二阶非线性微分方程周期解和孤立波的存在性

Existence of periodic solutions and solitary waves for second order nonlinear differential equations

考虑带有非局部项的二阶非线性微分方程-u″(x)+a(x)u(x)=f(x,u)∫+∞-∞W(x-s)|u(s)|2ds周期解和孤立波的存在性。将其转化为Banach空间上一个合适的算子的不动点问题,利用Krasnoselskii不动点定理以及所对应的格林函数的正性,分别获得上述二阶非线性微分方程至少存在一个周期解和一个孤立波的充分条件。

Abstract： Consider the existence of periodic solutions and solitary waves for second order nonlinear dif- ferential equations with nonlocal term：-u″（x）＋a（x）u（x）=f（x,u）∫＋∞-∞W（x-s）｜u（s）｜2dsFirst of all, the problem is reduced to a fixed point problem for a suitable operator on a Banach space. Then by using the Krasnoselskii fixed point theorem and the positivity of the Green＇s function, it is proven that there exists at least one periodic solution and at least one solitary wave.