## 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.
Reviewer: Cemil Tunç (Van)

### MSC:

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