Stability and boundedness of solutions of a certain system of third-order nonlinear delay differential equations. (English) Zbl 1351.34086

The author considers the following nonlinear differential system of the third order with variable delay \(r(t)\) \[ {X}''' + A{X}'' + B{X}' + H(X(t - r(t))) = P(t), \tag{1} \] 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, \[ J_H (X) = \left( {\frac{\partial h_i }{\partial x_j }} \right), \quad (i,\text{ }j = 1,\text{ }2,\dots,n), \] 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.


34K20 Stability theory of functional-differential equations
34K12 Growth, boundedness, comparison of solutions to functional-differential equations
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