scholarly journals Asymptotic Behavior of Solutions for Nonlinear Volterra Discrete Equations

2008 ◽  
Vol 2008 ◽  
pp. 1-18 ◽  
Author(s):  
E. Messina ◽  
Y. Muroya ◽  
E. Russo ◽  
A. Vecchio
Keyword(s):  
Asymptotic Behavior ◽  
Numerical Methods ◽  
Integral Equations ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Volterra Integral Equations ◽  
Asymptotic Behavior Of Solutions ◽  
Nonlinear Difference Equations ◽  
Discrete Equations

We consider nonlinear difference equations of unbounded order of the formxi=bi−∑j=0iai,jfi−j(xj),  i=0,1,2,…,wherefj(x)  (j=0,…,i)are suitable functions. We establish sufficient conditions for the boundedness and the convergence ofxiasi→+∞. Some of these conditions are interesting mainly for studying stability of numerical methods for Volterra integral equations.

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 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 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 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.

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