Its Applications to Nonlinear Matrix Equations
By using the fixed point theorem for monotone maps in a normal cone, we prove a uniqueness theorem for the positive definite solution of the matrix equation X=Q+A⁎f(X)A, where f is a monotone map on the set of positive definite matrices. Then we apply the uniqueness theorem to a special equation X=k...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2015-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2015/167049 |
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| Summary: | By using the fixed point theorem for monotone maps in a normal cone, we prove a uniqueness theorem for the positive definite solution of the matrix equation X=Q+A⁎f(X)A, where f is a monotone map on the set of positive definite matrices. Then we apply the uniqueness theorem to a special equation X=kQ+A⁎(X^-C)qA and prove that the equation has a unique positive definite solution when Q^≥C and k>1 and 0<q<1. For this equation the basic fixed point iteration is discussed. Numerical examples show that the iterative method is feasible and effective. |
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| ISSN: | 2314-4629 2314-4785 |