Dynamic behavior of vector solutions of a class of 2-D neutral differential systems
Arun Kumar Tripathy, Shibanee Sahu
Received October 19, 2023. Published online July 18, 2024.
Abstract: This work deals with the analysis pertaining some dynamic behavior of vector solutions of first order two-dimensional neutral delay differential systems of the form $\frac{{\rm d}}{{\rm d}t} \begin{bmatrix} u(t)+pu(t-\tau)\\
v(t)+pv(t-\tau)\end{bmatrix} = \begin{bmatrix} a & b \\
c & d
\end{bmatrix} \begin{bmatrix} u(t-\alpha)\\
v(t-\beta)
\end{bmatrix}.$
The effort has been made to study $\frac{{\rm d}}{{\rm d}t} \begin{bmatrix} x(t)-p(t)h_1(x(t-\tau))\\
y(t)-p(t)h_2(y(t-\tau)) \end{bmatrix} + \begin{bmatrix} a(t) & b(t)\\
c(t) & d(t) \end{bmatrix} \begin{bmatrix} f_1(x(t-\alpha))\\
f_2(y(t-\beta)) \end{bmatrix} =0,$
where $p,a,b,c,d,h_1,h_2,f_1,f_2 \in C(\mathbb{R},\mathbb{R})$; $\alpha,\beta,\tau\in\mathbb{R}^+$. We verify our results with the examples.
Keywords: oscillation; nonoscillation; nonlinear system of neutral differential equations; asymptotically stable; Banach's fixed point theorem
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Affiliations: Arun Kumar Tripathy (corresponding author), Shibanee Sahu, Department of Mathematics, Sambalpur University, Jyoti Vihar, Burla, Sambalpur, Odisha 768019, India, e-mail: arun_tripathy70@rediffmail.com, shibaneesahu100@gmail.com
