Global bifurcation of a class of periodic boundary-value problems

We prove that λ=0λ=0 is a global bifurcation point of the second-order periodic boundary-value problem (p(t)x′(t))′−λx(t)−λ2x′(t)−f(t,x(t),x′(t),x″(t));x(0)=x(1),x′(0)=x′(1)(p(t)x′(t))′−λx(t)−λ2x′(t)−f(t,x(t),x′(t),x″(t));x(0)=x(1),x′(0)=x′(1). We study this equation under hypotheses for which it may be solved explicitly for x″(t)x″(t). However, it is shown that the explicitly solved equation does not satisfy the usual conditions that are sufficient to conclude global bifurcation. Thus, we need to study the implicit equation with regard to global bifurcation.