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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

${\left(-1\right)}^{m}{u}^{{{\Delta }}^{2m}}\left(t\right)=\lambda f\left(t,u\left(\sigma \left(t\right)\right)\right),$
${u}^{{{\Delta }}^{2i}}\left(0\right)={u}^{{{\Delta }}^{2i}}\left(\sigma \left(1\right)\right)=0,\phantom{\rule{2.em}{0ex}}0\le i\le m-1,$

on a measure chain $𝕋$. 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 $𝕋$ 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:
 34B45 Boundary value problems for ODE on graphs and networks 39A99 Difference equations 34L05 General spectral theory for OD operators