Existence of nonzero nonnegative solutions of Sturm-Liouville boundary value problems and applications

Abstract

Sufficient conditions for the boundary value problems (BVPs) of linear Sturm-Liouville (S-L) homogeneous equations subject to the separated boundary conditions (BCs) to have only zero solution are provided in this paper for the first time. Some previous papers and classical books used the assertion that the BVPs have only zero solution as a hypothesis and did not provide any sufficient conditions to ensure that the assertion holds. The sufficient conditions obtained in this paper are a key toward obtaining both the Green's functions to such BVPs and uniqueness of solutions for the linear S-L nonhomogeneous BVPs including the one-dimensional elliptic BVPs. New results on the existence of nonzero nonnegative or strictly positive solutions for the BVPs of nonlinear S-L equations with the separated BCs are obtained by using the fixed point index theory for nowhere normal-outward maps in Banach spaces. The new results allow the nonlinearities in the S-L BVPs to take negative values and have no lower bounds and are applied to deal with the logistic type population models which contain such nonlinearities.