We study the existence and uniqueness of solutions to a particular class of two-point boundary value problems involving fourth-order ordinary differential equations. Such problems have exciting applications for modeling the deflections of beams. The primary tools employed in this study include the application of Banach's and Rus's fixed point theorems. Our theoretical results are applied to elastic beam deflections when the beam is subjected to a loading force and both ends are clamped. The existence and uniqueness of solutions to the models are guaranteed for certain classes of linear and nonlinear loading forces.