Liénard系统无穷边值问题解的存在性

Existence of Bounded Solutions on the Real line for Lienard System

讨论Lienard系统无穷边值问题单调解和非单调解的存在性。利用平面动力系统理论,通过对称变换或拟对称变换比较系统所定义的向量场并构造系统的不变区域,以此证明系统连结轨道的存在性,获得边值问题解存在的一系列充分条件。特别地,当源函数为双稳函数时,系统存在无穷多单调解。

The existence of monotone and non_monotone solutions of boundary value problem on the real line for Liénard equation is studied. Applying the theory of planar dynamical systems and the comparison method of vector fields defined by Liénard system and the system given by symmetric transformation or quasi_symmetric transformation, the invariant regions of the system are constructed. The existence of connecting orbits can be proved. A lot of sufficient conditions to guarantee the existence of solutions of the boundary value problem are obtained. Especially, when the source function is bi_stable, the existence of infinitely many monotone solusion is obtained.