In this paper, we introduce a modified implicit relation and obtain some new fixed point results for σ-implicit type contractive conditions in relational metric-like spaces. We present some nontrivial examples to illustrative facts and compare our results with the related work. We also discuss sufficient conditions for the existence of a unique positive definite solution of the non-linear matrix equation U = D +∑m i=1 A∗iG(U)Ai, where D is an n × n Hermitian positive definite matrix, A1, A2, …, Am are n × n matrices, and G is a non-linear self-mapping of the set of all Hermitian matrices which is continuous in the trace norm. Finally, we discuss a couple of examples, convergence and error analysis, average CPU time analysis and visualization of solution in surface plot. © 2021 the Author(s), licensee AIMS Press.