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.

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

By applying the Leggett-Williams fixed point theorem in a suitably constructed cone, we obtain the existence of at least three unbounded positive solutions for a boundary value problem on the half line. Our result improves and complements some of the work in the literature.

This paper discusses the asymptotic behavior of solutions of the Liénard equation, especially the global behavior of unbounded solutions, and also gives a class of sufficient and necessary conditions for the orbit of a solution to intersect the vertical isocline.

In this paper, we aim to study the global solvability of the following system of third order nonlinear neutral delay differential equations $$\frac{d}{dt}\left\{r_i\left(t\right)\frac{d}{dt}\left[{\lambda }_i\left(t\right)\frac{d}{dt}\left(x_i\left(t\right)-f_i(t,x_1(t-{\sigma }_{i1}),x_2(t-{\sigma }_{i2}),x_3(t-{\sigma }_{i3}))\right)\right]\right\}+\frac{d}{dt}\left[r_i\left(t\right)\frac{d}{dt}{g}_i(t,x_1\left(p_{i1}\left(t\right)\right),x_2\left(p_{i2}\left(t\right)\right),x_3\left(p_{i3}\left(t\right)\right))\right]+\frac{d}{dt}{h}_i(t,x_1\left(q_{i1}\left(t\right)\right),x_2\left(q_{i2}\left(t\right)\right),x_3\left(q_{i3}\left(t\right)\right))=l_i(t,x_1\left({\eta }_{i1}\left(t\right)\right),x_2\left({\eta }_{i2}\left(t\right)\right),x_3\left({\eta }_{i3}\left(t\right)\right)),t\ge t_0,i\in \{1,2,3\}$$ in the following bounded closed and convex set $$\Omega (a,b)=\left\{x\left(t\right)=(x_1\left(t\right),x_2\left(t\right),x_3\left(t\right))\in C([t_0,+\infty ),{ℝ}^3):a\left(t\right)\le x_i\left(t\right)\le b\left(t\right),\forall t\ge t_0,i\in \{1,2,3\}\right\},$$ where ${\sigma }_{ij}>0$, $r_i,{\lambda }_i,a,b\in C([t_0,+\infty ),{ℝ}^{+})$, $f_i,g_i,h_i,l_i\in C([t_0,+\infty )\times {ℝ}^3,ℝ)$, $p_{ij},q_{ij},{\eta }_{ij}\in C([t_0,+\infty ),ℝ)$ for $i,j\in \{1,2,3\}$. By applying the Krasnoselskii fixed point theorem, the Schauder fixed point theorem, the Sadovskii fixed point theorem and the Banach contraction principle, four existence results of uncountably many bounded positive solutions of the system are established.