Global structure of positive solutions for a nonlinear second order periodic boundary value problem

Abstract:

In this paper,~we study the global structure of positive solution for second-order periodic boundary value problem $$ \left\{\begin{array}{ll} u''-k^{2}u+\lambda a(t)f(u)=0,~~t\in[0,2\pi],\\[2ex] u(0)=u(2\pi),~u'(0)=u'(2\pi). \end{array} \right. $$ where~$k>0$~is a constant,~$\lambda$~is positive parameter,~$a\in C([0,2\pi],[0,\infty))$~and~$a(t)\not\equiv 0$~on any subinterval of~$[0,2\pi]$,~$f\in C([0,\infty),~[0,\infty))$.~The proof of the main results is based on the Rabinowitz global bifurcation theorems and a approach by approximation.