Positive solutions for a class of semipositone periodic boundary value problems via bifurcation theory

In this paper, we are concerned with the existence of positive solutions of nonlinear periodic boundary value problems like − u′′ + q(x)u = λ f (x, u), x ∈ (0, 2π), u(0) = u(2π), u′(0) = u′(2π), where q ∈ C([0, 2π], [0, ∞)) with q 6≡ 0, f ∈ C([0, 2π]×R+, R), λ > 0 is the bifurcation parameter. By using bifurcation theory, we deal with both asymptotically linear, superlinear as well as sublinear problems and show that there exists a global branch of solutions emanating from infinity. Furthermore, we proved that for λ near the bifurcation value, solutions of large norm are indeed positive.