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.

A convexity theorem for the period function T of Hamiltonian systems with separable variables is proved. We are interested in systems with non-monotone T. This result is applied to proving the uniqueness of critical orbits for second order ODE's.

The paper deals with the periodic boundary value problem (1) $L_{4}x\left(t\right)+a\left(t\right)x\left(t\right)\in F(t,x\left(t\right))$, $t\in J=[a,b]$, (2) $L_{i}x\left(a\right)=L_{i}x\left(b\right)$, $i=0,1,2,3$, where $L_{0}x\left(t\right)=a_{0}x\left(t\right)$, $L_{i}x\left(t\right)=a_i\left(t\right)L_{i-1}x\left(t\right)$, $i=1,2,3,4$, $a_0\left(t\right)=a_4\left(t\right)=1$, $a_i\left(t\right)$, $i=1,2,3$ and $a\left(t\right)$ are continuous on $J$, $a\left(t\right)\ge 0$, $a_i\left(t\right)>0$, $i=1,2$, $a_1\left(t\right)=a_3\left(t\right)·F(t,x):J\times R\to${nonempty convex compact subsets of $R$}, $R=(-\infty ,\infty )$. The existence of such periodic solution is proven via Ky Fan’s fixed point theorem.

The paper deals with the impulsive boundary value problem $$\frac{d}{dt}\left[\phi \left(y^{\text{'}}\left(t\right)\right)\right]=f(t,y\left(t\right),y^{\text{'}}\left(t\right)),y\left(0\right)=y\left(T\right),y^{\text{'}}\left(0\right)=y^{\text{'}}\left(T\right),y(t_i+)=J_i\left(y\left(t_i\right)\right),y^{\text{'}}(t_i+)=M_i\left(y^{\text{'}}\left(t_i\right)\right),i=1,...m.$$ The method of lower and upper solutions is directly applied to obtain the results for this problems whose right-hand sides either fulfil conditions of the sign type or satisfy one-sided growth conditions.