Positive Solutions for Third-Order Boundary Value Problems with Indefinite Weight

In this paper, we initially employ the disconjugacy theory to establish some sufficient conditions for the disconjugacy of y''' + beta y '' + alpha y' = 0. We then utilize Elias's spectrum theory to demonstrate the spectrum structure of the linear operator y''' + beta y '' + alpha y' coupled with the boundary conditions y(0) = y(1) = y' (1) = 0. Ultimately, by utilizing the acquired results, we ascertain the existence of positive solutions for the corresponding nonlinear third-order problem with an indefinite weight, based on the principles of bifurcation theory and the Leray-Schauder fixed point theorem.