Almost oscillatory three-dimensional dynamical system

Akin-Bohner Elvan, Lawrence, Došlá Zuzana, Lawrence Bonita

Description

In this article, we investigate oscillation and asymptotic properties for 3D systems of dynamic equations. We show the role of nonlinearities and we apply our results to the adjoint dynamic systems. 2010 Mathematics Subject Classification: 39A10 In this article, we investigate oscillation and asymptotic properties for 3D systems of dynamic equations. We show the role of nonlinearities and we apply our results to the adjoint dynamic systems. 2010 Mathematics Subject Classification: 39A10

Sujets

• Time scale
• Oscillation

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 4
• Langue: English

Extrait

Akin-Bohner et al . Advances in Difference Equations 2012, 2012 :46 http://www.advancesindifferenceequations.com/content/2012/1/46

R E S E A R C H Open Access Almost oscillatory three-dimensional dynamical system Elvan Akin-Bohner 1* , Zuzana Do š lá 2 and Bonita Lawrence 3

* Correspondence: [e-mail] 1 Missouri University of Science and Abstract Technology, 310 Rolla Building, In this article, we investigate oscillation and asymptotic properties for 3D systems of Missouri 65409-0020, USA dynamic equations. We show the role of nonlinearities and we apply our results to Full list of author information is available at the end of the article the adjoint dynamic systems. 2010 Mathematics Subject Classification: 39A10 Keywords: Time scales, Oscillation, Three-dimensional dynamical system

1 Introduction In this article, we investigate 3D dynamical systems of the form ⎪ ⎧ ⎨ x y   ( ( t t ) ) = = a b ( ( t t ) ) f g ( ( y z ( ( t t ) )) ) (1) ⎩ ⎪ z  t = λ c t h x t on a time scale , i.e., a closed subset of real numbers, λ = ± 1, a , b : T → [0, ∞ (not identically zero) and c : T → [0, ∞ are rd-continuous functions such that ∞ ∞  a ( τ ) τ =  b ( τ ) τ = ∞ , T ∈ (2)

and , , h : R → are continuous functions such that uf u > 0, ug u > 0, and uh u > 0 for u  = 0 Here we would like to indicate that none of the functions f , g or h are monotonic. Sometimes we will assume that functions f, g , and h satisfy f ( u ) ≥ F , g ( u ) ≥ G  h γ ( ( u ) ) ≥ H , for all u  = 0 (3)  α ( u )  β ( u ) u where F , G , H are positive constants and F a , F b , and F g are odd power functions, i. e.,  ( u ) = | u | p sgn u ( p > 0), p ∈ { α , β , γ } Here, we consider only unbounded time scales. The theory of time scales is initiated by Stefan Hilger [1] his PhD dissertation in 1988 in order to unify continuous and dis- crete analysis. The theory of dynamic equations on time scales helps us not only to © 2012 Akin-Bohner et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.