Existence and uniqueness of solutions of boundary value problems for two dimensional systems of nonlinear differential equations

Abstract

The paper considers the nonlinear system x ′ = f ( t , x , y ) , y ′ = g ( t , x , y ) x’ = f(t,x,y),y’ = g(t,x,y) with linear and nonlinear two point boundary conditions. With a Lipschitz condition, an interval of uniqueness for linear boundary conditions is determined using a comparison theorem. A corresponding existence theorem is established. Under the assumption of uniqueness, a general existence theorem is established for quite general nonlinearities in the functions and in the boundary conditions. Examples are provided. The results extend previous work on second order scalar differential equations.