On the Stability of Volterra Difference Equations of Convolution Type

In ⁴4. S. Elaydi. Stability of Volterra difference equations of convolution type, Dynamical Systems. Nankai Ser. Pure Appl. Math. Theoret. Phys., 4 (1993), 66-72., S. Elaydi obtained a characterization of the stability of the null solution of the Volterra difference equation

x_{n} = \sum_{i = 0}^{n - 1} a_{n - i} x_{i}, n \geq 1,

by localizing the roots of its characteristic equation

1 - \sum_{n = 1}^{\infty} a_{n} z^{n} = 0 .

The assumption that (an ) ∈ ℓ¹ was the single hypothesis considered for the validity of that characterization, which is an insufficient condition if the ratio R of convergence of the power series of the previous equation equals one. In fact, when R = 1, this characterization conflicts with a result obtained by Erdo¨ s et al. in ⁸8. P. Erdös, W. Feller & H. Pollard. A property of power series with positive coefficients. Bull. Amer. Math. Soc., 55 (1949), 201-204.. Here, we analyze the R = 1 case and show that some parts of that characterization still hold. Furthermore, studies on stability for the R < 1 case are presented. Finally, we study some results related to stability via finite approximation.

Keywords:
difference equation; stability; convolution

Authorship

H.P. OQUENDO

Universidade Federal do Paraná, Brasil. E-mail: higidio@ufpr.brUniversidade Federal do ParanáBrazilBrazilUniversidade Federal do Paraná, Brasil. E-mail: higidio@ufpr.br

J.R.R. BARBOSA ^**Corresponding author: José Renato R. Barbosa - E-mail: jrrb@ufpr.br.

Universidade Federal do Paraná, Brasil. E-mail: higidio@ufpr.brUniversidade Federal do ParanáBrazilBrazilUniversidade Federal do Paraná, Brasil. E-mail: higidio@ufpr.br

P.S. PACHECO

Universidade Federal Tecnológica do Paraná, Brasil. E-mail: patricias@utfpr.edu.brUniversidade Federal Tecnológica do ParanáBrazilBrazilUniversidade Federal Tecnológica do Paraná, Brasil. E-mail: patricias@utfpr.edu.br

*Corresponding author: José Renato R. Barbosa - E-mail: jrrb@ufpr.br.

SCIMAGO INSTITUTIONS RANKINGS

Universidade Federal do Paraná, Brasil. E-mail: higidio@ufpr.brUniversidade Federal do ParanáBrazilBrazilUniversidade Federal do Paraná, Brasil. E-mail: higidio@ufpr.br

Tables | Formulas

Tables (1)
Formulas (55)

Thumbnail

n r n L n (1 − r n ) n 1 1 - - 2 1.067 - - 3 1.012 - - 4 0.913 0.24716 0.00005 5 0.781 0.04963 0.00050 6 0.667 0.00024 0.00137

(1.1)

(1.2)

(1.3)

(1.4)

(1.5)

(2.1)

(2.2)

(2.3)

(2.4)

x_{n} = \sum_{i = 0}^{n - 1} a_{n - i} x_{i}, n \geq 1,

1 - \sum_{n = 1}^{\infty} a_{n} z^{n} = 0 .

x_{n} = \sum_{i = 0}^{n - 1} a_{n - i} x_{i}, n \geq 1,

1 - \sum_{n = 1}^{\infty} a_{n} z^{n} = 0 .

n	r _n	L _n	(1 − r _n ) ⁿ
1	1	-	-
2	1.067	-	-
3	1.012	-	-
4	0.913	0.24716	0.00005
5	0.781	0.04963	0.00050
6	0.667	0.00024	0.00137

(1.1)

x_{n} = \sum_{i = 0}^{n - 1} a_{n - i} x_{i}, n \geq 1,

x (z) : = \sum_{n = 0}^{\infty} x_{n} z^{n}, a (z) : = \sum_{n = 1}^{\infty} a_{n} z^{n} .

(1.2)

x (z) (1 - a (z)) = 1 .

(1.3)

1 - \sum_{n = 1}^{\infty} a_{n} z^{n} = 0

B_{r} (z_{0}) = {z \in ℂ : | z - z_{0} | < r}, r > 0 .

