Address:
Corresponding author: R.N. Rath, Flat-A 203, Center Point Apartment, Sailasree Vihar Bhubaneswar 751021 Former Professor of Mathematics, VSSUT, BURLA, Sambalpur, 768018, Orissa, India, and Former Principal Khallikote Autonomous College, Berhampur, Odisha, India, 760001
Department of Mathematics, Trident Academy of Technology, Bhubaneswar, Odisha, India,
Deputy Registrar, BPUT, Rourkella, Odisha, India
E-mail:
radhanathmath@yahoo.co.in
kalicharan.in@gmail.com
rath.subhendu@gmail.com
Abstract: In this article, we obtain sufficient conditions so that every solution of neutral delay differential equation \[ \big (y(t)- \sum _{i=1}^k p_i(t) y(r_i(t))\big )^{(n)}+ v(t)G( y(g(t)))-u(t)H(y(h(t))) = f(t) \] oscillates or tends to zero as $t\rightarrow \infty $, where, $n \ge 1$ is any positive integer, $p_i$, $r_i\in C^{(n)}([0,\infty ),\mathbb{R})$ and $p_i$ are bounded for each $i=1,2,\dots ,k$. Further, $f\in C([0, \infty ), \mathbb{R})$, $g$, $h$, $v$, $u \in C([0, \infty ), [0, \infty ))$, $G$ and $H \in C(\mathbb{R},\mathbb{R})$. The functional delays $r_i(t)\le t$, $g(t)\le t$ and $h(t)\le t$ and all of them approach $\infty $ as $t\rightarrow \infty $. The results hold when $u\equiv 0$ and $f(t)\equiv 0$. This article extends, generalizes and improves some recent results, and further answers some unanswered questions from the literature.
AMSclassification: primary 34C10; secondary 34C15, 34K40.
Keywords: oscillation, non-oscillation, neutral equation, asymptotic behaviour.
DOI: 10.5817/AM2022-2-65