We consider the non-linear matrix equation (NME) of the form U = Q + k i=1 A * i ℏ(U)A i , where Q is an n × n Hermitian positive definite matrix, A 1 , A 2 ,. .. , A m are n × n matrices, and ℏ is a non-linear self-mapping of the set of all Hermitian matrices which are continuous in the trace norm. We discuss sufficient conditions ensuring the existence of a unique positive definite solution of the given NME. In order to do this, we introduce Θ w-contractive conditions involving modified simulation functions in relational metric spaces and derive fixed points results based on them, followed by two suitable examples. In order to demonstrate the obtained conditions, we consider three different sets of matrices. Three different types of examples (including randomly generated matrix and a complex matrix) are given, together with convergence and error analysis, as well as average CPU time analysis with different dimensions bar graphs, and visualization of solutions in surface plot.