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This article investigates nonlinear, second-order ordinary differential equations subject to various two-point boundary conditions. A condition is introduced that ensures a priori bounds on the derivatives of solutions to the problem. In particular, quadratic growth conditions on the right-hand side of the differential equation are not employed. The ideas are then applied to ensure the existence of at least one solution. The main tools involve differential inequalities and fixed-point methods. [ABSTRACT FROM AUTHOR]