Common positive solutions for two non-linear matrix equations using fixed point results in b-metric-like spaces

Abstract

We consider the system of non-linear matrix equations (NMEs) of the form X=B1+∑i=1kAi∗f(X)Ai,X=B2+∑i=1kAi∗g(X)Ai,where B1,B2 are two n× n Hermitian positive definite matrices, A1, A2,.., Am are n× n matrices, and f, g are two non-linear self-mappings of the set of all Hermitian matrices which are continuous in the trace norm. We discuss a sufficient condition ensuring the existence of a unique positive definite solution of the given system. We demonstrate this sufficient condition for two different systems of NMEs. In order to do this, we introduce the notion of generalized orbital α-admissible pair of mappings and deduce common fixed points results for these mappings in b-metric-like spaces under so-called FG-contractive conditions. This is also demonstrated by a suitable example.