(1.4)

a_{n} = \frac{1}{n (n + 1)}, n \geq 1,

\gcd {n \in ℕ : a_{n} > 0} = 1, \sum_{n = 1}^{\infty} a_{n} = 1 and \sum_{n = 1}^{\infty} n a_{n} = \infty .

(1.5)

\frac{1}{R} = \underset{n \to \infty}{limsup} \sqrt[n]{| a_{n} |},

(2.1)

a_{n} = \frac{p^{n - 1}}{n (n + 1)}, n \geq 1,

\lim_{ρ \to ρ_{0}^{-}} | x (ρ e^{i θ}) | = \infty .

limsup \sqrt[n]{| x_{n} |} \geq \frac{1}{ρ_{0}} > 1 .

| a (z) | = |\sum_{n = 1}^{\infty} \frac{p^{n - 1}}{n (n + 1)} z^{n}| \leq \frac{1}{p} < 1, \forall z \in \bar{B_{1 / p} (0)},

f_{ρ} (θ) = x (ρ e^{i θ}) e^{- i n θ}, f (θ) = x (e^{i θ}) e^{- i n θ} .

\lim_{ρ \to 1^{-}} \int_{0}^{2 π} x (ρ e^{i θ}) e^{- i n θ} d θ = \int_{0}^{2 π} x (e^{i θ}) e^{- i n θ} d θ .

x_{n} = \frac{1}{2 π i} \int_{| z | = ρ} \frac{x (z)}{z^{n + 1}} d z = \frac{1}{2 π ρ^{n}} \int_{0}^{2 π} x (ρ e^{i θ}) e^{- i n θ} d θ .

x_{n} = \frac{1}{2 π} \int_{0}^{2 π} x (e^{i θ}) e^{- i n θ} d θ .

y (z) = \sum_{n = 0}^{\infty} y_{n} z^{n}, p (z) = \sum_{n = 0}^{\infty} p_{n} z^{n} .

| p (z) | \leq (1 - z) \sum_{n = 0}^{\infty} | y_{n} | z^{n} \leq C (1 - z) \sum_{n = 0}^{\infty} z^{n} = C . □

(2.2)

1 - a (z) = (1 - e^{- i θ_{1}} z)^{m_{1}} \dots (1 - e^{- i θ_{s}} z)^{m_{s}} q (z),

L^{1} : = \{q (z) = \sum_{n = 0}^{\infty} q_{n} z^{n} : (q_{n}) \in ℓ^{1}\} .

(2.3)

[q (z)]^{- 1} = \hat{q} (z) = \sum_{n = 0}^{\infty} {\hat{q}}_{n} z^{n} \in L^{1} .

(2.4)

{[\prod_{j = 1}^{s} (1 - e^{- i θ_{j}} z)]}^{- 1} = \sum_{n = 0}^{\infty} α_{n} z^{n} with α_{n} = \sum_{j = 1}^{s} (\frac{e^{i \ pbrack \sum_{μ \neq j} θ_{μ}}}{\prod_{μ \neq j} (e^{i θ_{μ}} - e^{i θ_{j}})}) e^{- i θ_{j} n} .

x (z) = (\sum_{n = 0}^{\infty} α_{n} z^{n}) (\sum_{n = 0}^{\infty} {\hat{q}}_{n} z^{n}) .

| x_{n} | \leq C \sum_{k = 0}^{\infty} | {\hat{q}}_{k} |,

p (z) (1 - z)^{m_{1} - 1} q (e^{i θ_{1}} z) = 1, \forall z \in B_{1} (0),

p (z) = (1 - z) {(1 - e^{i (θ_{1} - θ_{2})} z)}^{m_{2}} \dots {(1 - e^{i (θ_{1} - θ_{s})} z)}^{m_{s}} x (e^{i θ_{1}} z) .

1 = \lim_{\underset{z \in ℝ}{z \to 1^{-}}} x (e^{i θ_{1}} z) (1 - a (e^{i θ_{1}} z)) = \lim_{\underset{z \in ℝ}{z \to 1^{-}}} p (z) (1 - z)^{m_{1} - 1} q (e^{i θ_{1}} z) = 0,

q (z) = \sum_{n = 0}^{\infty} q_{n} z^{n} : = \frac{1 - a (z)}{1 - z} = (1 - a (z)) (\sum_{n = 0}^{\infty} z^{n}) .

q_{n} = 1 - \sum_{k = 1}^{n} a_{k} = \sum_{k = n + 1}^{\infty} a_{k} .

