构造一类八阶周期边值问题极值解的单调性方法

利用单调性技巧研究周期边值问题: ^u(8)(t)=f(t,u(t),u⁽⁴⁾(t)),u⁽ⁱ⁾(0)=u⁽ⁱ⁾(2π), i=0,1,…,7,〖WTBX〗其中f(t,u,v)为Caratheodory函数. 证明如果上述周期边值问题有上解和下解 , 分别表为β(t)和α(t), 并且有β(t)≤α(t), 则可构造2个单调序列｛β_j ｝和｛ αj｝, β_j≤α_{j, 使之于［0,2π］上分别 单调一致收敛于上述问题的极值解. 从而证明了上述周期边值问题解的存在性.}

A Monotone Method for Constructing Extremal Solutions to an Eighth Order Periodic Boundary Value Problems

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.