Abstract: The present paper deals with the eighth order periodic boundary value problem of the following form, u⁽⁸⁾(t)=f(t,u(t),u⁽⁴⁾(t)), u⁽ⁱ⁾(0)=u⁽ⁱ⁾(2π), i=0,1,…,7.where f(t,u,v) is a Caratheodory function.It is proved that if there exist upper and lower solutions to the periodic boundary value problem, represented by β(t) and α(t) respectively, and β (t)≤α(t), then the monotone sequences of functions ｛β_j｝ and ｛α_j｝, β_j≤α_j, can be constructed so that the sequences converge uniformly on ［0,2π］ to the extremal solutions of the problem and hence the solutions to the problem is obtained.

Key words: monotone method, periodic boundary valule problem, extr emal solution