In this paper, we establish a relation-theoretic version of the results presented by Kim et al. (Journal of Nonlinear and Convex Analysis, 16 (9) (2015), 1779-1786). To showcase the versatility of our results, we furnish some illustrative examples. Furthermore, we exhibit an application of our results to establish sufficient conditions for the existence of a positive definite common solution to a pair of nonlinear matrix equations.