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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 $$(-1)^m u^{\Delta^{2m}}(t)= \lambda f(t, u(\sigma(t))),$$ $$u^{\Delta^{2i}}(0)= u^{\Delta^{2i}}(\sigma(1))= 0,\qquad 0\le i\le m-1,$$ on a measure chain $\bbfT$. Results on existence and nonexistence of positive solutions are obtained for $\lambda$ evaluated in different intervals. Under certain assumptions, the complete scenario for all $\lambda> 0$ is established. This work develops and improves many known results in the literature even for the case that $\bbfT$ 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:
 34B45 Boundary value problems for ODE on graphs and networks 39A99 Difference equations 34L05 General spectral theory for OD operators
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