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On Ran–Reurings Fixed Point Theorem

  • Praveen Agarwal
  • Mohamed Jleli
  • Bessem Samet
Chapter

Abstract

In order to study the existence of solutions to a certain class of nonlinear matrix equations, Ran and Reurings [38] established an extension of Banach contraction principle to metric spaces equipped with a partial order. In this chapter, we present another proof of Ran–Reurings fixed point theorem using Banach contraction principle. Next, we present some applications of this result to the solvability of some classes of matrix equations.

References

  1. 1.
    Bai, Z.Z., Guo, X.X., Yin, J.F.: On two iteration methods for the quadratic matrix equation. Inter. J. Numer. Anal. Mod. 2, 114–122 (2005)MathSciNetGoogle Scholar
  2. 2.
    Benner, P., Fabbender, H.: On the solution of the rational matrix equation \(X=Q+LX^{-1}L^{*}\). EURASIP J. Adv. Signal Pro. 1, 1–10 (2007)MathSciNetGoogle Scholar
  3. 3.
    Berzig, M.: Solving a class of matrix equations via Bhaskar-Lakshmikantam coupled fixed point theorem. Appl. Math. Lett. 25, 1638–1643 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Berzig, M., Duan, X., Samet, B.: Positive definite solution of the matrix equation \(X=Q-A^*X^{-1}A+B^*X^{-1}B\) via Bhaskar-Lakshmikantham fixed point theorem. Math Sci. 6(27), 1–6 (2012)Google Scholar
  5. 5.
    Berzig, M., Samet, B.: Solving systems of nonlinear matrix equations involving Lipshitzian mappings. Fixed Point Theory Appl. 89, 2011 (2011)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Berzig, M., Samet, B.: An extension of coupled fixed point’s concept in higher dimension and applications. Comput. Math. Appl. 63(8), 1319–1334 (2012)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Berzig, M., Samet, B.: Positive fixed points for a class of nonlinear operators and applications. Positivity 17(2), 235–255 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bhaskar, T.G., Lakshmikantham, V.: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 65, 1379–1393 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bhatia, R.: Matrix Analysis. Graduate Texts in Mathematics, vol. 169. Springer, New York (1997)CrossRefGoogle Scholar
  10. 10.
    Duan, X.F., Liao, A.P.: On the existence of Hermitian positive definite solutions of the matrix equation \(X^{s}+A^{*}X^{-t}A=Q\). Linear Algebra Appl. 429, 673–687 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Duan, X.F., Liao, A.P.: On the nonlinear matrix equation \(X+A^{*}X^{-q}A=Q (q\ge 1)\). Math. Comput. Mod. 49, 936–945 (2009)CrossRefGoogle Scholar
  12. 12.
    Duan, X.F., Liao, A.P., Tang, B.: On the nonlinear matrix equation \(X-\sum \limits _{i=1}^{m}A^{*}_{i}X^{\delta _{i}}A_{i}=Q\). Linear Algebra Appl. 429, 110–121 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    El-Sayed, S.M.: Ran, ACM.: On an iterative method for solving a class of nonlinear matrix equations. SIAM J. Matrix Anal. Appl. 23, 632–645 (2001)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Engwerda, J.C.: On the existence of a positive definite solution of the matrix equation \(X+A^{T}X^{-1}A=I\). Linear Algebra Appl. 194, 91–108 (1993)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Engwerda, J.C., Ran, A.C.M., Rijkeboer, A.L.: Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation \(X+A^{*}X^{-1}A=Q\). Linear Algebra Appl. 186, 255–275 (1993)Google Scholar
  16. 16.
    Ferrante, A., Levy, B.C.: Hermitian solutions of the equations \(X=Q+NX^{-1}N^{*}\). Linear Algebra Appl. 247, 359–373 (1996)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Fital, S., Guo, C.H.: A note on the fixed-point iteration for the matrix equations \(X\pm A^{*}X^{-1}A=I\). Linear Algebra Appl. 429, 2098–2112 (2008)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Guo, C.H., Lancaster, P.: Iterative solution of two matrix equations. Math. Comput. 68, 1589–1603 (1999)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Guo, D., Lakshmikantham, V.: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal. 11, 623–632 (1987)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Guo, D.: Fixed points of mixed monotone operators with applications. Appl. Anal. 31, 215–224 (1988)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Guo, D.: Existence and uniqueness of positive fixed point for mixed monotone operators and applications. Appl. Anal. 46, 91–100 (1992)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Guo, D., Cho, Yeol Je., Zhu, J.: Partial Ordering Methods in Nonlinear Problems. Nova Publishers, New York (2004)Google Scholar
  23. 23.
