Eigenvalue comparisons for differential equations on a measure chain.
Chuan Jen Chyan, John M. Davis, Johnny Henderson, William Yin · 1998
The theory of $mathbf{u_0}$-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 $$ y^{DeltaDelta}(t)+lambda p(t)y(sigma(t))=0, qquad tin[0,1], $$ 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.