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 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.

scholarly journals Implicit linear difference equations over a non-Archi-medean ring

2021 ◽  
Keyword(s):  
Power Series ◽  
Difference Equation ◽  
Unique Solution ◽  
Formal Power Series ◽  
Difference Equations ◽  
Linear Difference Equation ◽  
Ring Of Integers ◽  
Ring Of Polynomials ◽  
The Difference ◽  
Formal Power

Over any field an implicit linear difference equation one can reduce to the usual explicit one, which has infinitely many solutions ~ one for each initial value. It is interesting to consider an implicit difference equation over any ring, because the case of implicit equation over a ring is a significantly different from the case of explicit one. The previous results on the difference equations over rings mostly concern to the ring of integers and to the low order equations. In the present article the high order implicit difference equations over some other classes of rings, particularly, ring of polynomials, are studied. To study the difference equation over the ring of integer the idea of considering p-adic integers ~ the completion of the ring of integers with respect to the non-Archimedean p-adic valuation was useful. To find a solution of such an equation over the ring of polynomials it is naturally to consider the same construction for this ring: the ring of formal power series is a completion of the ring of polynomials with respect to a non-Archimedean valuation. The ring of formal power series and the ring of p-adic integers both are the particular cases of the valuation rings with respect to the non-Archimedean valuations of some fields: field of Laurent series and field of p-adic rational numbers respectively. In this article the implicit linear difference equation over a valuation ring of an arbitrary field with the characteristic zero and non-Archimedean valuation are studied. The sufficient conditions for the uniqueness and existence of a solution are formulated. The explicit formula for the unique solution is given, it has a form of sum of the series, converging with respect to the non-Archimedean valuation. Difference equation corresponds to an infinite system of linear equations. It is proved that in a case the implicit difference equation has a unique solution, it can be found using Cramer rules. Also in the article some results facilitating the finding the polynomial solution of the equation are given.

scholarly journals On the Characterization of a Class of Difference Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-12
Author(s):  
Muhammed Altun
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Toeplitz Matrices ◽  
Complex Banach Space ◽  
Asymptotic Behavior Of Solutions ◽  
Complex Numbers ◽  
Fixed Sequence ◽  
Lower Triangular ◽  
The Difference

We focus on the behavior of solutions of the difference equation , , where () is a fixed sequence of complex numbers, and () is a fixed sequence in a complex Banach space. We give the general solution of this difference equation. To examine the asymptotic behavior of solutions, we compute the spectra of operators which correspond to such type of difference equations. These operators are represented by upper triangular or lower triangular infinite banded Toeplitz matrices.

scholarly journals Accurate solution estimates for vector difference equations

2003 ◽  
Vol 2003 (48) ◽  
pp. 3059-3066
Author(s):  
Rigoberto Medina
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Reaction Diffusion ◽  
Accurate Solution ◽  
Region Of Attraction ◽  
Constant Matrix ◽  
Partial Reaction ◽  
Volterra Difference Equation ◽  
Vector Difference ◽  
Solution Estimates

Accurate estimates for the norms of the solutions of a vector difference equation are derived. They give us stability conditions and bounds for the region of attraction of the stationary solution. Our approach is based on estimates for the powers of a constant matrix. We also discuss applications of our main results to partial reaction-diffusion difference equations and to a Volterra difference equation.

A New Look at the Stability Analysis of Delay-Differential Equations

2005 ◽  
Author(s):  
Tama´s Kalma´r-Nagy
Keyword(s):  
Difference Equation ◽  
Differential Equations ◽  
Characteristic Equation ◽  
Delay Differential Equations ◽  
Linear Delay ◽  
The Laplace Transform ◽  
Method Of Steps ◽  
The Difference ◽  
The Stability ◽  
Delay Differential

