scholarly journals Asymptotic solutions of nondiagonal linear difference systems

2000 ◽  
Vol 31 (3) ◽  
pp. 175-192
Author(s):  
Raul Naulin ◽  
Manuel Pinto
Keyword(s):  
Linear System ◽  
Asymptotic Integration ◽  
Asymptotic Solutions ◽  
Difference System ◽  
Perturbed System ◽  
Diagonal System ◽  
Change Of Variables ◽  
Difference Systems ◽  
The Matrix ◽  
Linear Difference Systems

This paper, relying on dichotomic properties of the matrix difference system $ W(n+1)=A(n)W(n)A^{-1}(n)$, gives conditions under which a perturbed system $ y(n+1)=(A(n)+B(n))y(n)$, by means of a nonautonomous change of variables $ y(n)=S(n)x(n)$, can be reduced to the form $ x(n+1)=A(n)x(n)$. From this, a theory of asymptotic integration of the perturbed system follows, where the linear system $ x(n+1)=A(n)x(n)$ is nondiagonal. As a consequence of these results, we prove that the diagonal system $ x(n+1)=\Lambda(n)x(n)$ has a Levinson dichotomy iff system $ W(n+1)=\Lambda(n)W(n)\Lambda^{-1}(n)$ has an ordinary dichotomy.

scholarly journals Asymptotic solutions of nondiagonal linear difference systems

2000 ◽  
Vol 31 (2) ◽  
pp. 109-122
Author(s):  
A. M. Jarrah ◽  
E. Malkowsky
Keyword(s):  
Asymptotic Solutions ◽  
Dual Spaces ◽  
Difference Systems ◽  
Alpha Beta ◽  
Linear Difference Systems

In this paper we shall give the $ \alpha$-, $ \beta$-, $ \gamma$- and $ f$-duals of the sets $ w_{0}^{p}(\Lambda)$, $ w_{\infty}^{p}(\Lambda)$, $ c_{0}^{p}(\Lambda)$, $ c^{p}(\Lambda)$ and $ c_{\infty}^{p}(\Lambda)$. Furthermore, we shall determine the continuous dual spaces of the sets $ w_{0}^{p}(\Lambda)$, $ c_{0}^{p}(\Lambda)$ and $ c^{p}(\Lambda)$.

scholarly journals Existence of Homoclinic Orbits for a Class of Asymptoticallyp-Linear Difference Systems withp-Laplacian

2011 ◽  
Vol 2011 ◽  
pp. 1-17
Author(s):  
Qiongfen Zhang ◽  
X. H. Tang
Keyword(s):  
Critical Point ◽  
Critical Point Theory ◽  
Mountain Pass Theorem ◽  
Homoclinic Orbits ◽  
Point Theory ◽  
Homoclinic Solutions ◽  
Mountain Pass ◽  
Difference System ◽  
Difference Systems ◽  
Linear Difference Systems

By applying a variant version of Mountain Pass Theorem in critical point theory, we prove the existence of homoclinic solutions for the following asymptoticallyp-linear difference system withp-LaplacianΔ(|Δu(n-1)|p-2Δu(n-1))+∇[-K(n,u(n))+W(n,u(n))]=0, wherep∈(1,+∞),n∈ℤ,u∈ℝN,K,W:ℤ×ℝN→ℝare not periodic inn, and W is asymptoticallyp-linear at infinity.

On the Necessary and Sufficient Conditions for the Stability of Linear Generalized Ordinary Differential, Linear Impulsive and Linear Difference Systems

2009 ◽  
Vol 16 (4) ◽  
pp. 597-616
Author(s):  
Shota Akhalaia ◽  
Malkhaz Ashordia ◽  
Nestan Kekelia
Keyword(s):  
Linear System ◽  
Difference Equations ◽  
Closed Interval ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Difference Systems ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Generalized Ordinary Differential Equations ◽  
Lyapunov Sense

Abstract Necessary and sufficient conditions are established for the stability in the Lyapunov sense of solutions of a linear system of generalized ordinary differential equations 𝑑𝑥(𝑡) = 𝑑𝐴(𝑡) · 𝑥(𝑡) + 𝑑𝑓(𝑡), where and are, respectively, matrix- and vector-functions with bounded total variation components on every closed interval from . The results are realized for the linear systems of impulsive, ordinary differential and difference equations.

scholarly journals On the oscillation of a nonlinear two-dimensional difference systems

2001 ◽  
Vol 32 (3) ◽  
pp. 201-209 ◽  
Author(s):  
E. Thandapani ◽  
B. Ponnammal
Keyword(s):  
Asymptotic Behavior ◽  
Sufficient Conditions ◽  
Real Sequence ◽  
Two Dimensional ◽  
Difference System ◽  
Nonoscillatory Solutions ◽  
Difference Systems ◽  
All Solutions ◽  
Necessary And Sufficient ◽  
Strongly Sublinear

The authors consider the two-dimensional difference system$$ \Delta x_n = b_n g (y_n) $$ $$ \Delta y_n = -f(n, x_{n+1}) $$where $ n \in N(n_0) = \{ n_0, n_0+1, \ldots \} $, $ n_0 $ a nonnegative integer; $ \{ b_n \} $ is a real sequence, $ f: N(n_0) \times {\rm R} \to {\rm R} $ is continuous with $ u f(n,u) > 0 $ for all $ u \ne 0 $. Necessary and sufficient conditions for the existence of nonoscillatory solutions with a specified asymptotic behavior are given. Also sufficient conditions for all solutions to be oscillatory are obtained if $ f $ is either strongly sublinear or strongly superlinear. Examples of their results are also inserted.

Removing apparent singularities of linear difference systems

2021 ◽  
Vol 102 ◽  
pp. 86-107
Author(s):  
Moulay A. Barkatou ◽  
Maximilian Jaroschek
Keyword(s):  
Difference Systems ◽  
Linear Difference Systems

scholarly journals Stability and stabilization of linear positive systems on time scales

Positivity ◽  
2020 ◽  
Vol 24 (5) ◽  
pp. 1361-1372
Author(s):  
Zbigniew Bartosiewicz
Keyword(s):  
Control System ◽  
Linear System ◽  
Time Scale ◽  
Sufficient Conditions ◽  
Positive Control ◽  
Positive Systems ◽  
The Matrix ◽  
Exponentially Stable ◽  
Stability And Stabilization ◽  
Necessary And Sufficient

Abstract It is shown that a positive linear system on a time scale with a bounded graininess is uniformly exponentially stable if and only if the characteristic polynomial of the matrix defining the system has all its coefficients positive. Then this fact is used to find necessary and sufficient conditions of positive stabilizability of a positive control system on a time scale.

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