Comparison theorems for ordinary differential equations with general boundary conditions (Q1107691)

The authors prove a comparison theorem for the nonlinear two point boundary value problem \(dy/dt=A(t,y)\), \(t\in (a,b)\), \(f(\xi,y(a),y(b))=0\). They assume that A is continuous in t and satisfies a Lipschitz condition with respect to y. The proof makes use of a result due to the first author, which gives estimates for solutions of the initial value problem \(dy/dt=A(t,y)\), \(y(a)=\xi\) in terms of solutions of associated differential inequalities. The first result gives an estimate for the solutions to the problem with boundary conditions \(y(a)=g(\xi,v)\), \(v=y(b)=y(a)\), in terms of the solutions to associated differential inequalities. A second version of this result treats the boundary condition \(y(a)=g(\xi,y)\). Finally a generalization is given to the functional equation \(y=g(\xi,y)\). This result is obtained by a fixed point argument, using the previous results.