\sum_{k = 0}^{n - 1} q_{k} = \sum_{k = 1}^{n} k a_{k} + n \sum_{k = n + 1}^{\infty} a_{k} .

\sum_{k = 0}^{n - 1} | q_{k} | \leq \sum_{k = 1}^{n} k | a_{k} | + n \sum_{k = n + 1}^{\infty} | a_{k} | \leq \sum_{k = 1}^{\infty} k | a_{k} | \forall n \in ℕ .

\frac{1 - a (z)}{(1 - e^{- i θ_{1}} z) \dots (1 - e^{- i θ_{s}})} = \sum_{j = 1}^{s} \frac{β_{j} (1 - a (z))}{(1 - e^{- i θ_{j}} z)} = \sum_{j = 1}^{s} β_{j} q_{j} (z),

1 - a (z) = (1 - e^{- i θ_{1}} z) \dots (1 - e^{- i θ_{s}} z) q (z) .

| h (z) | < | f (z) | in | z | = ρ,

p_{n} (z) = z^{n} - a_{1} z^{n - 1} - \dots - a_{n - 1} z - a_{n},

r_{n} : = \max {| z | : p_{n} (z) = 0}

p (z) = (z - z_{1}) \dots (z - z_{n}) .

r_{n} < 1 \begin{matrix}  \end{matrix} a n d \begin{matrix}  \end{matrix} \sum_{i = n + 1}^{\infty} | a_{i} | < {(1 - r_{n})}^{n},

1 - r_{n} \leq 1 - | z_{i} | \leq | z | - | z_{i} | \leq | z - z_{i} |, i = 1, . . ., n .

s_{n} (z) : = 1 - \sum_{k = 1}^{n} a_{k} z^{k} .

|1 - a (z) - s_{n} (z)| = |\sum_{i = n + 1}^{\infty} a_{i} z^{i}| \leq \sum_{i = n + 1}^{\infty} |a_{i}| < {(1 - r_{n})}^{n} \leq |s_{n} (z)| .

(a_{n}) = (\frac{3}{2}, - \frac{9}{16}, \frac{β_{1}}{20}, \frac{β_{2}}{20^{2}}, \frac{β_{3}}{20^{3}}, \dots)

\sum_{i = 3}^{\infty} | a_{i} | = \frac{1}{19} < (1 - 3 / 4)^{2},

(a_{n}) = (1, - \frac{41}{36}, \frac{8}{9}, - \frac{34}{81}, \frac{16}{81}, - \frac{4}{81}, \frac{1}{2 \cdot 4^{6}}, \frac{1}{2^{2} \cdot 4^{6}}, \frac{1}{2^{3} \cdot 4^{6}}, \dots)

\sum_{i = n + 1}^{\infty} | a_{i} | < δ_{n} (ρ_{0}),

δ_{n} (ρ) : = | (1 - ρ | z_{1} |) (1 - ρ | z_{2} |) \dots (1 - ρ | z_{n} |) |,

| s_{n} (z) | = | 1 - z z_{1} | \dots | 1 - z z_{n} | \geq δ_{n} (ρ_{0}) .

| 1 - a (z) - s_{n} (z) | = | \sum_{i = n + 1}^{\infty} a_{i} z^{i} | \leq \sum_{i = n + 1}^{\infty} | a_{i} | < δ_{n} (ρ_{0}) \leq | s_{n} (z) | .

\sum_{i = n + 1}^{\infty} | a_{i} | < E,

a_{1} = 4, a_{2} = - 4, a_{n} = \frac{1}{2^{n - 1}}, n \geq 3 .

\sum_{k = 3}^{\infty} | a_{k} | = \frac{1}{2} < | 1 - 2 |^{2},

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On the Stability of Volterra Difference Equations of Convolution Type

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[1] *Corresponding author: José Renato R. Barbosa - E-mail: jrrb@ufpr.br.