<p>This paper is concerned with second-order causal differential equations with Riemann–Stieltjes integral boundary conditions. To obtain sufficient conditions for the existence of extremal solutions for such problems, a novel comparison principle is first formulated. Next, the existence and uniqueness of the solutions for the second-order linear differential equations are proved based on the fixed point theorem. Then, the monotone iterative technique is employed to investigate the existence of the extremal solution, with the relative results being derived on the basis of upper and lower solutions. Finally, an example is provided to verify the theoretical results.</p>