The problems related to periodic solutions of cellular neural networks (CNNs) involving $D$ operator and proportional delays are considered. We shall present Topology degree theory and differential inequality technique for obtaining the existence of periodic solution to the considered neural networks. Furthermore, Laypunov functional method is used for studying global asymptotic stability of periodic solutions to the above system.

In this paper we consider the equation $$y^{\text{'}\text{'}\text{'}}+q\left(t\right){y^{\text{'}}}^{\alpha }+p\left(t\right)h\left(y\right)=0,$$ where $p,q$ are real valued continuous functions on $[0,\infty )$ such that $q\left(t\right)\ge 0$, $p\left(t\right)\ge 0$ and $h\left(y\right)$ is continuous in $(-\infty ,\infty )$ such that $h\left(y\right)y>0$ for $y\ne 0$. We obtain sufficient conditions for solutions of the considered equation to be nonoscillatory. Furthermore, the asymptotic behaviour of these nonoscillatory solutions is studied.

We study relations between the almost specification property, the asymptotic average shadowing property and the average shadowing property for dynamical systems on compact metric spaces. We show implications between these properties and relate them to other important notions such as shadowing, transitivity, invariant measures, etc. We provide examples showing that compactness is a necessary condition for these implications to hold. As a consequence, we also obtain a proof that limit shadowing in...

The author considers the quasilinear differential equations $$\begin{array}{c}{\left(r\left(t\right)\varphi \left(x^{\text{'}}\right)\right)}^{\text{'}}+q\left(t\right)f\left(x\right)=0,t\ge a\\ \multicolumn{2}{c}{\text{and}}\\ {\left(r\left(t\right)\varphi \left(x^{\text{'}}\right)\right)}^{\text{'}}+F(t,x)=\pm g\left(t\right),t\ge a.\end{array}$$ By means of topological tools there are established conditions ensuring the existence of nonnegative asymptotic decaying solutions of these equations.

The paper describes asymptotic properties of a strongly nonlinear system $\dot{x}=f(t,x)$, $(t,x)\in ℝ\times {ℝ}^n$. The existence of an $\lfloor n/2\rfloor$ parametric family of solutions tending to zero is proved. Conditions posed on the system try to be independent of its linear approximation.