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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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