Nonlinear Boundary Value Problems and a Priori Bounds on Solutions

Results concerning the existence of solutions of multipoint boundary value problems are given. The results are based on a topological transversality method and rely on a priori bounds on solutions. Applications are made to conjugate type and focal type boundary value problems; the third order, 3-point boundary value problem $y''' = f(x,y,y',y'')$, $y(x_1 ) = r_1 $, $y(x_2 ) = r_2 $, $y(x_3 ) = r_3 $, where $x_1 < x_2 < x_3 $, is discussed.