scholarly journals Asymptotic Properties of Solutions to Second-Order Difference Equations of Volterra Type

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Janusz Migda ◽  
Małgorzata Migda ◽  
Magdalena Nockowska-Rosiak
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equations ◽  
Existence Of Solutions ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Type Equation ◽  
Asymptotic Behavior Of Solutions ◽  
Volterra Type ◽  
Order Difference ◽  
Prescribed Asymptotic Behavior

We consider the discrete Volterra type equation of the form Δ(rnΔxn)=bn+∑k=1nK(n,k)f(xk). We present sufficient conditions for the existence of solutions with prescribed asymptotic behavior. Moreover, we study the asymptotic behavior of solutions. We use o(ns), for given nonpositive real s, as a measure of approximation.

scholarly journals Asymptotic behavior of solutions to difference equations in Banach spaces

2021 ◽  
pp. 1-17
Author(s):  
Janusz Migda
Keyword(s):  
Asymptotic Behavior ◽  
Banach Spaces ◽  
Difference Equations ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Degree Of Approximation ◽  
Asymptotic Behavior Of Solutions ◽  
Vector Version ◽  
A New Technique ◽  
Prescribed Asymptotic Behavior

We investigate the asymptotic properties of solutions to higher order nonlinear difference equations in Banach spaces. We introduce a new technique based on a vector version of discrete L'Hospital's rule, remainder operator, and the regional topology on the space of all sequences on a given Banach space. We establish sufficient conditions for the existence of solutions with prescribed asymptotic behavior. Moreover, we are dealing with the problem of approximation of solutions. Our technique allows us to control the degree of approximation of solutions.

scholarly journals Asymptotic behavior of solutions of second-order difference equations of Volterra type

2019 ◽  
Vol 43 (5) ◽  
pp. 2203-2217
Author(s):  
Małgorzata MIGDA ◽  
Aldona DUTKIEWICZ
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equations ◽  
Second Order ◽  
Asymptotic Behavior Of Solutions ◽  
Volterra Type ◽  
Order Difference

Asymptotic behavior of solutions of linear homogeneous second-order difference equations

1999 ◽  
Vol 66 (2) ◽  
pp. 167-170
Author(s):  
A. D. Kozak ◽  
O. N. Novoselov
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equations ◽  
Second Order ◽  
Asymptotic Behavior Of Solutions ◽  
Order Difference

scholarly journals The asymptotic behavior of solutions of nonlinear second-order difference equations

2001 ◽  
Vol 14 (5) ◽  
pp. 611-616 ◽  
Author(s):  
E. Thandapani ◽  
S.L. Marian
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equations ◽  
Second Order ◽  
Asymptotic Behavior Of Solutions ◽  
Order Difference

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 The Asymptotic Behavior of Solutions of Second order Difference Equations With Damping Term

2018 ◽  
Vol 14 (2) ◽  
pp. 7806-7811
Author(s):  
Jai Kumar S ◽  
K. Alagesan
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Asymptotic Behavior Of Solutions ◽  
Damping Term ◽  
Order Difference ◽  
Keywords And Phrases ◽  
Second Order Difference Equation ◽  
Order Difference Equation

  The author presents some sufficient conditions for second order difference equation with damping term of the form                                                                             ^(an ^(xn + cxn-k)) + pn^xn + qnf(xn+1-l) = 0 An example is given to illustrate the main results. 2010 AMS Subject Classification: 39A11 Keywords and Phrases: Second order, difference equation, damping term.

scholarly journals Asymptotic behavior of solutions of second order difference equations with deviating argument

2018 ◽  
Vol 19 (2) ◽  
pp. 1047
Author(s):  
Janusz Migda ◽  
Magdalena Nockowska-Rosiak
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equations ◽  
Second Order ◽  
Asymptotic Behavior Of Solutions ◽  
Order Difference ◽  
Deviating Argument

scholarly journals Asymptotic Behavior of Higher Order Quasilinear Neutral Difference Equations

2016 ◽  
Vol 8 (6) ◽  
pp. 148
Author(s):  
V. Sadhasivam ◽  
Pon. Sundar ◽  
A. Santhi
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equations ◽  
Higher Order ◽  
Asymptotic Behavior Of Solutions ◽  
Functional Difference ◽  
Order Difference ◽  
Neutral Difference Equations ◽  
Odd Order ◽  
Functional Difference Equations

We study, the asymptotic behavior of solutions to a class of higher order quasilinear neutral difference equations under the assumptions that allow applications to even and odd-order difference equations with delayed and advanced arguments, as well as to functional difference equations with more complex arguments that may for instance, alternate infinitely between delayed and advanced types. New theorems extend a number of results reported in the literature. Illustrative examples are presented.

scholarly journals Asymptotic Behavior of Solutions of Delayed Difference Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-24 ◽  
Author(s):  
J. Diblík ◽  
I. Hlavičková
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equations ◽  
Special Class ◽  
General Theorem ◽  
Asymptotic Behavior Of Solutions ◽  
Order Difference ◽  
First Order ◽  
Discrete Equations

This contribution is devoted to the investigation of the asymptotic behavior of delayed difference equations with an integer delay. We prove that under appropriate conditions there exists at least one solution with its graph staying in a prescribed domain. This is achieved by the application of a more general theorem which deals with systems of first-order difference equations. In the proof of this theorem we show that a good way is to connect two techniques—the so-called retract-type technique and Liapunov-type approach. In the end, we study a special class of delayed discrete equations and we show that there exists a positive and vanishing solution of such equations.

scholarly journals Asymptotic Properties of Solutions to Discrete Volterra Monotone Type Equations

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 918
Author(s):  
Janusz Migda ◽  
Małgorzata Migda ◽  
Ewa Schmeidel
Keyword(s):  
Asymptotic Behavior ◽  
Bounded Solution ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Degree Of Approximation ◽  
The Other ◽  
Volterra Equations ◽  
Asymptotically Periodic ◽  
Monotone Type ◽  
Prescribed Asymptotic Behavior

We investigate the higher order nonlinear discrete Volterra equations. We study solutions with prescribed asymptotic behavior. For example, we establish sufficient conditions for the existence of asymptotically polynomial, asymptotically periodic or asymptotically symmetric solutions. On the other hand, we are dealing with the problem of approximation of solutions. Among others, we present conditions under which any bounded solution is asymptotically periodic. Using our techniques, based on the iterated remainder operator, we can control the degree of approximation. In this paper we choose a positive non-increasing sequence u and use o(un) as a measure of approximation.

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