Common Positive Solution of Two Nonlinear Matrix Equations Using Fixed Point Results

Hemant Kumar Nashine, Rajendra Pant and Reny George
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Hemant Kumar Nashine: Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam
Rajendra Pant: Department of Mathematics and Applied Mathematics, University of Johannesburg, Kingsway Campus, Auckland Park 2006, South Africa
Reny George: Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia

Mathematics, 2021, vol. 9, issue 18, 1-16

Abstract: We discuss a pair of nonlinear matrix equations (NMEs) of the form X = R 1 + ? i = 1 k A i * F ( X ) A i , X = R 2 + ? i = 1 k B i * G ( X ) B i , where R 1 , R 2 ? P ( n ) , A i , B i ? M ( n ) , i = 1 , ? , k , and the operators F , G : P ( n ) ? P ( n ) are continuous in the trace norm. We go through the necessary criteria for a common positive definite solution of the given NME to exist. We develop the concept of a joint Suzuki-implicit type pair of mappings to meet the requirement and achieve certain existence findings under weaker assumptions. Some concrete instances are provided to show the validity of our findings. An example is provided that contains a randomly generated matrix as well as convergence and error analysis. Furthermore, we offer graphical representations of average CPU time analysis for various initializations.

Keywords: positive definite matrix; nonlinear matrix equation; fixed point; relational metric space; Suzuki contraction (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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