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 u ''' (t) + k(1)u '' (t) + k(2)u ' (t) + k(3)u(t) = lambda a(t) f (u), t is an element of (0, omega), u(0) = u(omega), u ' = (0) = u ' (omega), u '' (0) = u '' (omega), where k(1), k(3) is an element of (0, infinity) and k(2) is an element of (0, (pi/omega)(2)) are constants. lambda is a positive parameter. The weight function a is an element of C([0, omega], R) may change sign. f is an element of C([0, infinity), R) with f (0) := lim(u -> 0+) f (u) > 0. We show that there exists a constant lambda* > 0, such that the above problem has at least one positive periodic solution for lambda is an element of (0, lambda*) 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 lambda(0) such that for all lambda is an element of (0, lambda(0)), 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.