Uniqueness implies existence of solutions for nonlinear focal-like boundary value problems

Given an appropriate growth condition for f and a uniqueness assumption on y ( n ) = 0 with respect to certain focal boundary value problems, it is shown that uniqueness of solutions to the nonlinear differential equation y ( n ) = f ( t , y , y ′ , … , y ( n − 1 ) ) , subject to boundary conditions of the form g i j ( y ( t j ) , … , y ( n − 1 ) ( t j ) ) = y i j implies existence of solutions.