## Stability and boundedness of solutions of some third-order nonlinear vector delay differential equation.(English)Zbl 1370.34129

In this article, the authors consider the following nonlinear differential system of third order with variable delay $$r(t)$$: $\left(G(X)X^\prime\right)'' + A{X}'' + B{X}' + H\left(X(t - r(t))\right) = P(t),\tag{1}$ where $$X \in \mathbb R^n$$, $$t \in [0,\infty )$$, $$\mathbb R^ + = [0,\infty )$$, $$r(t)$$ is a continuous differentiable function such that $$0 \leq r(t) \leq \gamma ,$$ $$\gamma$$ is a positive constant, and $${r}'(t) \leq \xi ,\text{ }(0 < \xi < 1), \quad A$$ and $$B$$ are $$n\times n -$$ constant symmetric matrices, $$G\text{, }H\text{ }:\mathbb R^n \to \mathbb R^n$$ and $$P:\mathbb R^ + \to \mathbb R^n$$ are continuous functions with $$H$$ is differentiable and $$G$$ is two times differentiable, and $$H(0) = 0.$$ They establish two new results on the stability of the zero solution and uniformly boundedness and uniformly ultimately boundedness of all solutions of equation (1) when $$P(t) \equiv 0$$ and $$P(t) \neq 0,$$respectively. The technique of the proof involves the Lyapunov-Krasovskii functional approach. An example is given for illustration of the main results.
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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