Hille–Nehari theorems for dynamic equations with a time scale independent critical constant

摘要

In this paper, we give Hille–Nehari test for nonoscillation/oscillation of the dynamic equation(rxΔ)Δ(t)+p(t)x(t)=0fort∈[t0,∞)T,where t0∈T, supT=∞, r∈Crd([t0,∞)T,R+) and p∈Crd([t0,∞)T,R0+). We show that the critical constant for this dynamic equation is 14 as in the well-known cases T=R and T=Z. We also present illustrating examples showing that the critical constant 14 is sharp on arbitrary time scales. With two different techniques, we extend our results to the dynamic equation(rxΔ)Δ(t)+p(t)xσ(t)=0fort∈[t0,∞)Tby preserving the constant 14. The second technique is new even for the discrete case T=Z.