Positive solutions for fractional-order boundary value problems with or without dependence of integer-order ones

This paper investigates the existence, non-existence, uniqueness, and multiplicity of positive solutions for a boundary value problem involving the Riemann-Liouville derivative of order $$\alpha \in (1,2]$$ . First, we establish existence and non-existence results for both sublinear and superlinear cases by characterizing the first eigenvalue $$\lambda _1(\alpha )$$ . Second, we prove the uniqueness of positive solutions; especially, for the superlinear case, we employ an approach by leveraging the non-degeneracy of solutions near $$\alpha =2$$ . Finally, we demonstrate the existence of at least three positive solutions for Hénon-type problems.