ON THE DIFFERENCE EQUATION OF THE POINCARÉ TYPE (Part 1)

Leonid A. Gutnik

Dedicated to the memory of professor A.O. Gelfond.

Sit ye near some old Cavern’s Mouth, and brood Until ye start, as if the sea nimphs quir’d! John Keats

Table of contents

§0. Foreword. §1. Difference operators. §2. Reformulation of the problem. §3. The general plan of the proof of the Theorem 5. §4. The polynomials with coefficients in a linear space. §5. A several constants. §6. Some special norms on ℂ^{n}. §7. Some remarks on contracted maps and fixed point Theorem. §8. End of the proof of the Theorem 5.

§0. Foreword.

As a participiant of Gelfond’s seminar at the Moscow University, I was always excited by his big intellect and the width of his interests, which were not limited only to the Trancendental Number Theory, where he had obtained the most famous results, but spread over many other topics: Function Theory, Analytical Number Theory, Integral Equations, History of Mathematics etc. . He paid a special attention to the Difference Equations, in particular, to the Difference Equations of Poincaré Type. His results (together with I.M. Kubenskaya) on this subject [2] are well-known; he had also inspired the works on this topic by his pupils M.A.Evgrafov [3] and G.A.Frejman [4].

Let the difference equation

(1)  ∑_{k=0}^{n} a_{k}(ν)y(ν + k) = 0,

with n ∈ ℕ, a_{k}(ν) ∈ ℂ for k = 0, …, n and ν ∈ ℕ − 1 be given. Let further a_{n}(ν) = 1, where ν ∈ ℕ − 1, and suppose there exists

(2)  a_{k}^{∼} = lim_{ν→∞} a_{k}(ν).

Typeset by AMS-TEX

Equation (1) satisfying condition (2) is called a difference equation of Poincaré type. The polynomial

(3)  T (z) = ∑_{k=0}^{n} a_{k}^{∼}z^{k}

is called the characteristic polynomial of the equation (1).

The main result here belongs to O.Perron [1].

Theorem 1 (Perron’s Theorem.) Let ρ_{1}, …, ρ_{s} are pairwise distinct absolute values of all the roots of the polynomial (3) and let e_{σ}, σ = 1, …, s, is the

sum of multyplicities of these roots with module equal to ρ_{σ}. So, ∑_{σ=0}^{s} e_{σ} = n. Let

ρ_{k} < ρ_{i} for 1 ≤ i < k ≤ s.

Then the space of solutions of the equation (1) have a basis Γ, which splits into s mutually disjoint classes Ω_{σ}, σ = 1, …, s, with the following property: if a solution y(ν) of the equation (1) is a linear combination of the elements of Γ, in which all the elements of Ω_{σ}, 1 ≤ σ ≤ q − 1, where q ∈ {1, …, s}, have coefficients equal to zero, but at least one element in Ω_{q} has in this linear combination the nonzero coefficient, then

(4)  lim sup_{ν→∞} |y(ν)|^{1/ν} = ρ_{q}.

The equality (4) may be reformulated so:

for any nonzero solution of the equation (1) and any ε ∈ (0, 1) there exist a positive constants γ_{0,0} and γ_{0,1} such that

|y(ν)| ≤ γ_{0,1}(ρ_{q} + ε)^{ν}

for any ν ∈ ℕ − 1 and the inequality

(5)  γ_{0,0}(ρ_{q}(1 − ε))^{ν} ≤ |y(ν)|

is fulfilled, when ν run over an infinite subset of ℕ − 1; because this infinite subset is undefined, the last assertion is insufficient for some problems of Diophantine Approximations. This gap was partially closed in [6] (Theorem 2 and Theorem 3).

