A new fixed point theorem in [formula omitted]-metric space and its application to solve a class of nonlinear matrix equations

Abstract

In this article we extend the notions of G-metric and b-metric and define a new metric called Gb-metric with coefficient b≥1. A fixed point theorem is proved in this metric space. We obtain parallel results of several existing fixed point theorems such as that of Banach, Geraghty and Boyd–Wong in Gb-metric space using our theorem. As an application of our fixed point theorem we provide a fixed point iteration to solve a class of nonlinear matrix equations of the form Xs+A∗G(X)A=Q, where s≥1, A is an n×n matrix, G is a continuous function from the set of all Hermitian positive definite matrices to the set of all Hermitian positive semi-definite matrices and Q is an n×n Hermitian positive definite matrix. It is noted that the error in estimated solution we get by following our method is lesser than the error we get with Ćirić’s fixed point iteration.