Oscillatory behavior of Nabla dynamic equations on time scales

Abstract

Consider the first-order nabla dynamic equations

$$x^{\mathrm{\nabla }}(t)-\sum _{i=1}^m{p}_i(t)x({\tau }_i(t))=0,t\in [t_0,\infty {)}_{\mathbb{𝕋}},$$

where $p_i(t)$ $(i=1,2,\dots ,m)\in C_{ld}([t_0,\infty {)}_{\mathbb{𝕋}},{ℝ}^{+})$, ${\tau }_i(t)(i=1,2,\dots ,m)\in C_{ld}({[t_0,\infty )}_{𝕋},𝕋)$ and ${\tau }_i(t)\ge t$. Under the assumption that the ${\tau }_i(t)$ are not necessarily monotone $(i=1,2,\dots ,m)$, we present new sufficient conditions for the oscillation of first-order nabla dynamic equations on time scales. We can say that this paper is the first in the literature in obtaining the oscillations of the solutions of the above equation. An example illustrating the results is also given.