Unique solutions of systems of second-order nonlinear two-point boundary-value problems

Abstract

A boundary-value problem in ordinary differential equations differs from an initial-value problem in that it must satisfy conditions at two or more points. An n th order differential equation can have n conditions at one point or one condition at n points. In fluid mechanics and trajectory mechanics, systems of second-order equations with conditions specified at two points are often encountered. This paper will consider only systems of this form. The question of existence and uniqueness of solutions of the two-point boundary-value problem (a scalar D.E.) y + f ( t , y , y ' ) = O , w i t h y ( a ) = A a n d y ( b ) = B , where f is continuous and satisfies the uniform Lipschitz condition I f ( t , y , y ' ) f ( t, z, z') I ~< K l y z I + L l y ' z' I, (1)