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Sufficient conditions are established for the global stability of solutions of certain third-order nonlinear differential equations. Our result improves on Tunc’s [10].

This paper extends some known results on the boundedness of solutions and the existence of periodic solutions of certain vector equations to matrix equations.

In this paper a number of known results on the stability and boundedness of solutions of some scalar third-order nonlinear delay differential equations are extended to some vector third-order nonlinear delay differential equations.

The paper studies the equation $$\stackrel{⃛}{X}+\Psi \left(\dot{X}\right)\ddot{X}+\Phi \left(X\right)\dot{X}+cX=P\left(t\right)$$ in two cases: (i) $P\left(t\right)\equiv 0$, (ii) $P\left(t\right)\ne 0$. In case (i), the global asymptotic stability of the solution $X=0$ is studied; in case (ii), the boundedness of all solutions is proved.

In this paper, we shall give sufficient conditions for the ultimate boundedness of solutions for some system of third order non-linear ordinary differential equations of the form $$\stackrel{\text{t}o8pt...}{Xwidth0ptheight5.46pt}+F\left(\ddot{X}\right)+G\left(\dot{X}\right)+H\left(X\right)=P(t,X,\dot{X},\ddot{X})$$ where $X,F\left(\ddot{X}\right)$, $G\left(\dot{X}\right)$, $H\left(X\right)$, $P(t,X,\dot{X},\ddot{X})$ are real $n$-vectors with $F,G$, $H:{ℝ}^n\to {ℝ}^n$ and $P:ℝ\times {ℝ}^n\times {ℝ}^n\times {ℝ}^n\to {ℝ}^n$ continuous in their respective arguments. We do not necessarily require that $F\left(\ddot{X}\right),G\left(\dot{X}\right)$ and $H\left(X\right)$ are differentiable. Using the basic tools of a complete Lyapunov Function, earlier results are generalized.

Sufficient conditions are established for ultimate boundedness of solutions of certain nonlinear vector differential equations of third-order. Our result improves on Tunc’s [C. Tunc, On the stability and boundedness of solutions of nonlinear vector differential equations of third order].

We prove the ultimate boundedness of solutions of some third order nonlinear ordinary differential equations using the Lyapunov method. The results obtained generalize earlier results of Ezeilo, Tejumola, Reissig, Tunç and others. The Lyapunov function used does not involve the use of signum functions as used by others.

We extend, in this paper, some known results on the boundedness of solutions of certain second order nonlinear scalar differential equations to system of second order nonlinear differential equations.

We consider certain class of second order nonlinear nonautonomous delay differential equations of the form $$a\left(t\right)x^{\text{'}\text{'}}+b\left(t\right)g(x,x^{\text{'}})+c\left(t\right)h\left(x(t-r)\right)m\left(x^{\text{'}}\right)=p(t,x,x^{\text{'}})$$ and $${\left(a\left(t\right)x^{\text{'}}\right)}^{\text{'}}+b\left(t\right)g(x,x^{\text{'}})+c\left(t\right)h\left(x(t-r)\right)m\left(x^{\text{'}}\right)=p(t,x,x^{\text{'}}),$$ where $a$, $b$, $c$, $g$, $h$, $m$ and $p$ are real valued functions which depend at most on the arguments displayed explicitly and $r$ is a positive constant. Different forms of the integral inequality method were used to investigate the boundedness of all solutions and their derivatives. Here, we do not require construction of the Lyapunov-Krasovski functional to establish our results. This work...