Even order nonlinear eigenvalue problems on a measure chain Original Research Article

We consider the even order nonlinear eigenvalue problem (−1)muΔ2m(t)=λf(t,u(σ(t))), uΔ2i(0)=uΔ2i(σ(1))=0,0⩽i⩽m−1, on a measure chain View the MathML source. Results on existence and nonexistence of positive solutions are obtained for λ evaluated in different intervals. Under certain assumptions, the complete scenario for all λ>0 is established. Our work develops and improves many known results in the literature even for the case that View the MathML source is the real number line. We also interpret our general results on measure chains to the discrete case which yields a new set of conditions for the existence and nonexistence of positive solutions of eigenvalue problems for difference equations.