Theorem 2. Let λ_{1}, …, λ_{n} are the zeros of the polynomial (3) and one can meet each root here as many times as the multiplicity of this root is. Let |λ_{i}| ≥ |λ_{k}| for 1 ≤ i ≤ k ≤ n. Let further the equality (2) is fulfilled. Let a_{n}(ν) ≠ 0 for ν ∈ ℕ − 1. Then there exists the fundamental system of solutions

(6)  y_{1}(ν), …, y_{n}(ν)

of the equation (1) such that, if

h(V) = max(|v_{1}|, …, |v_{n}|), y(V, ν) = ∑_{k=1}^{n} v_{k}y_{k}(ν),

k_{1}(V) = inf{k ∈ [1, n] ∩ ℕ: v_{k} ≠ 0}, k_{2}(V) = sup{k ∈ [1, n] ∩ ℕ: v_{k} ≠ 0},

where V ∈ ℂ^{n}⧹{0}, V = ( v_{1} ; ⋮ ; v_{n} ) , then the following conditions ar fulfilled:

(I) for any ε ∈ (0, 1) and any V ∈ ℂ^{n}⧹{0} there exists γ_{3}(V, ε) > 0 such that

γ_{3}(V, ε)(|λ_{k_{1}(V)}|(1 − ε))^{ν} ≤ ∑_{κ=0}^{n−1} |y(V, ν + κ)|,

where ν ∈ ℕ − 1; (II) for any ε ∈ (0, 1) there exist γ_{4}(ε) > 0 and γ_{5}(ε) > 0 such that

γ_{4}(ε)h(V)(|λ_{k_{2}(V)}|(1 − ε))^{ν} ≤ ∑_{κ=0}^{n−1} |y(V, ν + κ)|,

|y(V, ν)| ≤ γ_{5}(ε)h(V)(|λ_{k_{1}(V)}| + ε))^{ν},

where ν ∈ ℕ − 1, V ∈ ℂ^{n}.

Theorem 3. Let are fulfilled all the conditions of the Theorem 2. Let (6) is the declared in the Theorem 2 the fundamental system of solutions of the equation (1). Let y_{1}^{∼}(ν), …, y_{r}^{∼}(ν) be some linear independent over ℂ system of solutions

of the equation (1) and let y_{l}^{∼}(ν) = ∑_{k=1}^{n} η_{k,l}y_{k}(ν), where η_{k,l} ∈ ℂ, k = 1, …, n; l =

1, …, r; ν ∈ ℕ − 1. Let

𝔎 = {k ∈ [1, n] ∩ ℕ: {l ∈ [1, r] ∩ ℕ: η_{k,l} ≠ 0} ≠ ∅},

and let k_{3} = inf(𝔎), k_{4} = sup(𝔎). For any X ∈ ℂ^{r}, X = ( x_{1} ; ⋮ ; x_{r} ) , let

y^{∼}(X, ν) = x_{1}y_{1}^{∼}(ν) + … + x_{r}y_{r}^{∼}(ν), h(V) = max(|v_{1}|, …, |v_{n}|).

Then for any ε ∈ (0, 1) there exist γ_{6}(ε) > 0 and γ_{7}(ε) > 0 such that

γ_{6}(ε)h(X)(|λ_{k_{4}}h(X)|(1 − ε))^{ν} ≤ ∑_{κ=0}^{n−1} |y^{∼}(X, ν + κ)|,

|y^{∼}(X, ν)| ≤ γ_{7}(ε)h(X)(|λ_{k_{3}}| + ε))^{ν},

where ν ∈ ℕ − 1, X ∈ ℂ^{r}.

The following Theorem [8] may be considered as development of the above Theorem 2 and Theorem 3.

Theorem 4. Let s ∈ ℕ − 1, n ∈ ℕ,

a_{i}^{∼} ∈ ℂ, a_{i}(ν) ∈ ℂ, a_{n}(ν) = 1, a_{i}(ν) − a_{i}^{∼} = O(1/ν)

for ν ∈ ℕ and 0 ≤ i ≤ n. For m ∈ ℕ let V_{m} denotes linear over ℂ space of solutions y = y(ν) of the equation

