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Solution of a class of cross-coupled nonlinear matrix equations

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  • Nashine, Hemant Kumar
  • Bose, Snehasish

Abstract

The cross-coupled nonlinear matrix equations play an important role in decision making of a variety of dynamical systems and control theory [1]. In this paper we solve the cross-coupled nonlinear matrix equations of the formX=Q1+∑i=1mAi*Fi(X)Ai−∑j=1nBj*Gj(Y)Bj,Y=Q2+∑k=1pCk*F˜k(Y)Ck−∑l=1qDl*G˜l(X)Dl,where Q1, Q2 are n × n Hermitian positive definite matrices, Ai, Bj, Ck, Dl’s are n × n matrices, and F1,…,Fm,F˜1,…,F˜p are order-preserving mappings and G1,…,Gn,G˜1,…,G˜q are order-reversing mappings from the set of n × n Hermitian positive definite matrices to itself. Our approach is based on a new fixed point result discussed in the framework of G-metric spaces, followed by some examples, that distinguishes it from the previously used methods.

Suggested Citation

  • Nashine, Hemant Kumar & Bose, Snehasish, 2019. "Solution of a class of cross-coupled nonlinear matrix equations," Applied Mathematics and Computation, Elsevier, vol. 362(C), pages 1-1.
    • Handle: RePEc:eee:apmaco:v:362:y:2019:i:c:4
    DOI: 10.1016/j.amc.2019.06.048
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    References listed on IDEAS

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    1. Engwerda, J.C., 1993. "On the existence of a positive definite solution of the matrix equation X = ATX-1A = I," Other publications TiSEM 9d762863-0dfe-4aeb-8a13-5, Tilburg University, School of Economics and Management.
    2. Engwerda, J.C. & Ran, A.C.M. & Rijkeboer, A.L., 1992. "Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation X+A*X-1A=Q," Other publications TiSEM cbc6bc1e-3bbf-49a4-8222-d, Tilburg University, School of Economics and Management.
    3. Asmaa M. Al-Dubiban, 2013. "On the Iterative Method for the System of Nonlinear Matrix Equations," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-7, March.
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