scholarly journals Asymptotic Convergence of the Solutions of a Discrete Equation with Two Delays in the Critical Case

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
L. Berezansky ◽  
J. Diblík ◽  
M. Růžičková ◽  
Z. Šutá
Keyword(s):  
Critical Case ◽  
Critical Value ◽  
Discrete Equation ◽  
Asymptotic Convergence ◽  
All Solutions ◽  
Two Delays ◽  
Convergent Solution

A discrete equationΔy(n)=β(n)[y(n−j)−y(n−k)]with two integer delayskandj, k>j≥0is considered forn→∞. We assumeβ:ℤn0−k∞→(0,∞), whereℤn0∞={n0,n0+1,…},  n0∈ℕandn∈ℤn0∞. Criteria for the existence of strictly monotone and asymptotically convergent solutions forn→∞are presented in terms of inequalities for the functionβ. Results are sharp in the sense that the criteria are valid even for some functionsβwith a behavior near the so-called critical value, defined by the constant(k−j)−1. Among others, it is proved that, for the asymptotic convergence of all solutions, the existence of a strictly monotone and asymptotically convergent solution is sufficient.

scholarly journals Asymptotic Convergence of the Solutions of a Dynamic Equation on Discrete Time Scales

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
J. Diblík ◽  
M. Růžičková ◽  
Z. Šmarda ◽  
Z. Šutá
Keyword(s):  
Time Scales ◽  
Discrete Time ◽  
Dynamic Equation ◽  
Time Scale ◽  
Asymptotic Convergence ◽  
Main Attention ◽  
All Solutions ◽  
Convergent Solution

The paper investigates a dynamic equationΔy(tn)=β(tn)[y(tn−j)−y(tn−k)]forn→∞, wherekandjare integers such thatk>j≥0, on an arbitrary discrete time scaleT:={tn}withtn∈ℝ,n∈ℤn0−k∞={n0−k,n0−k+1,…},n0∈ℕ,tn<tn+1,Δy(tn)=y(tn+1)−y(tn), andlimn→∞tn=∞. We assumeβ:T→(0,∞). It is proved that, for the asymptotic convergence of all solutions, the existence of an increasing and asymptotically convergent solution is sufficient. Therefore, the main attention is paid to the criteria for the existence of an increasing solution asymptotically convergent forn→∞. The results are presented as inequalities for the functionβ. Examples demonstrate that the criteria obtained are sharp in a sense.

Induction Suspension of Conductive Microring and Its Gyroscopic Stabilization

2021 ◽  
Author(s):  
Dmitrii Skubov ◽  
Ivan Popov ◽  
Pavel Udalov
Keyword(s):  
Magnetic Field ◽  
Steady State ◽  
Eddy Current ◽  
Equilibrium Position ◽  
Critical Case ◽  
Conservative System ◽  
Critical Value ◽  
Main Task ◽  
Gyroscopic Stabilization

Abstract The main task of our work is determination of possible levitation of micro-ring with eddy current in magnetic field of down ring with set alternating current and determination of critical value of «ohmic» damping separated field of parameters, at which motions of suspension ring transit from divergent to meeting to steady-state equilibrium position. I. e. in this critical case the motion practically coincides with motions of conservative system. The possibility of gyroscopic stabilization of suspension ring taking into account initial set rotation is considered. Thereby it can serve as contactless micro-gyroscope.

scholarly journals Fluctuation limit of branching processes with immigration and estimation of the means

2005 ◽  
Vol 37 (02) ◽  
pp. 523-538 ◽  
Author(s):  
M. Ispány ◽  
G. Pap ◽  
M. C. A. van Zuijlen
Keyword(s):  
Least Squares ◽  
Critical Case ◽  
Branching Processes ◽  
Critical Value ◽  
Subcritical Case ◽  
Asymptotically Normal ◽  
Fluctuation Limit ◽  
Conditional Least Squares ◽  
Type Process ◽  
Conditional Least Squares Estimator

We investigate a sequence of Galton-Watson branching processes with immigration, where the offspring mean tends to its critical value 1 and the offspring variance tends to 0. It is shown that the fluctuation limit is an Ornstein-Uhlenbeck-type process. As a consequence, in contrast to the case in which the offspring variance tends to a positive limit, it transpires that the conditional least-squares estimator of the offspring mean is asymptotically normal. The norming factor is n 3/2, in contrast to both the subcritical case, in which it is n 1/2, and the nearly critical case with positive limiting offspring variance, in which it is n.

scholarly journals Convergence of solutions for two delays Volterra integral equations in the critical case

2010 ◽  
Vol 23 (10) ◽  
pp. 1162-1165 ◽  
Author(s):  
Eleonora Messina ◽  
Yoshiaki Muroya ◽  
Elvira Russo ◽  
Antonia Vecchio
Keyword(s):  
Integral Equations ◽  
Critical Case ◽  
Volterra Integral Equations ◽  
Convergence Of Solutions ◽  
Two Delays

scholarly journals Global Dynamics Behaviors of Viral Infection Model for Pest Management

