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Nonuniqueness results for ordinary differential equations

Josef Kalas — 1998

Czechoslovak Mathematical Journal

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.

Asymptotic behaviour of a difference equation with complex-valued coefficients

Josef Kalas — 2005

Archivum Mathematicum

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.

Asymptotic properties of an unstable two-dimensional differential system with delay

Josef Kalas — 2006

Mathematica Bohemica

The asymptotic behaviour of the solutions is studied for a real unstable two-dimensional system x ' ( t ) = 𝖠 ( t ) x ( t ) + 𝖡 ( t ) x ( t - r ) + h ( t , x ( t ) , x ( t - r ) ) , where r > 0 is a constant delay. It is supposed that 𝖠 , 𝖡 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 → ∞ are given. The method of investigation is based on the transformation of the real system considered to one...

Asymptotic behaviour of a two-dimensional differential system with a nonconstant delay under the conditions of instability

Josef Kalas;  Josef Rebenda — 2011

Mathematica Bohemica

We present several results dealing with the asymptotic behaviour of a real two-dimensional system x ' ( t ) = 𝖠 ( t ) x ( t ) + ∑ k = 1 m 𝖡 k ( t ) x ( θ k ( t ) ) + h ( t , x ( t ) , x ( θ 1 ( t ) ) , ⋯ , x ( θ m ( t ) ) ) with bounded nonconstant delays t - θ k ( t ) ≥ 0 satisfying lim t → ∞ θ k ( t ) = ∞ , under the assumption of instability. Here 𝖠 , 𝖡 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...

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