scholarly journals Weighted Asymptotically Periodic Solutions of Linear Volterra Difference Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Josef Diblík ◽  
Miroslava Růžičková ◽  
Ewa Schmeidel ◽  
Małgorzata Zbąszyniak
Keyword(s):  
Difference Equation ◽  
Periodic Solutions ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Volterra Difference Equation ◽  
Asymptotically Periodic ◽  
Volterra Difference Equations ◽  
Asymptotically Periodic Solutions

A linear Volterra difference equation of the formx(n+1)=a(n)+b(n)x(n)+∑i=0nK(n,i)x(i),wherex:N0→R,a:N0→R,K:N0×N0→Randb:N0→R∖{0}isω-periodic, is considered. Sufficient conditions for the existence of weighted asymptotically periodic solutions of this equation are obtained. Unlike previous investigations, no restriction on∏j=0ω-1b(j)is assumed. The results generalize some of the recent results.

scholarly journals Existence of asymptotically periodic solutions of scalar Volterra difference equations

2009 ◽  
Vol 43 (1) ◽  
pp. 51-61 ◽  
Author(s):  
Josef Diblík ◽  
Miroslava Růžičková ◽  
Ewa Schmeidel
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Difference Equation ◽  
Periodic Solutions ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Volterra Difference Equation ◽  
Asymptotically Periodic ◽  
Volterra Difference Equations ◽  
Asymptotically Periodic Solutions

Abstract There is used a version of Schauder’s fixed point theorem to prove the existence of asymptotically periodic solutions of a scalar Volterra difference equation. Along with the existence of asymptotically periodic solutions, sufficient conditions for the nonexistence of such solutions are derived. Results are illustrated on examples.

Bounded solutions of nonlinear discrete volterra equations

2016 ◽  
Vol 66 (5) ◽  
Author(s):  
Małgorzata Migda ◽  
Janusz Migda
Keyword(s):  
Periodic Solutions ◽  
Volterra Equation ◽  
Sufficient Conditions ◽  
Volterra Equations ◽  
Real Constant ◽  
Bounded Solutions ◽  
All Solutions ◽  
Asymptotically Periodic ◽  
Asymptotically Periodic Solutions

AbstractWe give sufficient conditions, under which for every real constant, there exists a solution of the nonlinear discrete Volterra equationconvergent to this constant. We give also conditions under which all solutions are asymptotically constant. Sufficient conditions for the existence of asymptotically periodic solutions of the above equation are also derived.

scholarly journals On the Stability of Volterra Difference Equations of Convolution Type

2018 ◽  
Vol 18 (3) ◽  
pp. 337
Author(s):  
Higidio Portillo Oquendo ◽  
Jose Renato Ramos Barbosa ◽  
Patricia Sánez Pacheco
Keyword(s):  
Power Series ◽  
Difference Equation ◽  
Characteristic Equation ◽  
Difference Equations ◽  
Null Solution ◽  
Volterra Difference Equation ◽  
Finite Approximation ◽  
The Stability ◽  
Volterra Difference Equations

In \cite{Elaydi-10}, S.\ Elaydi obtained a characterization of the stability ofthe null solution of the Volterra difference equation\beqaex_n=\sum_{i=0}^{n-1} a_{n-i} x_i\textrm{,}\quad n\geq 1\textrm{,}\eeqaeby localizing the roots of its characteristic equation\beqae1-\sum_{n=1}^{\infty}a_nz^n=0\textrm{.}\eeqaeThe assumption that $(a_n)\in\ell^1$ was the single hypothesis considered for the validity of that characterization, which is an insufficient condition if theratio $R$ of convergence of the power series of the previous equation equals one. In fact, when $R=1$, this characterization conflicts with a result obtainedby Erd\"os et al in \cite{Erdos}. Here, we analyze the $R=1$ case and show thatsome parts of that characterization still hold. Furthermore, studies on stability for the $R<1$ case are presented. Finally, we state some new results related to stability via finite approximation.

scholarly journals Existence and Global Asymptotic Behavior of S -Asymptotically ω -Periodic Solutions for Evolution Equation with Delay

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Hong Qiao ◽  
Qiang Li ◽  
Tianjiao Yuan
Keyword(s):  
Asymptotic Behavior ◽  
Evolution Equation ◽  
Periodic Solutions ◽  
Sufficient Conditions ◽  
Compact Semigroup ◽  
Existence Results ◽  
Asymptotically Periodic ◽  
Abstract Evolution Equation ◽  
Equation With Delay ◽  
Asymptotically Periodic Solutions

This paper is concerned with the abstract evolution equation with delay. Firstly, we establish some sufficient conditions to ensure the existence results for the S -asymptotically periodic solutions by means of the compact semigroup. Secondly, we consider the global asymptotic behavior of the delayed evolution equation by using the Gronwall-Bellman integral inequality involving delay. These results improve and generalize the recent conclusions on this topic. Finally, we give an example to exhibit the practicability of our abstract results.

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