Singular and Nonsingular Third-Order Periodic Boundary Value Problems with Indefinite Weight

We are concerned with the existence of positive periodic solutions of third-order periodic boundary value problems where k1, k3 ∈ (0,∞) and k2 ∈ (0, (π ω )2) are constants. λ is a positive parameter. The weight function a ∈ C([0, ω],R) may change sign. f ∈ C([0,∞),R) with f (0) := limu→0+ f (u) 0. We show that there exists a constant λ ∗ 0, such that the above problem has at least one positive periodic solution for λ ∈ (0, λ ∗ ) where f (0) is bounded. This result is based upon bifurcation theory and Leray–Schauder fixed point theorem. On the other hand, by using Krasnoselskii’s fixed point theorem in a cone, we show that there exists a positive constant λ0 such that for all λ ∈ (0, λ0), the above problem has at least one positive periodic solutionwhere f (0) is unbounded, namely that f has a singularity at u = 0. And this result is applicable to weak as well as strong singularities.