Admissibility for discrete Volterra equations

Song, Yihong; Baker, Christopher T. H.

doi:10.1080/10236190600563260

Cited by 22 publications

(33 citation statements)

References 20 publications

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“…If, in addition, lim n→∞ B(n, j) = 0 for each j ≥ 0, then lim n→∞ R(n, j) = 0 for each j ≥ 0. For details, see [13] or [15]. where v ∈ (0, 1), then {R(n, m)} satisfies conditions Assumption H3 and H4.…”

Section: Assumption H4 the Resolvent {R(n M)} Satisfiesmentioning

confidence: 99%

“…In [13], the discrete Paley-Wiener theorem for convolution equations has been extended to non-convolution equations. From the results in [13], we know that if sup n≥0 n j=0 |B(n, j)| < 1, then sup n≥0 n j=0 |R(n, j)| < ∞. If, in addition, lim n→∞ B(n, j) = 0 for each j ≥ 0, then lim n→∞ R(n, j) = 0 for each j ≥ 0.…”

Section: Assumption H4 the Resolvent {R(n M)} Satisfiesmentioning

confidence: 99%

“…, by the discrete Paley-Wiener theorem (see [13]). In this case, one readily sees that the Assumptions H2, H3 and H4 are satisfied.…”

Section: B(n R)y(r)mentioning

confidence: 99%

“…We assume det(I − λB(n, n)) = 0. As in Section 4, we define the resolvent R λ (n, m) as the unique solution of the following two equations (0)) = 0 and det(I − λZ{K}(z)) = 0 for |z| ≥ 1, by virtue of the discrete Paley-Wiener Theorem (see [13]), where Z{K}(z) is the Z-transform of {K(n)} n≥0 .…”

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confidence: 99%

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Periodic solutions of discrete Volterra equations

Baker

Song²

2004

Mathematics and Computers in Simulation

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In this paper, we investigate periodic solutions of linear and nonlinear discrete Volterra equations of convolution or non-convolution type with unbounded memory.For linear discrete Volterra equations of convolution type, we establish Fredholm's alternative theorem and for equations of non-convolution type, and we prove that a unique periodic solution exists for a particular bounded initial function under appropriate conditions. Further, this unique periodic solution attracts all other solutions with bounded initial function. All solutions of linear discrete Volterra equations with bounded initial functions are asymptotically periodic under certain conditions. A condition for periodic solutions in the nonlinear case is established.

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Section: Assumption H4 the Resolvent {R(n M)} Satisfiesmentioning

confidence: 99%

Section: Assumption H4 the Resolvent {R(n M)} Satisfiesmentioning

confidence: 99%

“…, by the discrete Paley-Wiener theorem (see [13]). In this case, one readily sees that the Assumptions H2, H3 and H4 are satisfied.…”

Section: B(n R)y(r)mentioning

confidence: 99%

mentioning

confidence: 99%

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Periodic solutions of discrete Volterra equations

Baker

Song²

2004

Mathematics and Computers in Simulation

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“…We consider the nonlinear system of Volterra difference equations In recent years, there has been an increasing interest in the study of the asymptotic behavior of the solutions of both convolution and non-convolution-type linear and nonlinear Volterra difference equations (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and references therein). Appleby et al [2], under appropriate assumptions, have proved that the solutions of the discrete linear Volterra equation converge to a finite limit, which in general is non-trivial.…”

Section: Introductionmentioning

confidence: 99%

On the boundedness of the solutions in nonlinear discrete Volterra difference equations

Győri

Awwad

2012

Adv Differ Equ

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In this article, we investigate the boundedness property of the solutions of linear and nonlinear discrete Volterra equations in both convolution and non-convolution case. Strong interest in these kind of discrete equations is motivated as because they represent a discrete analogue of some integral equations. The most important result of this article is a simple new criterion, which unifies and extends several earlier results in both discrete and continuous cases. Examples are also given to illustrate our main theorem.

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Stability and robust stability of positive linear Volterra difference equations

Ngoc¹,

Naito²,

Shin³

et al. 2008

Intl J Robust & Nonlinear

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We first introduce a class of positive linear Volterra difference equations. Then, we offer explicit criteria for uniform asymptotic stability of positive equations. Furthermore, we get a new Perron-Frobenius theorem for positive linear Volterra difference equations. Finally, we study robust stability of positive equations under structured perturbations and affine perturbations. Two explicit stability bounds with respect to these perturbations are given. POSITIVE LINEAR VOLTERRA DIFFERENCE EQUATIONS 553 for many interesting problems in Mathematics, Physics, Economics, Biology, etc. Moreover, in general, obtained results of problems for classes of positive systems are often very interesting, see [1,2,[4][5][6][7][8]. In recent times, problems of positive systems have attracted a lot of attention from researchers, see .In this paper, we first introduce a class of positive linear Volterra difference equations of the convolution typeThen, we study stability and robust stability of positive linear Volterra difference equations. It is important to note that problems of stability and robust stability of linear Volterra difference equations have been studied extensively for a long time, see [29-41] and references therein. However, to the best of our knowledge, aspects of positivity of linear Volterra difference equations have not been exploited in the literature and the main purpose of the present paper is to fill this gap. This paper belongs to a series of our papers on positive systems, which have been published only in recent times, see [5][6][7][16][17][18][19][20][21][22][23][24][25][26][27].The organization of this paper is as follows. In the following section, we give some notations and preliminary results that will be used in the sequel. In Section 3, we first introduce a class of positive linear Volterra difference equations of convolution type. Then, we give some explicit criteria for uniform asymptotic stability of positive equations. In Section 4, we present a new Perron-Frobenius theorem for positive linear Volterra difference equations. As a particular case of the obtained result, we get back the Perron-Frobenius theorem for positive polynomial matrices. In the last section, we study robust stability of positive equations under structured perturbations and affine perturbations. Two explicit stability bounds with respect to structured perturbations and affine perturbations are given. To the best of our knowledge, most of the results of this paper are new. PRELIMINARIES

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Admissibility for discrete Volterra equations

Cited by 22 publications

References 20 publications

Periodic solutions of discrete Volterra equations

Periodic solutions of discrete Volterra equations

On the boundedness of the solutions in nonlinear discrete Volterra difference equations

Stability and robust stability of positive linear Volterra difference equations

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