global bifurcation of positive solutions of a second-order periodic boundary value problem with indefinite weight

in this paper, we are concerned with the global structure and stability of positive solutions of periodic boundary value problem

−u′′(t) + q(t)u(t) = λa(t)f (u(t)), 0 < t < 2π,

u(0) = u(2π), u′(0) = u′(2π),

where q ∈ c(r, [0,∞)) is of periodic 2π and q(t) ̸≡ 0, t ∈ [0, 2π]; a ∈ c(r, r) is of periodic 2π and changes sign. the proof of our main results are based on bifurcation techniques.