It is shown that the method of steps for linear delay-differential equations combined with the Laplace-transform can be used to determine the stability of the equation. The result of the method is an infinite dimensional difference equation whose stability corresponds to that of the transcendental characteristic equation. Truncations of this difference equation are used to construct numerical stability charts. The method is demonstrated on a first and second order delay equation. Correspondence between the transcendental characteristic equation and the difference equation is proved for the first order case.

scholarly journals Subexponential Solutions of Linear Volterra Difference Equations

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Martin Bohner ◽  
Nasrin Sultana
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Rate Of Convergence ◽  
Difference Equations ◽  
Renewal Equation ◽  
Volterra Difference Equations

AbstractWe study the asymptotic behavior of the solutions of a scalar convolution sum-difference equation. The rate of convergence of the solution is found by determining the asymptotic behavior of the solution of the transient renewal equation.

On oscillation and nonoscillation properties of Emden-Fowler difference equations

2009 ◽  
Vol 7 (2) ◽  
Author(s):  
Mariella Cecchi ◽  
Zuzana Došlá ◽  
Mauro Marini
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Asymptotic Properties ◽  
Linear Case ◽  
Nonoscillatory Solutions ◽  
Oscillation And Nonoscillation

AbstractA characterization of oscillation and nonoscillation of the Emden-Fowler difference equation $$ \Delta (a_n \left| {\Delta x_n } \right|^\alpha sgn\Delta x_n ) + b_n \left| {x_{n + 1} } \right|^\beta sgnx_{n + 1} = 0 $$ is given, jointly with some asymptotic properties. The problem of the coexistence of all possible types of nonoscillatory solutions is also considered and a comparison with recent analogous results, stated in the half-linear case, is made.

On dichotomic maps for a class of differential-difference equations

1991 ◽  
Vol 117 (3-4) ◽  
pp. 317-328 ◽  
Author(s):  
L. A. V. Carvalho ◽  
K. L. Cooke
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Difference Equations ◽  
Positive Definite ◽  
Initial Condition ◽  
Null Solution ◽  
Differential Difference Equation

SynopsisStability and asymptotic stability of the null solution of the differential-difference equation (E)x′(t) = f(x(t), x(t − r)), f: RNxRN → RN, f(0, 0) = 0, are studied by means of an extension of the Liapunov–Razumikhin method. Let V: RN → R be a differentiate map, let C = C(+ −r, 0=, RN), and let x(t, ψ) denote the solution of (E) with initial condition ψ in C at t = 0. For t ≧ 0 let xt(ψ) be defined by xt,(ψ)(θ) = x(t + θ, ψ), −r ≦θ ≦0. Let V′ (ψ) be the variation of V along the solution x(t, ψ). We say that V is dichotomic with respect to (E) if there exist T ≧0 and Ω, a neighbourhood of the origin in C, such that if ψ is in the closure of the set where V′ (xT(ψ)) >; 0, then V(x(T, ψ)) ≦ V(x(s, ψ)) for some s, −r ≦ sT. It is proved that if V is positive definite, continuously differentiable, and dichotomic, then the null solution of (E) is stable. A concept of strict dichotomic map is introduced and used to prove asymptotic stability. A number of examples are given to illustrate the applications of the method.

scholarly journals NONEXISTENCE OF POSITIVE SOLUTIONS OF A NONLINEAR PARTIAL DIFFERENCE EQUATION

1997 ◽  
Vol 28 (1) ◽  
pp. 51-58
Author(s):  
SHU-TANG LIU ◽  
SUI-SUN CHENG
Keyword(s):  
Difference Equation ◽  
Positive Solutions ◽  
Characteristic Equation ◽  
Difference Equations ◽  
Necessary Conditions ◽  
Partial Difference Equation ◽  
Partial Difference ◽  
Existence Of Positive Solutions ◽  
Positive Roots

Necessary conditions are derived for the existence of positive solutions of a class of nonlinear partiai' difference equations. The technique used to derive these conditions is based on the nonexistence of positive roots of an associated characteristic equation.

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