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

Title:

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

Creator:

Kalas, Josef and Rebenda, Josef

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:a9bbbb57-e2f8-4544-bfaf-451e1d3911cb
uuid:a9bbbb57-e2f8-4544-bfaf-451e1d3911cb
doi:10.21136/MB.2011.141584

Subject:

delayed differential equations, asymptotic behaviour, boundedness of solutions, Lyapunov method, and Ważewski topological principle

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

We present several results dealing with the asymptotic behaviour of a real twodimensional system x ′ (t) = A(t)x(t) + ∑ Pm k=1 Bk(t)x(θk(t)) + h(t, x(t), x(θ1(t)), . . . , x(θm(t))) with bounded nonconstant delays t − θk(t) ≥ 0 satisfying limt→∞ θk(t) = ∞, under the assumption of instability. Here A, Bk 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 complex-valued coefficients. Asymptotic properties are studied by means of a suitable Lyapunov-Krasovskii functional and the Wa˙zewski topological principle. The results generalize some previous ones, where the asymptotic properties for two-dimensional systems with one constant or nonconstant delay were studied.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

215-224

Source:

Mathematica bohemica | 2011 Volume:136 | Number:2

Harvested from:

CDK

Metadata only:

false