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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.

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