(7)  ∑_{k=0}^{n} a_{k}(ν)y(ν + k) = 0,

where ν ∈ m + ℕ − 1. Let the absolute values of all the roots of the polynomial a_{0}^{∼}z^{0} + … + a_{n}^{∼}z^{n} are among the numbers {ρ_{i}: 1 ≤ i ≤ 1 + s}, such that ρ_{s+1} = 0 and ρ_{j} < ρ_{i} for 1 ≤ i < j ≤ s + 1. Let e_{i} and k_{i} denote respectively the sum of and the maximum of the multiplicities of those roots, whose absolute value is equal to ρ_{i}, 1 ≤ i ≤ 1 + s. For given y = y(ν) in V_{m}, let ω_{n,y}(ν) = max(|y(ν)| , …, |y(ν + n − 1)|). Then there exist A > 0, m ∈ ℕ, and the subspaces V_{m,1}^{∨}, …, V_{m,s+1}^{∨} such that

V_{m} = V_{m,1}^{∨} ⊕ …, ⊕ V_{m,s+1}^{∨}, dim_{ℂ}(V_{m,i}^{∨}) = e_{i}, 1 ≤ i ≤ s + 1.

and, moreover, the spaces

V_{m,j}^{∧} = V_{m,j}^{∨} ⊕ … ⊕ V_{m,s+1}^{∨}, 1 ≤ j ≤ s + 1

have the folloving property:

if y ∈ V_{m,θ}^{∧}\V_{m,θ+1}^{∧} for some θ ∈ {1, ..., s}, then

exp(−A(ln(ν) + ν^{1−1/k_{θ}}))β ≤ ω_{n,y}(ν)(ρ_{θ})^{−ν} ≤ exp(A(ln(ν) + ν^{1−1/k_{θ}}))ω_{n,y}(m),

where β = β(y) > 0 depends only from equation and y (but does not depend from ν); if k^{∗} := k_{s+1} > 0 and y ∈ V_{m,s+1}^{∨}(= V_{m,s+1}^{∧}), then

|y(ν)| ≤ (A/ν)^{ν/k^{∗}}ω_{n,y}(m).

The main part of Perron’s proof of his Theorem is the following

Perron’s decomposition Lemma ([1], Hilfsatz 3). Let

(8)  T (z) = T_{1}(z)T_{2}(z),

where T (z) is characteristic polynomial (3),

(9)  T_{1}(z) = ∑_{α=0}^{p} b_{α}^{∼}z^{α}, T_{2}(z) = ∑_{β=0}^{q} u_{β}^{∼}z^{β},

b_{p}^{∼} = u_{q}^{∼} = a_{n}^{∼} = 1 and absolute value of each root of T_{1}(z) is greater than the absolute value of each root of T_{2}(z).

Then there exist m ∈ ℕ, b_{α}(ν) ∈ ℂ, α = 0, …, p, ν ∈ ℕ + m − 1,

and u_{β}(ν) ∈ ℂ, β = 0, …, q, ν ∈ ℕ + m − 1 such that

(10)  lim_{ν→∞} b_{α}(ν) = b_{α}^{∼}, α = 0, …, p, b_{p}(ν) = 1, b_{0}(ν) ≠ 0,

(11)  lim_{ν→∞} u_{β}(ν) = u_{β}^{∼}, β = 0, …, q, u_{q}(ν) = 1,

where ν ∈ ℕ + m − 1, and the connected with the equation (1) the equation

(12)  ∑_{k=0}^{n} a_{k}(ν)y(ν + k) = 0,

where ν ∈ ℕ − 1 + m, is equivalent to the equation

∑_{α=0}^{p} b_{α}(ν)y(ν + α) = r(ν),

where ν ∈ ℕ − 1 + m and r(ν) satisfies to the equation

∑_{β=0}^{q} u_{β}(ν)r(ν + β) = 0

with ν ∈ ℕ−1+m. The following supplement to the Perron’s decomposition Lemma was proved in [7].

Theorem 5. Let the functin ξ(x) is defined on [0, +∞), let ξ(x) > 0 for x ∈ [0, +∞), let ξ(x) decreases together with increasing of x ∈ [0, +∞) and let

lim_{x→∞}(ξ(x)) = 0. Let

(13)  ξ(x/2) = O(ξ(x)),

when x → ∞,

(14)  lim_{x→0}(log(ξ(x)))/x = 0.

