Displaying similar documents to “Asymptotic behaviour of a two-dimensional differential system with a nonconstant delay under the conditions of instability”

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

Josef Kalas (2006)

Mathematica Bohemica

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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...

Asymptotic estimation of the convergence of solutions of the equation x ˙ ( t ) = b ( t ) x ( t - τ ( t ) )

Josef Diblík, Denis Khusainov (2001)

Archivum Mathematicum

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The main result of the present paper is obtaining new inequalities for solutions of scalar equation x ˙ ( t ) = b ( t ) x ( t - τ ( t ) ) . Except this the interval of transient process is computed, i.e. the time is estimated, during which the given solution x ( t ) reaches an ε - neighbourhood of origin and remains in it.

Asymptotic behavior of solutions of neutral nonlinear differential equations

Jozef Džurina (2002)

Archivum Mathematicum

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In this paper we study asymptotic behavior of solutions of second order neutral functional differential equation of the form x ( t ) + p x ( t - τ ) ' ' + f ( t , x ( t ) ) = 0 . We present conditions under which all nonoscillatory solutions are asymptotic to a t + b as t → ∞ , with a , b ∈ R . The obtained results extend those that are known for equation u ' ' + f ( t , u ) = 0 .

Convergence to equilibria in a differential equation with small delay

Mihály Pituk (2002)

Mathematica Bohemica

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Consider the delay differential equation x ˙ ( t ) = g ( x ( t ) , x ( t - r ) ) , ( 1 ) where r > 0 is a constant and g ℝ 2 → ℝ is Lipschitzian. It is shown that if  r is small, then the solutions of (1) have the same convergence properties as the solutions of the ordinary differential equation obtained from (1) by ignoring the delay.