scholarly journals Asymptotic Stability of Non-Autonomous Systems and a Generalization of Levinson’s Theorem

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1213
Author(s):  
Min-Gi Lee
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Stability ◽  
Linear Systems ◽  
Fixed Point Theory ◽  
Autonomous Systems ◽  
Zero Solution ◽  
Coefficient Matrix ◽  
Point Theory ◽  
Levinson’S Theorem ◽  
Indispensable Tool

We study the asymptotic stability of non-autonomous linear systems with time dependent coefficient matrices { A ( t ) } t ∈ R . The classical theorem of Levinson has been an indispensable tool for the study of the asymptotic stability of non-autonomous linear systems. Contrary to constant coefficient system, having all eigenvalues in the left half complex plane does not imply asymptotic stability of the zero solution. Levinson’s theorem assumes that the coefficient matrix is a suitable perturbation of the diagonal matrix. Our objective is to prove a theorem similar to Levinson’s Theorem when the family of matrices merely admits an upper triangular factorization. In fact, in the presence of defective eigenvalues, Levinson’s Theorem does not apply. In our paper, we first investigate the asymptotic behavior of upper triangular systems and use the fixed point theory to draw a few conclusions. Unless stated otherwise, we aim to understand asymptotic behavior dimension by dimension, working with upper triangular with internal blocks adds flexibility to the analysis.

AN ASYMPTOTIC STABILITY THEOREM FOR QUASILINEAR DELAY DIFFERENTIAL EQUATIONS WITH OSCILLATING COEFFICIENTS

2013 ◽  
Vol 24 (11) ◽  
pp. 1350092 ◽  
Author(s):  
NGUYEN TIEN DUNG
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Fixed Point Theory ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Asymptotic Behavior Of Solutions ◽  
Variable Delays ◽  
Nonlinear Anal ◽  
Oscillating Coefficients ◽  
Several Delays

In this paper, we provide new necessary and sufficient conditions of the asymptotic stability for a class of quasilinear differential equations with several delays and oscillating coefficients. Our results are established by means of fixed point theory and improve those obtained in [J. R. Graef, C. Qian and B. Zhang, Asymptotic behavior of solutions of differential equations with variable delays, Proc. London Math. Soc.81 (2000) 72–92; B. Zhang, Fixed points and stability in differential equations with variable delays, Nonlinear Anal.63 (2005) e233–e242].

scholarly journals Asymptotic behavior of recursions via fixed point theory

2008 ◽  
Vol 337 (2) ◽  
pp. 1125-1141 ◽  
Author(s):  
Kristine Ey ◽  
Christian Pötzsche
Keyword(s):  
Fixed Point ◽  
Asymptotic Behavior ◽  
Fixed Point Theory ◽  
Point Theory

scholarly journals Stability Conditions of Second Order Integrodifferential Equations with Variable Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Dingheng Pi
Keyword(s):  
Fixed Point Theory ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Point Theory ◽  
Stability Conditions ◽  
Asymptotically Stable ◽  
Variable Delay ◽  
Functional Differential ◽  
Necessary And Sufficient

We investigate integrodifferential functional differential equationsẍ+f(t,x,ẋ)ẋ+∫t-r(t)t‍a(t,s)g(x(s))ds=0with variable delay. By using the fixed point theory, we obtain conditions which ensure that the zero solution of this equation is stable under an exponentially weighted metric. Then we establish necessary and sufficient conditions ensuring that the zero solution is asymptotically stable. We will give an example to apply our results.