Let further all the conditions of the Perron’s decomposition Lemma are fulfilled and together with (2) the more strong condition holds: a_{k}(ν) − a_{k}^{∼} = O(ξ(ν)), k = 0, …, n, when ν → ∞. Then for b_{α}(ν), α = 0, …, p, and u_{β}(ν) , β = 0, …, q, from the assertion of the Perron’s decomposition Lemma the following conditions are fulfilled (cf. with (10) and (11)):

b_{α}(ν) − b_{α}^{∼} = O(ξ(ν)), α = 0, …, p, b_{0}(ν) = 1, b_{q}(ν) ≠ 0,
u_{β}(ν) − u_{β}^{∼} = O(ξ(ν)), β = 0, …, q, u_{0}(ν) = 1,

My aim is to present the proofs of the theorems 2, 3, and 4. Here I prove the Theorem 5.

Acknowledgments. I express my deepest thanks for help and support of my work to Professors B.Z.Moroz,I.I.Piatetski-Shapiro,A.G.Aleksandrov,P.Bundshuh and S.G.Gindikin.

§1. Difference operators.

Let m ∈ ℕ − 1 and 𝔙_{m} = ℂ^{ℕ−1+m} be the algebra over ℂ of all the maps of ℕ − 1 + m into ℂ; let 1_{m} and 0_{m} are respectively the unity element and the zero element of this algebra. Let R = 𝔙_{m}, . For each f ∈ R let 𝔪_{f} be the operator on R, which turns each a ∈ R into fa. Let 𝔐_{R} = {𝔪_{f}: f ∈ R}. Then 𝔐_{R} as the ring consisting of all the R− endomorphisms of the commutative ring R with the unity element is isomorph to R. Let 𝔐_{R}^{∨} is the ring of all the ℂ−endomorphisms of the additive group of the ℂ−algebra R. If ϕ ∈ 𝔐_{R}^{∨} and ψ ∈ 𝔐_{R}^{∨}, then ϕ ∘ ψ denotes the element in 𝔐_{R}^{∨} such that (ϕ ∘ ψ)a = ϕ(ψa) for any a ∈ R. On R = 𝔙_{m} acts the ℂ−linear operator ▽, which tansforms each y = y(ν) ∈ 𝔙_{m} into ▽y such that (▽y)(ν) = y(ν + 1) for all ν ∈ ℕ − 1 + m. Let R < ▽ > be the subring of the ring 𝔐_{R}^{∨}, generated by ▽ and 𝔐_{R}. Since ▽ ∘ 𝔪_{f} = 𝔪_{▽f} ∘ ▽, it follows

that each D ∈ R < ▽ > may be represented in the form D = ∑_{k=0}^{n(D)} 𝔪_{𝔠_{k}(D)} ∘ ▽^{k},

with n(D) ∈ ℕ − 1, ▽^{0} = 𝔪_{1_{R}} and 𝔠_{k}(D) ∈ R for k = 0 , …, n(D); in particular, ▽^{k} = 𝔪_{1_{R}}▽^{k} for any k ∈ ℕ − 1; if we put 𝔠_{k}(D) = 0 for k ∈ n(D) + ℕ, then

(15)  D = ∑_{k=0}^{∞} 𝔪_{𝔠_{k}(D)} ∘ ▽^{k}.

Let R|_{n} < ▽ >= {D ∈ R < ▽ >: (k ∈ ℕ + n) ⇒ (𝔠_{k}(D) = 0)}, where n ∈ ℕ − 1. Each element D ∈ R|_{n} < ▽ > may be represented in the form

(16)  D = ∑_{k=0}^{n} 𝔪_{𝔠_{k}(D)} ∘ ▽^{k},

Clearly,

(17)  D = ∑_{k=0}^{n} 𝔪_{𝔞_{k}(D)} ∘ (△)^{k},

where D ∈ R|_{n} < ▽ >, △ = ▽−𝔪_{1_{R}}, △^{0} = 𝔪_{1_{R}} and 𝔞_{k}(D) ∈ R for k = 0 , …, n; the action of D on 1, ν , …, ν^{n} shows that 𝔞_{k}(D) in (17) and

𝔠_{k}(D) = ∑_{κ=k}^{n} \binom{κ}{k}𝔞_{κ}(D)

in (16) for k = 0 , …, n are defined uniquely and are the ℂ−linear maps of the space R|_{n} < ▽ > into R. We name by difference polynomials the elements of R < ▽ > . Let

