## Eigenvalue comparisons for differential equations on a measure chain

### Summary

Summary: The theory of $u_{0}$-positive operators with respect to a cone in a Banach space is applied to eigenvalue problems associated with the second order $\Delta$-differential equation (often referred to as a differential equation on a measure chain) given by $y^{\Delta\Delta}(t)+\lambda p(t)y(\sigma(t))=0, t\in[0,1]$, satisfying the boundary conditions $y(0)=0=y(\sigma^2(1))$. The existence of a smallest positive eigenvalue is proven and then a theorem is established comparing the smallest positive eigenvalues for two problems of this type.

34B99, 39A99

measure chain