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Eigenvalue intervals for even-order Sturm-Liouville dynamic equations. (English) Zbl 1084.34025
Summary: We study the existence of eigenvalue intervals for the even-order dynamic equation on time scales $$(-1)^n x^{(\Delta\nabla)^n}(t)=\lambda c(t) f(x(t)),\quad t\in [a,b],$$ satisfying the boundary conditions $$\alpha_{i+1} x^{(\Delta\nabla)^i}(a)- \beta_{i+1} x^{(\Delta\nabla)^i\Delta}(a)= 0,\ \gamma_{i+1} x^{(\Delta\nabla)^i}(b)+ \delta_{i+1} x^{(\Delta\nabla)^i\Delta}(b)= 0$$ for $0\le i\le n- 1$, where $f$ is a positive function and $c$ is a nonnegative function that is allowed to vanish on some subintervals of $[a, b]$ of the time scale. The methods involve applications of Krasnoselskii’s fixed-point theorem for operators on a cone in a Banach space.

34B24Sturm-Liouville theory
39A10Additive difference equations