Stability and boundedness of solutions of a certain system of third-order nonlinear delay differential equations (Q2817422)

The author considers the following nonlinear differential system of the third order with variable delay \(r(t)\) NEWLINE\[NEWLINE{X}''' + A{X}'' + B{X}' + H(X(t - r(t))) = P(t), \tag{1}NEWLINE\]NEWLINE where \(X \in \mathbb R ^n\), \(t \in [0,\infty )\), \(\mathbb R^ + = [0,\infty )\), \(r(t)\) is continuous and bounded differentiable function, \(A\) and \(B\) are \(n\times n - \) constant symmetric matrices, \(\text{ }H\text{ }:\mathbb R^n \to \mathbb R^n\) is continuous differentiable functions with \(H(0) = 0\) such that the Jacobian matrix \(J_H (X)\) exist and is symmetric and continuous, that is, NEWLINE\[NEWLINE J_H (X) = \left( {\frac{\partial h_i }{\partial x_j }} \right), \quad (i,\text{ }j = 1,\text{ }2,\dots,n), NEWLINE\]NEWLINE exists and is symmetric and continuous, where \((x_1 ,x_2 ,\dots,x_n )\) and \((h_i )\) are components of \(X\) and \(H,\) respectively, \(P:\mathbb R^+ \to \mathbb R^n\) is a continuous function and the primes in equation (1) indicate differentiation with respect to \(t, \quad t \geq t_0 \geq 0.\) The author obtains sufficient conditions for the asymptotical stability of the zero solution of equation (1) when \(P(t) \equiv 0\) and boundedness of all solutions of equation (1) when \(P(t) \neq 0\), respectively. Two theorems are proved on the stability and boundedness of solutions. An example is given for the illustrations.