一类三阶周期边值共振问题解的存在性
Existence of Solutions for a Class of Third-Order Periodic Boundary Value Problems at Resonance

本文运用了~Lyapunov-Schmidt~过程和紧向量场方程的解集连通理论为三阶周期边值共振问题 $$ \left\{\begin{array}{ll} v'''(t)=f(t,v(t)),~~\ \ \ t\in [0,T],\\[2ex] v^{(i)}(0)-v^{(i)}(T)=0 ,\ \ \ i=0,1,2 \end{array} \right.\eqno $$ 发展上下解方法,~并且得到其解的存在性结果,~其中函数~$f: [0,T]\times \mathbb{R}\rightarrow \mathbb{{R}}$~连续且有界.
By using the Lyapunov-Schmidt procedure and the connectivity theory of the solution set of compact vector fields,~we develope the method of upper and lower solutions and obtain the existence of solutions for a third-order periodic boundary value problem at resonance~ $$ \left\{\begin{array}{ll} v'''(t)=f(t,v(t)),~~\ \ \ t\in [0,T],\\[2ex] v^{(i)}(0)-v^{(i)}(T)=0 ,\ \ \ i=0,1,2, \end{array} \right.\eqno $$ where~$f: [0,T]\times \mathbb{R}\rightarrow \mathbb{R}$~is continuous and bounded