scholarly journals Asymptotic Behavior of Solutions to Half-Linearq-Difference Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-12
Author(s):  
Pavel Řehák
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Positive Solutions ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Special Limit ◽  
Bounded Functions ◽  
Necessary And Sufficient

We derive necessary and sufficient conditions for (some or all) positive solutions of the half-linearq-difference equationDq(Φ(Dqy(t)))+p(t)Φ(y(qt))=0,t∈{qk:k∈N0}withq>1,Φ(u)=|u|α−1sgn⁡uwithα>1, to behave likeq-regularly varying orq-rapidly varying orq-regularly bounded functions (that is, the functionsy, for which a special limit behavior ofy(qt)/y(t)ast→∞is prescribed). A thorough discussion on such an asymptotic behavior of solutions is provided. Related Kneser type criteria are presented.

scholarly journals Positive strongly decreasing solutions of Emden-Fowler type second-order difference equations with regularly varying coefficients

Filomat ◽  
2019 ◽  
Vol 33 (9) ◽  
pp. 2751-2770
Author(s):  
Aleksandra Kapesic ◽  
Jelena Manojlovic
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Nonlinear Difference Equation ◽  
Order Difference ◽  
Varying Coefficients ◽  
Regularly Varying ◽  
Necessary And Sufficient

Positive decreasing solutions of the nonlinear difference equation ?(pn|?xn|?-1?xn)=qn|xn+1|?-1xn+1, n ? 1, ? > ? > 0, are studied under the assumption that p; q are regularly varying sequences. Necessary and sufficient conditions are established for the existence of regularly varying strongly decreasing solutions and it is shown that the asymptotic behavior of all such solutions is governed by a unique formula.

scholarly journals On nonoscillatory solutions tending to zero of third-order nonlinear differential equations

2011 ◽  
Vol 48 (1) ◽  
pp. 135-143 ◽  
Author(s):  
Ivan Mojsej ◽  
Alena Tartal’ová
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Third Order ◽  
Nonoscillatory Solutions ◽  
The Third ◽  
Necessary And Sufficient

Abstract The aim of this paper is to present some results concerning with the asymptotic behavior of solutions of nonlinear differential equations of the third-order with quasiderivatives. In particular, we state the necessary and sufficient conditions ensuring the existence of nonoscillatory solutions tending to zero as t → ∞.

scholarly journals Some Qualitative Behavior of Solutions of General Class of Difference Equations

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 585 ◽  
Author(s):  
Osama Moaaz ◽  
Dimplekumar Chalishajar ◽  
Omar Bazighifan
Keyword(s):  
Periodic Solution ◽  
Asymptotic Behavior ◽  
Asymptotic Stability ◽  
Difference Equations ◽  
General Class ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Qualitative Behavior ◽  
Local Asymptotic Stability ◽  
Necessary And Sufficient

In this work, we consider the general class of difference equations (covered many equations that have been studied by other authors or that have never been studied before), as a means of establishing general theorems, for the asymptotic behavior of its solutions. Namely, we state new necessary and sufficient conditions for local asymptotic stability of these equations. In addition, we study the periodic solution with period two and three. Our results essentially extend and improve the earlier ones.

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.

Asymptotic behavior of the records of multivariate random sequences in a norm sense

2020 ◽  
Vol 70 (6) ◽  
pp. 1457-1468
Author(s):  
Haroon M. Barakat ◽  
M. H. Harpy
Keyword(s):  
Asymptotic Behavior ◽  
Weak Convergence ◽  
Sufficient Conditions ◽  
Record Values ◽  
Necessary And Sufficient Conditions ◽  
Random Sequences ◽  
Necessary And Sufficient

AbstractIn this paper, we investigate the asymptotic behavior of the multivariate record values by using the Reduced Ordering Principle (R-ordering). Necessary and sufficient conditions for weak convergence of the multivariate record values based on sup-norm are determined. Some illustrative examples are given.

scholarly journals Existence of a Period-Two Solution in Linearizable Difference Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
E. J. Janowski ◽  
M. R. S. Kulenović
Keyword(s):  
Difference Equation ◽  
Linear Equation ◽  
Difference Equations ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Minimal Period ◽  
Real Numbers ◽  
Existence And Nonexistence ◽  
The Difference ◽  
Necessary And Sufficient

Consider the difference equationxn+1=f(xn,…,xn−k),n=0,1,…,wherek∈{1,2,…}and the initial conditions are real numbers. We investigate the existence and nonexistence of the minimal period-two solution of this equation when it can be rewritten as the nonautonomous linear equationxn+l=∑i=1−lkgixn−i,n=0,1,…,wherel,k∈{1,2,…}and the functionsgi:ℝk+l→ℝ. We give some necessary and sufficient conditions for the equation to have a minimal period-two solution whenl=1.

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