Open access

DOI: 10.24193/fpt-ro.2018.1.21

Authors: Sk Monowar Hossein, Snehasish Bose and Kallol Paul

Abstract: In this paper we consider a pair of nonlinear matrix equations of the form X=Q₁+(Y^*XY)^r₁, Y=Q₂+(X^*YX)^r₂, where Q₁, Q₂ are n x n Hermitian positive definite matrices, r₁, r₂ ∈ and prove the existence and uniqueness of positive definite solutions of these equations. We provide an algorithm to approach the solution. We present a coupled fixed point theorem for non-decreasing mapping and show that a particular case of our nonlinear matrix equations also can be solved by using the derived coupled fixed point theorem. Also we show that by replacing Y with Y^-1 in first equation and X with X^-1 in second equation and taking Q₁=Q₂ and r₁=r₂, the reduced system can be solved using the coupled fixed point theorem of Berinde [5].

Key Words and Phrases: Fixed point, partially ordered set, matrix equation, Thompson metric.

2010 Mathematics Subject Classification: 15A24, 47H10, 47H09.

Published on-line: February 1st, 2018.

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