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Keywords:
differential equation; non-monotone argument; oscillatory solution; nonoscillatory solution; Grönwall inequality
differential equation; non-monotone argument; oscillatory solution; nonoscillatory solution; Grönwall inequality
Summary:
Consider the first-order linear delay (advanced) differential equation$$ x'(t)+p(t)x( \tau (t)) =0\quad (x'(t)-q(t)x(\sigma (t)) =0),\quad t\geq t_{0}, $$ where $p$ $(q)$ is a continuous function of nonnegative real numbers and the argument $\tau (t)$ $(\sigma (t))$ is not necessarily monotone. Based on an iterative technique, a new oscillation criterion is established when the well-known conditions$$ \limsup \limits _{t\rightarrow \infty }\int _{\tau (t)}^{t}p(s) {\rm d}s>1\quad \biggl (\limsup \limits _{t\rightarrow \infty }\int _{t}^{\sigma (t)}q(s) {\rm d}s>1\bigg ) $$ and $$ \liminf _{t\rightarrow \infty }\int _{\tau (t)}^{t}p(s) {\rm d}s>\frac {1}{\rm e}\quad \biggl (\liminf _{t\rightarrow \infty }\int _{t}^{\sigma (t)}q(s) {\rm d}s>\frac {1}{\rm e}\bigg ) $$ are not satisfied. An example, numerically solved in MATLAB, is also given to illustrate the applicability and strength of the obtained condition over known ones.
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