In the present paper we give general nonuniqueness results which cover most of the known nonuniqueness criteria. In particular, we obtain a generalization of the nonuniqueness theorem of Chr. Nowak, of Samimi’s nonuniqueness theorem and of Stettner’s nonuniqueness criterion.

The asymptotic behaviour for solutions of a difference equation ${z}_{n}=f(n,{z}_{n})$, where the complex-valued function $f(n,z)$ is in some meaning close to a holomorphic function $h$, and of a Riccati difference equation is studied using a Lyapunov function method. The paper is motivated by papers on the asymptotic behaviour of the solutions of differential equations with complex-valued right-hand sides.

The asymptotic behaviour of the solutions is studied for a real unstable two-dimensional system ${x}^{\text{'}}\left(t\right)=\U0001d5a0\left(t\right)x\left(t\right)+\U0001d5a1\left(t\right)x(t-r)+h(t,x\left(t\right),x(t-r))$, where $r>0$ is a constant delay. It is supposed that $\U0001d5a0$, $\U0001d5a1$ and $h$ are matrix functions and a vector function, respectively. Our results complement those of Kalas [Nonlinear Anal. 62(2) (2005), 207–224], where the conditions for the existence of bounded solutions or solutions tending to the origin as $t\to \infty $ are given. The method of investigation is based on the transformation of the real system considered to one...

We present several results dealing with the asymptotic behaviour of a real two-dimensional system ${x}^{\text{'}}\left(t\right)=\U0001d5a0\left(t\right)x\left(t\right)+{\sum}_{k=1}^{m}{\U0001d5a1}_{k}\left(t\right)x\left({\theta}_{k}\left(t\right)\right)+h(t,x\left(t\right),x\left({\theta}_{1}\left(t\right)\right),\cdots ,x\left({\theta}_{m}\left(t\right)\right))$ with bounded nonconstant delays $t-{\theta}_{k}\left(t\right)\ge 0$ satisfying ${lim}_{t\to \infty}{\theta}_{k}\left(t\right)=\infty $, under the assumption of instability. Here $\U0001d5a0$, ${\U0001d5a1}_{k}$ and $h$ are supposed to be matrix functions and a vector function, respectively. The conditions for the instable properties of solutions together with the conditions for the existence of bounded solutions are given. The methods are based on the transformation of the real system considered to one equation with...

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.

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