非线性三阶两点边值问题变号解的逐次逼近方法

Successively Approximate Method of Sign-changing Solution to a Nonlinear Third-order Two-point Boundary Value Problem

本文研究了一个非线性三阶两点边值问题变号解的存在性与逐次逼近,其中非线性项关于空间变元单调增并且关于时间变元奇异.利用Green函数,将该问题转化为一个等价积分方程,其中相伴积分算子是全连续并且增的.在适当的条件下借助于全连续增算子构造了两个逐次迭代序列.这些序列从常值函数开始并且一致收敛于此问题的变号解.结论说明这种变号解的存在性仅仅依赖于非线性项在某个有界集合上的增长,而与非线性项在这个集合以外的状态无关.最后,数值算例证实新的逼近方法对于数值计算是有效的.

In this paper, the existence and the successive approximation of sign-changing so-lutions are studied for a nonlinear third-order two-point boundary value problem, in which the nonlinear term is monotone increasing in the space variable and is singular in the time variable. By employing the Green function, the problem is transformed into an integral equation in which the associated integral operator is completely continuous and increasing. Under some suitable conditions, two successively iterative sequences are constructed by applying the completely continuous increasing operator. The sequences start with the constant functions and uniformly converge to the sign-changing solutions of the problem. The result indicates that the existence of sign-changing solutions only depends on the growth of the nonlinear term on a bounded set and is independent of the states of nonlinearity outside the set. Finally, the numerical example demonstrates that the new approximate method is effective for the numerical simulation.