We use the super-subsolution principle, maximal principle and approximation theory to know the existence, monotonicity with respect toλ, the boundary behaviour and the regularity of solutions.

Some sets of suitable conditions are given by using the explosive super-subsolution method,the inner estimation theory and the perturbed technique,which guarantee,respectively,the existence of explosive positive solutions.

The upper and lower solution method of nonlocal problem for the first order ordinary differential equation;

Based on upper and lower solutions,the existence and uniqueness theorem are established.

Based on the upper and lower solution, the uniqueness theorem is established.

The method of upper and lower solutions for higher order P-Laplace equation boundary value problems;

The upper and lower solutions of m-point boundary value problems at resonance and topological degree;

Squeeze Rule and the Upper and Lower Solutions Method for Differential Equation;

By constructing proper upper-lower solutions,the existence of travelling wave solutions was proved.

In this paper we give the periodic upper-lower solution method for proving the existence of periodic solution of ordinary differential system, using this method we also get nonzero periodic solutions of mutualism models and competitive models in ecology.

By putting in impulsion at suitable time,the instruction of upper-lower solutions in every interval and the control of impulsive source and reactive function,the extinct time of solution can be controlled into a desired time interval.

By using the method of upper and lower solutions,we obtain the existence of solutions for boundary value problem of second order integro differential equations of Volterra type in a normal cone.