Existence of Solutions for a Class of Third-Order Periodic Boundary Value Problems at Resonance

Abstract:

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.

Key words:Lyapunov-Schmidt procedure;~Connected set;~Disordered lower and upper solutions;~Resonance;~ Existence