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Keywords:
oscillation; nonoscillation; nonlinear system of neutral differential equations; asymptotically stable; Banach's fixed point theorem
oscillation; nonoscillation; nonlinear system of neutral differential equations; asymptotically stable; Banach's fixed point theorem
Summary:
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.
References:
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