On the behavior of the solutions for linear autonomous neutral delay difference equations

Kordonis, Ioannis; Philos, Ch. G.

doi:10.1080/102361908808184

Cited by 18 publications

(19 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For some related results, we refer to Graef and Qian [5]. Moreover, the discrete analogues of the results in [6,11] have been given by Kordonis and Philos [7] and Kordonis et al [8], respectively. The results in [7,8] concern difference equations with discrete variable.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

An asymptotic result for some delay difference equations with continuous variable

Philos

Purnaras

2004

Adv Differ Equ

View full text Add to dashboard Cite

We consider a nonhomogeneous linear delay difference equation with continuous variable and establish an asymptotic result for the solutions. Our result is obtained by the use of a positive root with an appropriate property of the so-called characteristic equation of the corresponding homogeneous linear (autonomous) delay difference equation. More precisely, we show that, for any solution, the limit of a specific integral transformation of it, which depends on a suitable positive root of the characteristic equation, exists as a real number and it is given explicitly in terms of the positive root of the characteristic equation and the initial function.

show abstract

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

An asymptotic result for some delay difference equations with continuous variable

Philos

Purnaras

2004

Adv Differ Equ

View full text Add to dashboard Cite

show abstract

“…Note that if f in Theorem 3.7 is linear, then the value of the limit (3.12) can be given explicitly in terms of the initial data (x 0 ,x −1 ,...,x −k ) (see [2] or [4] for details).…”

Section: Resultsmentioning

confidence: 99%

Asymptotic estimates and exponential stability for higher-order monotone difference equations

Liz¹,

Pituk²

2005

Adv Diff Equ

View full text Add to dashboard Cite

Asymptotic estimates are established for higher-order scalar difference equations and inequalities the right-hand sides of which generate a monotone system with respect to the discrete exponential ordering. It is shown that in some cases the exponential estimates can be replaced with a more precise limit relation. As corollaries, a generalization of discrete Halanay-type inequalities and explicit sufficient conditions for the global exponential stability of the zero solution are given.

show abstract

“…Since l : C D → R is continuous (see (6) Remark. The property that for most solutions the asymptotic behavior can be determined in terms of a simple real dominant characteristic value has been observed in many other linear systems (see, e.g., [2], [3], [4], [5], [7], [8] for related results).…”

Section: Theoremmentioning

confidence: 99%

Cesàro summability in a linear autonomous difference equation

Pituk

2005

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. For a linear autonomous difference equation with a unique real eigenvalue λ 0 , it is shown that for every solution x the ratio of x and the eigensolution corresponding to λ 0 is Cesàro summable to a limit which can be expressed in terms of the initial data. As a consequence, for most solutions the Lyapunov characteristic exponent is equal to λ 0 . The proof is based on a Tauberian theorem for the Laplace transform.Given r > 0, let C = C([−r, 0], R) denote the Banach space of all continuous functions from [−r, 0] into R equipped with the supremum norm,Consider the linear homogeneous difference equationwhere the difference operator D : C → R is given byand the symbol x t ∈ C is defined by x t (θ) = x(t+θ) for θ ∈ [−r, 0]. Throughout the paper, we assume that the kernel η : [−r, 0] → R is a nonconstant nondecreasing function such that η(0) = 0 and η is continuous from the left at each θ ∈ (−r, 0]. The integral in (2) is a Riemann-Stieltjes integral. With Eq.(1), we associate an initial condition of the formThe monotonicity and left continuity of η at zero imply that Var [s,0] − and hence the difference operator D is atomic at zero (see [6] for a definition). By known existence theorems (see, e.g., [6, Chap. 12]), the initial value problem (1)-(3) has a unique solution on [−r, ∞). We shall write x(t) = x(t, ψ) for the unique solution of (1) and (3).

show abstract

On the behavior of the solutions for linear autonomous neutral delay difference equations

Cited by 18 publications

References 6 publications

An asymptotic result for some delay difference equations with continuous variable

An asymptotic result for some delay difference equations with continuous variable

Asymptotic estimates and exponential stability for higher-order monotone difference equations

Cesàro summability in a linear autonomous difference equation

Contact Info

Product

Resources

About