This paper deals with the second order nonlinear neutral differential inequalities $\left(A_{\nu }\right)$: ${(-1)}^{\nu }x\left(t\right)\{z^{\text{'}\text{'}}\left(t\right)+{(-1)}^{\nu }q\left(t\right)f\left(x\left(h\left(t\right)\right)\right)\}\le 0,$$t\ge t_0\ge 0...$

A sufficient condition for the nonoscillation of nonlinear systems of differential equations whose left-hand sides are given by $n$-th order differential operators which are composed of special nonlinear differential operators of the first order is established. Sufficient conditions for the oscillation of systems of two nonlinear second order differential equations are also presented.

The purpose of this paper is to obtain oscillation criterions for the differential system of neutral type.

The paper deals with the higher-order ordinary differential equations and the analogous higher-order difference equations and compares the corresponding fundamental concepts. Important dissimilarities appear for the moving frame method.

Further extension of the Levinson transformation theory is performed for partially dissipative periodic processes via the fixed point index. Thus, for example, the periodic problem for differential inclusions can be treated by means of the multivalued Poincaré translation operator. In a certain case, the well-known Ważewski principle can also be generalized in this way, because no transversality is required on the boundary.

Let $f(t,x)$ be a vector valued function almost periodic in $t$ uniformly for $x$, and let $\mathrm{m}od\left(f\right)=L_1\oplus L_2$ be its frequency module. We say that an almost periodic solution $x\left(t\right)$ of the system $$\dot{x}=f(t,x),t\in ℝ,x\in D\subset {ℝ}^n$$ is irregular with respect to $L_2$ (or partially irregular) if $(\mathrm{m}od\left(x\right)+L_1)\cap L_2=\left\{0\right\}$. Suppose that $f(t,x)=A\left(t\right)x+X(t,x),$ where $A\left(t\right)$ is an almost periodic $(n\times n)$-matrix and $\mathrm{m}od\left(A\right)\cap \mathrm{m}od\left(X\right)=\left\{0\right\}.$ We consider the existence problem for almost periodic irregular with respect to $\mathrm{m}od\left(A\right)$ solutions of such system. This problem is reduced to a similar problem for a system of smaller dimension, and sufficient conditions...

We present a three species model describing the degradation of substrate by two competing populations of microorganisms in a marine sediment. Considering diffusion to be the main transport process, we obtain a reaction diffusion system (RDS) which we study in terms of spontaneous pattern formation. We find that the conditions for patterns to evolve are likely to be fulfilled in the sediment. Additionally, we present simulations that are consistent with experimental data from the literature. We...

Two theorems about period doubling bifurcations are proved. A special case, where one multiplier of the homogeneous solution is equal to +1 is discussed in the Appendix.