    Hasanov, V.I.: Positive definite solutions of the matrix equations \(X\pm A^{*}X^{-q}A=Q\). Linear Algebra Appl. 404, 166–182 (2005)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Hasanov, V.I., El-Sayed, S.M.: On the positive definite solutions of the nonlinear matrix equations \(X+A^{*}X^{-\delta }A=Q\). Linear Algebra Appl. 412, 154–160 (2006)MathSciNetCrossRefGoogle Scholar
  25. 25.
    He, Y.M., Long, J.H.: On the Hermitian positive definite solution of the nonlinear matrix equation \(X+\sum \limits _{i=1}^{m}A_{i}^{*}X^{-1}A_{i}=I\). Appl. Math. Comput. 216, 3480–3485 (2010)MathSciNetGoogle Scholar
  26. 26.
    Ivanov, I.G.: On positive definite solutions of the family of matrix equations \(X+A^{*}X^{-n}A=Q\). J. Comput. Appl. Math. 193, 277–301 (2006)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Ivanov, I.G., Hasanov, V.I., Uhilg, F.: Improved methods and starting values to solve the matrix equations \(X\pm A^{*}X^{-1}A=I\) iteratively. Math. Comput. 74, 263–278 (2004)CrossRefGoogle Scholar
  28. 28.
    Ivanov, I.G., El-Sayed, S.M.: Properties of positive definite solutions of the equation \(X+A^{*}X^{-2}A=I\). Linear Algebra Appl. 279, 303–316 (1998)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Liu, X.G., Gao, H.: On the positive definite solutions of the matrix equation \(X^{s}\pm A^{*}X^{-t}A=I_{n}\). Linear Algebra Appl. 368, 83–97 (2003)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Long, J.H., Hu, X.Y., Zhang, L.: On the Hermitian positive definite solution of the nonlinear matrix equation \(X+A_{1}^{*}X^{-1}A_{1}+A_{2}^{*}X^{-1}A_{2}=I\). Bull. Braz. Math. Soc. 222, 645–654 (2008)Google Scholar
  31. 31.
    Meini, B.: Efficient computation of the extreme solutions of \(X+A^{*}X^{-1}A=Q\) and \(X-A^{*}X^{-1}A=Q\). Math. Comput. 71, 1189–1204 (2001)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Monsalve, M., Raydan, M.: A new inversion-free method for a rational matrix equation. Linear Algebra Appl. 433, 64–71 (2010)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Nieto, J.J., Rodríguez-López, R.: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. (Engl. Ser.) 23, 2205–2212 (2007)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Opoitsev, V.I.: Heterogenic and combined-concave operators. Syber. Math. J. 16, 781–792 (1975)Google Scholar
  35. 35.
    Opoitsev, V.I.: Dynamics of collective behavior. III. Heterogenic systems. Avtomat. i Telemekh. 36, 124–138 (1975)Google Scholar
  36. 36.
    Peng, Z.Y., El-Sayed, S.M.: On positive definite solution of a nonlinear matrix equation. Numer. Linear Algebra Appl. 14, 99–113 (2007)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Peng, Z.Y., El-Sayed, S.M., Zhang, X.L.: Iterative methods for the extremal positive definite solution of the matrix equation \(X+A^{*}X^{-\alpha }A=Q\). J. Comput. Appl. Math. 200, 520–527 (2007)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Ran, A.C.M., Reurings, M.C.B.: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 132, 1435–1443 (2004)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Samet, B.: Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces. Nonlinear Anal. 72, 4508–4517 (2010)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Samet, B.: Ran-Reurings fixed point theorem is an immediate consequence of the Banach contraction principle. J. Nonlinear Sci. Appl. 9, 873–875 (2016)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Samet, B., Karapinar, E., Aydi, H., Cojbasic Rajic, V.: Discussion on some coupled fixed point theorems. Fixed Point Theory Appl. 2013, 50 (2013)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Samet, B., Vetro, C.: Coupled fixed point, f-invariant set and fixed point of N-order. Ann. Funct. Anal. 1(2), 46–56 (2010)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Samet, B., Vetro, C.: Coupled fixed point theorems for multi-valued nonlinear contraction mappings in partially ordered metric spaces. Nonlinear Anal. 74(12), 4260–4268 (2011)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Xu, S.F.: Perturbation analysis of the maximal solution of the matrix equation \(X+A^{*}X^{-1}A=P\). Linear Algebra Appl. 336, 61–70 (2001)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Yong, J.M., Zhou, Z.Y.: Stochastic Controls. Hamiltonian Systems and HJB Equations. Springer, New York (1999)CrossRefGoogle Scholar
  46. 46.
    Zhan, X.Z.: Computing the extreme positive definite solutions of a matrix equation. SIAM J. Sci. Comput. 17, 632–645 (1996)CrossRefGoogle Scholar
  47. 47.
    Zhan, X.Z., Xie, J.J.: On the matrix equation \(X+A^{T}X^{-1}A=I\). Linear Algebra Appl. 247, 337–345 (1996)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of MathematicsAnand International College of EngineeringJaipurIndia
  2. 2.Department of Mathematics, College of SciencesKing Saud UniversityRiyadhSaudi Arabia

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