On an open problem regarding the spectral radius of the derivatives of a function and of its iterates

2019 ◽  
Vol 28 (1) ◽  
pp. 33-39
Author(s):  
VASILE BERINDE ◽  
◽  
STEFAN MARUSTER ◽  
IOAN A. RUS ◽  
◽  
...  
Keyword(s):  
Asymptotic Stability ◽  
Spectral Radius ◽  
Open Problem ◽  
Global Asymptotic Stability ◽  
Fixed Point Theory ◽  
Point Theory ◽  
General Context ◽  
Iterative Approximation ◽  
The Relationship ◽  
Derivatives Of

The main aim of this note is to investigate empirically the relationship between the spectral radius of the derivative of a function f : Rm → Rm and the spectral radius of the derivatives of its iterates, which is done by means of some numerical experiments for mappings of two and more variables. In this way we give a partial answer to an open problem raised in [Rus, I. A., Remark on a La Salle conjecture on global asymptotic stability, Fixed Point Theory, 17 (2016), No. 1, 159–172] and [Rus, I. A., A conjecture on global asymptotic stability, communicated at the Workshop ”Iterative Approximation of Fixed Points”, SYNASC2017, Timis¸oara, 21-24 September 2017] and also illustrate graphically the importance and difficulty of this problem in the general context. An open problem regarding the domains of convergence is also proposed.

Stability by Fixed Point Theory for Nonlinear Delay Difference Equations

2009 ◽  
Vol 16 (4) ◽  
pp. 683-691
Author(s):  
Chuhua Jin ◽  
Jiaowan Luo
Keyword(s):  
Fixed Point ◽  
Difference Equations ◽  
Fixed Point Theory ◽  
Zero Solution ◽  
Point Theory ◽  
The Stability ◽  
Delay Difference Equations ◽  
Delay Difference ◽  
Nonlinear Delay

Abstract We study the stability of the zero solution of nonlinear delay difference equations by fixed point theory. An example is given to illustrate our theory.

scholarly journals Stability by fixed point theory of impulsive differential equations with delay

2019 ◽  
Vol 57 (2) ◽  
pp. 18-33
Author(s):  
Halimi Berrezoug ◽  
Jorge Losada ◽  
Juan J. Nieto ◽  
Abdelghani Ouahab
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Point Theory ◽  
Impulsive Differential Equations ◽  
Zero Solution ◽  
Point Theory ◽  
Asymptotically Stable ◽  
Equations With Delay

Abstract In this paper we ensure that for some class of impulsive differential equations with delay the zero solution is asymptotically stable by means of fixed point theory.

scholarly journals FIXED POINTS AND STABILITY OF A CLASS OF NONLINEAR DELAY INTEGRO-DIFFERENTIAL EQUATIONS WITH VARIABLE DELAYS

2017 ◽  
pp. 031
Author(s):  
Hocine Gabsi ◽  
Abdelouaheb Ardjouni ◽  
Ahcene Djoudi
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Points ◽  
Fixed Point Theory ◽  
Zero Solution ◽  
Point Theory ◽  
Second Order ◽  
Asymptotically Stable ◽  
Variable Delays ◽  
Nonlinear Delay

In this work we study a class of second order nonlinear neutral integro-differential equations    x(t)+f(t,x(t),x(t))x(t)+∑_{j=1}^{N}∫_{t-τ_{j}(t)}^{t}a_{j}(t,s)g_{j}(s,x(s))ds    +∑_{j=1}^{N}b_{j}(t)x′(t-τ_{j}(t))=0,with variable delays and give some new conditions ensuring that the zero solution is asymptotically stable by means of the fixed point theory. Our work extends and improves previous results in the literature such as, D. Pi <cite>pi2,pi3</cite> and T. A. Burton <cite>b12</cite>. An example is given to illustrate our claim.

scholarly journals Fixed Points and Stability of a Class of Integrodifferential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Dingheng Pi
Keyword(s):  
Fixed Point ◽  
Fixed Point Theory ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Point Theory ◽  
Asymptotically Stable ◽  
Variable Delay ◽  
Functional Differential ◽  
Necessary And Sufficient

We study a class of integrodifferential functional differential equationsx¨+f(t,x,x˙)x˙+∑j=1N∫t-rj(t)taj(t,s)gj(s,x(s))ds=0with variable delay. By using the fixed point theory, we establish necessary and sufficient conditions ensuring that the zero solution of this equation is asymptotically stable.

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