Existence of Solutions to a Second Order Coupled System with Nonlinear Coupled Boundary Conditions

We study existence of solution in the presence of upper and lower solutions of some second-order nonlinear coupled ordinary differential system (ODS for short) depending on first order derivatives with nonlinear coupled boundary conditions (CBCs for short). Our method for nonlinear coupled system with nonlinear CBCs is new and it unifies the treatment of many different second order problems. Nagumo condition is used to define bound for the derivative of the solution. Coupled lower and upper solutions, Arzela-Ascoli theorem and Schauder's fixed point theorem play an important role in establishing the arguments.