Volterra discrete equations: summability of the fundamental matrix

Vecchio, Antonia

doi:10.1007/s002110100270

Cited by 18 publications

(18 citation statements)

References 16 publications

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“…The following lemma (see also [8,19,35,36]) shows that the solution of (2.1) can be represented by using the fundamental matrix as well as by using the resolvent; compare…”

Section: Resolvent and Fundamental Matricesmentioning

confidence: 99%

“…The resolvent R (n, m) for the kernel B (n + 1, m) = (I − B(n + 1, n + 1)) −1 B(n + 1, m) is defined (see, e.g., [17,36]) by…”

Section: Resolvent and Fundamental Matricesmentioning

confidence: 99%

“…To the best of our knowledge, relatively few papers ( [5,6,8,9,[25][26][27][30][31][32]35,36], in particular) deal with discrete Volterra ("summation") equations. Much of the general qualitative theory remains to be developed, though there are some results in the literature concerning boundedness of solutions, frequently based upon specific applications.…”

Section: Introductionmentioning

confidence: 99%

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Linearized stability analysis of discrete Volterra equations

Song¹,

Baker

2004

Journal of Mathematical Analysis and Applications

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Various different types of stability are defined, in a unified framework, for discrete Volterra equations of the type x(n) = f (n) + n j =0 K(n, j, x(n)) (n 0). Under appropriate assumptions, stability results are obtainable from those valid in the linear case (K(n, j, x(n)) = B(n, j )x(j )), and a linearized stability theory is studied here by using the fundamental and resolvent matrices. Several necessary and sufficient conditions for stability are obtained for solutions of the linear equation by considering the equations in various choices of Banach space B, the elements of which are sequences of vectors (x(n), f (n) ∈ E d , B(n, j ) : E d → E d , n, j 0, etc.). We show that the theory, including a number of new results as well as results already known, can be presented in a systematic framework, in which results parallel corresponding results for classical Volterra integral equations of the second kind.

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“…The following lemma (see also [8,19,35,36]) shows that the solution of (2.1) can be represented by using the fundamental matrix as well as by using the resolvent; compare…”

Section: Resolvent and Fundamental Matricesmentioning

confidence: 99%

“…The resolvent R (n, m) for the kernel B (n + 1, m) = (I − B(n + 1, n + 1)) −1 B(n + 1, m) is defined (see, e.g., [17,36]) by…”

Section: Resolvent and Fundamental Matricesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Linearized stability analysis of discrete Volterra equations

Song¹,

Baker

2004

Journal of Mathematical Analysis and Applications

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“…Theorem 2.2 and Remark 2.2 will be useful to obtain sufficient condition that the zero solution of (1.1) is uniformly asymptotically stable (cf. [1][2][3][4][5]10,15,16,19,20]). Finally in this section, under the condition…”

Section: Remark 22 For the Linear Casementioning

confidence: 99%

“…Using the Liapunov technique, Crisci et al [1][2][3] and Vecchio [19,20] have analyzed the various stability of numerical methods. They are, for example, the first-order and some higher order backward differential formulas, and the implicit Euler methods without requiring the summability of the kernel.…”

Section: Introductionmentioning

confidence: 99%