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Even order nonlinear eigenvalue problems on a measure chain. (English) Zbl 1024.34021

Summary: The authors consider the even-order nonlinear eigenvalue problem

(-1) m u Δ 2m (t)=λf(t,u(σ(t))),
u Δ 2i (0)=u Δ 2i (σ(1))=0,0im-1,

on a measure chain 𝕋. 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. This work develops and improves many known results in the literature even for the case that 𝕋 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.

34B45Boundary value problems for ODE on graphs and networks
39A99Difference equations
34L05General spectral theory for OD operators