Summary: The authors consider the even-order nonlinear eigenvalue problem
on a measure chain . Results on existence and nonexistence of positive solutions are obtained for evaluated in different intervals. Under certain assumptions, the complete scenario for all is established. This work develops and improves many known results in the literature even for the case that is the real number line. The authors also interpret their general results on measure chains to the discrete case which yields a new set of conditions for the existence and nonexistence of positive solutions to eigenvalue problems for difference equations.