When the delay terms $\tau_{i}(t)$ $(i=1,2,\ldots ,m)$ are not necessarily monotone, we present new sufficient conditions for the oscillation of first-order delay dynamic equations on time scales.
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Objective & Hypothesis:In this paper, we consider the first-order dynamic equation as the following:$$x^{\Delta}(t)+\sum\limits_{i=1}^m p_i(t)x(\tau_i(t))=0,\,\,t\in[t_0,\infty)_{\mathbb{T}}$$where $p_{i}\in C_{rd}\left( [t_{0},\infty )_{\mathbb{T}},\mathbb{R}^{+}\right) ,$ $\tau _{i}\in C_{rd}\left( [t_{0},\infty )_{\mathbb{T}},\mathbb{T}\right) $ $(i=1,2,\ldots ,m)$ and $\tau_i(t)\leq t,\,\, \lim_{t\to\infty}\tau_i(t)=\infty$.
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Key Findings & Data:In this paper, we consider the first-order dynamic equation as the following:$$x^{\Delta}(t)+\sum\limits_{i=1}^m p_i(t)x(\tau_i(t))=0,\,\,t\in[t_0,\infty)_{\mathbb{T}}$$where $p_{i}\in C_{rd}\left( [t_{0},\infty )_{\mathbb{T}},\mathbb{R}^{+}\right) ,$ $\tau _{i}\in C_{rd}\left( [t_{0},\infty )_{\mathbb{T}},\mathbb{T}\right) $ $(i=1,2,\ldots ,m)$ and $\tau_i(t)\leq t,\,\, \lim_{t\to\infty}\tau_i(t)=\infty$.
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Conclusion & Impact:When the delay terms $\tau_{i}(t)$ $(i=1,2,\ldots ,m)$ are not necessarily monotone, we present new sufficient conditions for the oscillation of first-order delay dynamic equations on time scales.
In this paper, we consider the first-order dynamic equation as the following:$$x^{\Delta}(t)+\sum\limits_{i=1}^m p_i(t)x(\tau_i(t))=0,\,\,t\in[t_0,\infty)_{\mathbb{T}}$$where $p_{i}\in C_{rd}\left( [t_{0},\infty )_{\mathbb{T}},\mathbb{R}^{+}\right) ,$ $\tau _{i}\in C_{rd}\left( [t_{0},\infty )_{\mathbb{T}},\mathbb{T}\right) $ $(i=1,2,\ldots ,m)$ and $\tau_i(t)\leq t,\,\, \lim_{t\to\infty}\tau_i(t)=\infty$. When the delay terms $\tau_{i}(t)$ $(i=1,2,\ldots ,m)$ are not necessarily monotone, we present new sufficient conditions for the oscillation of first-order delay dynamic equations on time scales.
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