• Pitambar Das

Articles written in Proceedings – Mathematical Sciences

• Oscillation of odd order delay differential equations

In this paper, we obtain the sufficient and necessary conditions for all solutions of the odd-order nonlinear delay differential equation.x(n)+Q(t)f(x(g(t)))=0 to be oscillatory. In particular, ifn=1, Q(t)&gt;0, f(x)=xα, where α∈(0,1) and is a ratio of odd integers andg(t)=t−ϑ for some ϑ&gt;0, then every solution of (*) oscillates if and only if ∫Q(s)ds=∞.

• Oscillation in odd-order neutral delay differential equations

Consider the odd-order functional differential equation$$\left( {x\left( t \right) - ax\left( {t - \tau } \right)} \right)^n + p\left( t \right)f\left( {x\left( {t - \sigma } \right)} \right) = 0$$ where 0≤α&lt;1, τ, σ∈(0, ∞),pC([0, ∞), (0, ∞)),fC1(R,R) such thatf is increasing,xf(x)&gt;0 forx≠0 andf satisfies a generalized linear condition$$\mathop {\lim \inf }\limits_{x \to 0} \left| {\frac{{df}}{{dx}}} \right| = 1$$ Here we prove that every solution of (*) oscillates if$$\mathop {\lim \inf }\limits_{x \to 0} \int_{t - \sigma /n}^t {\sigma ^{n - 1} p\left( s \right)ds &gt; \frac{1}{e}\left( {1 - a} \right)\left( {n - 1} \right)!\left( {\frac{n}{{n - 1}}} \right)^{n - 1} }$$ This result generalizes a recent result of Gopalsamyet al. [6].

• Proceedings – Mathematical Sciences

Volume 131, 2021
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Posted on July 25, 2019