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The authors use the upper and lower solution method to study the existence of solutions of nonlinear mixed two-point boundary value problems for third-order nonlinear differential equation  y′′′=f(x,y,y′,y′′),  y′(b)=h(y′(a)),  p(y(a),y(b),y′(a),y′(b))=0,  g(y(a),y(b),y′(a),y′(b),y′′(a),y′′(b))=0. Some new existence results are obtained by developing the upper and lower solution method. Some applications are also presented.