The paper is concentrated on Professor Miloš Ráb and his contribution to the theory of oscillatory properties of solutions of second and third order linear differential equations, the theory of differential equations with complex coefficients and dynamical systems, and the theory of nonlinear second order differential equations. At the beginning, we take a brief look at the most important moments in his life. Afterwards, we describe his scientific activities on mentioned theories.

We study asymptotic properties of solutions for a system of second differential equations with $p$-Laplacian. The main purpose is to investigate lower estimates of singular solutions of second order differential equations with $p$-Laplacian ${\left(A\left(t\right){\Phi }_p\left(y^{\text{'}}\right)\right)}^{\text{'}}+B\left(t\right)g\left(y^{\text{'}}\right)+R\left(t\right)f\left(y\right)=e\left(t\right)$. Furthermore, we obtain results for a scalar equation.

The system of nonlinear differential equations $$x^{\text{'}}+p_1\left(t\right)x^{{\alpha }_1}+q_1\left(t\right)y^{{\beta }_1}=0,y^{\text{'}}+p_2\left(t\right)x^{{\alpha }_2}+q_2\left(t\right)y^{{\beta }_2}=0,A$$ is under consideration, where ${\alpha }_i$ and ${\beta }_i$ are positive constants and $p_i\left(t\right)$ and $q_i\left(t\right)$ are positive continuous functions on $[a,\infty )$. There are three types of different asymptotic behavior at infinity of positive solutions $\left(x\right(t),y(t\left)\right)$ of (). The aim of this paper is to establish criteria for the existence of solutions of these three types by means of fixed point techniques. Special emphasis is placed on those solutions with both components decreasing to zero as $t\to \infty$, which can be...

We solve the problem of the existence and uniqueness of coexistence states for the classical predator-prey model of Lotka-Volterra with diffusion in the scalar case.

Connections between uniform exponential expansiveness and complete admissibility of the pair $(c_0(\mathbb{N},X),c_0(\mathbb{N},X))$ are studied. A discrete version for a theorem due to Van Minh, Räbiger and Schnaubelt is presented. Equivalent characterizations of Perron type for uniform exponential expansiveness of evolution families in terms of complete admissibility are given.

We give characterizations for uniform exponential stability and uniform exponential instability of linear skew-product flows in terms of Banach sequence spaces and Banach function spaces, respectively. We present a unified approach for uniform exponential stability and uniform exponential instability of linear skew-product flows, extending some stability theorems due to Neerven, Datko, Zabczyk and Rolewicz.

We consider nonautonomous competitive Kolmogorov systems, which are generalizations of the classical Lotka-Volterra competition model. Applying Ahmad and Lazer's definitions of lower and upper averages of a function, we give an average condition which guarantees that all but one of the species are driven to extinction.

A two species non-autonomous competitive phytoplankton system with Beddington-DeAngelis functional response and the effect of toxic substances is proposed and studied in this paper. Sufficient conditions which guarantee the extinction of a species and global attractivity of the other one are obtained. The results obtained here generalize the main results of Li and Chen [Extinction in two dimensional nonautonomous Lotka-Volterra systems with the effect of toxic substances, Appl. Math. Comput. 182(2006)684-690]....