2009 ◽  
Vol 2009 ◽  
pp. 1-16 ◽  
Author(s):  
Chunjin Wei ◽  
Lansun Chen
Keyword(s):  
Small Amplitude ◽  
Critical Value ◽  
Infection Model ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Virus Particles ◽  
All Solutions ◽  
Pest Eradication ◽  
Viral Infection Model

According to biological strategy for pest control, a mathematical model with periodic releasing virus particles for insect viruses attacking pests is considered. By using Floquet's theorem, small-amplitude perturbation skills and comparison theorem, we prove that all solutions of the system are uniformly ultimately bounded and there exists a globally asymptotically stable pest-eradication periodic solution when the amount of virus particles released is larger than some critical value. When the amount of virus particles released is less than some critical value, the system is shown to be permanent, which implies that the trivial pest-eradication solution loses its stability. Further, the mathematical results are also confirmed by means of numerical simulation.

Asymptotic convergence of the solutions of a discrete equation with several delays

2012 ◽  
Vol 218 (9) ◽  
pp. 5391-5401 ◽  
Author(s):  
J. Diblík ◽  
M. Růžičková ◽  
Z. Šutá
Keyword(s):  
Discrete Equation ◽  
Asymptotic Convergence ◽  
Several Delays

scholarly journals Multiple Solutions for Nonlinear Dirichlet Problems with Concave Terms

2013 ◽  
Vol 113 (2) ◽  
pp. 206 ◽  
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Dirichlet Problem ◽  
Multiple Solutions ◽  
Critical Case ◽  
Critical Value ◽  
Constant Sign ◽  
Smooth Solutions ◽  
Dirichlet Problems ◽  
Truncation Techniques ◽  
Semilinear Problem ◽  
Concave Term

We consider a nonlinear parametric Dirichlet problem with parameter $\lambda>0$, driven by the $p$-Laplacian and with a concave term $\lambda|u|^{q-2}u$, $1<q<p$ and a Carathéodory perturbation $f(z,\zeta)$ which is asymptotically $(p-1)$-linear at infinity. Using variational methods combined with Morse theory and truncation techniques, we show that there is a critical value $\lambda^*>0$ of the parameter such that for $\lambda\in (0,\lambda^*)$ the problem has five nontrivial smooth solutions, four of constant sign (two positive and two negative) and the fifth nodal. In the semilinear case ($p=2$), we show that there is a sixth nontrivial smooth solution, but we cannot provide information about its sign. Finally for the critical case $\lambda=\lambda^*$, we show that the nonlinear problem ($p\ne 2$) still has two nontrivial constant sign smooth solutions and the semilinear problem ($p=2$) has three nontrivial smooth solutions, two of which have constant sign.

scholarly journals Fluctuation limit of branching processes with immigration and estimation of the means

2005 ◽  
Vol 37 (2) ◽  
pp. 523-538 ◽  
Author(s):  
M. Ispány ◽  
G. Pap ◽  
M. C. A. van Zuijlen
Keyword(s):  
Least Squares ◽  
Critical Case ◽  
Branching Processes ◽  
Critical Value ◽  
Subcritical Case ◽  
Asymptotically Normal ◽  
Fluctuation Limit ◽  
Conditional Least Squares ◽  
Type Process ◽  
Conditional Least Squares Estimator

We investigate a sequence of Galton-Watson branching processes with immigration, where the offspring mean tends to its critical value 1 and the offspring variance tends to 0. It is shown that the fluctuation limit is an Ornstein-Uhlenbeck-type process. As a consequence, in contrast to the case in which the offspring variance tends to a positive limit, it transpires that the conditional least-squares estimator of the offspring mean is asymptotically normal. The norming factor is n3/2, in contrast to both the subcritical case, in which it is n1/2, and the nearly critical case with positive limiting offspring variance, in which it is n.

Asymptotic theory for a critical class of fourth-order differential equations

1982 ◽  
Vol 383 (1785) ◽  
pp. 465-476 ◽  
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Critical Case ◽  
Asymptotic Theory ◽  
All Solutions ◽  
Exponential Character ◽  
Independent Variable ◽  
Fourth Order Equations ◽  
Critical Class ◽  
Asymptotic Forms

A recently formulated asymptotic theory for differential equations with large coefficients that are like powers of the independent variable is developed here for fourth-order equations with general coefficients. In this general approach a critical case is identified that forms a borderline between situations where all solutions have asymptotically a certain exponential character in terms of the coefficients and where only two solutions have this character. The asymptotic forms of a fundamental set of solutions are obtained in the critical case.

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