Nonoscillation of first-order dynamic equations with several delays
The authors study the delay dynamic equation with positive variable coefficients and multiple delays, of the form: \[ x^\Delta(t) + \sum_{i \in [1,n]_{\mathbb{T}}} A_i(t) x(\alpha_i(t)) = 0 , \quad\text{for }t \in [t_0, \infty)_{\mathbb{T}}, \] where \(\mathbb{T}\) is a time scale, and establish that nonoscillation is equivalent to the existence of a positive solution for the generalized characteristic inequality and to the positivity of the fundamental solution. A few comparison theorems between the solutions of two different equations and two different solutions of the same equation are proved. Several examples of non standard time scales are also provided to illustrate the nonoscillatory criterion for dynamic equations that are different from delay differential and classical difference equations. It is also shown that, if the above dynamic equation is nonoscillatory, then it has no slowly oscillating solutions. A few important open problems are suggested.
scientific article; zbMATH DE number 5791591