(18)  ϕD = ∑_{k=0}^{∞} 𝔪_{ϕ𝔠_{k}(D)} ∘ ▽^{k},

where ϕ ∈ 𝔐_{R}^{∨}, D ∈ R < ▽ >; in particular, D_{1}D = ∑_{k=0}^{∞} 𝔪_{D_{1}𝔠_{k}(D)} ∘ ▽^{k}, where

D ∈ R < ▽ >, D_{1} ∈ R < ▽ > . Clearly, D_{1}D not always coincides with D_{1} ∘ D. The equality (18) determines on R < ▽ > the structure of the left 𝔐_{R}^{∨}-module. Clearly,

(19)  ▽^{s} ∘ D = (▽^{s}D) ∘ ▽^{s},

for any D ∈ R < ▽ > and s ∈ ℕ − 1. For each D ∈ 𝔙_{m} < ▽ > the elements 𝔠_{k}(D) are defined only for k ∈ ℕ − 1; let

(20)  𝔠_{k}(D) = 0_{m}

for k ∈ −ℕ. Let D_{1} ∈ 𝔙_{m}|_{p} < ▽ > and D_{2} ∈ 𝔙_{m}|_{q} < ▽ > . In view of (19) and (20),

D_{2} ∘ D_{1} = ∑_{s=0}^{q} 𝔪_{𝔠_{s}(D_{2})} ∘ ▽^{s} ∘ D_{1} = ∑_{s=0}^{q} 𝔪_{𝔠_{s}(D_{2})} ∘ (▽^{s}D) ∘ ▽^{s} =

∑_{s=0}^{q} ∑_{k=0}^{p} 𝔪_{𝔠_{s}(D_{2})} ∘ 𝔪_{▽^{s}𝔠_{k}(D_{1})} ∘ ▽^{k+s} = ∑_{s=0}^{q} ∑_{k=0}^{p} 𝔪_{𝔠_{s}(D_{2})▽^{s}𝔠_{k}(D_{1})} ∘ ▽^{k+s} =

∑_{κ=0}^{p+q} ∑_{s=0}^{q} 𝔪_{𝔠_{s}(D_{2})▽^{s}𝔠_{κ−s(D_{1})}} ∘ ▽^{κ}.

Therefore, for D = D_{2} ∘ D_{1}, we have

(21)  𝔠_{κ}(D_{2} ∘ D_{1}) = ∑_{s=0}^{q} 𝔠_{s}(D_{2}) ▽^{s} 𝔠_{κ−s}(D_{1}),

where κ = 0, …, p + q. In particular,

(22)  𝔠_{κ}(D_{2} ∘ D_{1}) = 𝔠_{q}(D_{2}) ▽^{q} 𝔠_{p}(D_{1}),
𝔠_{0}(D_{2} ∘ D_{1}) = 𝔠_{0}(D_{2})𝔠_{0}(D_{1}).

If

(23)  𝔠_{p}(D_{1}) = 1_{m}, 𝔠_{q}(D_{2}) = 1_{m},

then, in view of (22), 𝔠_{p+q}(D_{2} ∘ D_{1}) = 1_{m} . If (23) is fulfilled, then

(24)  𝔠_{κ}(D_{2} ∘ D_{1}) = ▽^{q}𝔠_{κ−q}(D_{1}) + ∑_{s=0}^{q−1} 𝔠_{s}(D_{2}) ▽^{s} 𝔠_{κ−s}(D_{1}),

where κ ∈ ℤ. Let

(25)  a_{k} = 𝔠_{k}(D_{2} ∘ D_{1}), u_{k} = 𝔠_{k}(D_{2}), b_{k} = 𝔠_{k}(D_{1}),

where k ∈ ℤ. Then (24) may be rewritten in the form

(26)  a_{κ} = ▽^{q}b_{κ−q} + ∑_{s=0}^{q−1} u_{s} ▽^{s} b_{κ−s},

where κ ∈ ℤ. In view of (25),

(27)  a_{κ} = ∑_{s=0}^{q−1} u_{s} ▽^{s} b_{κ−s},

where κ = 0, …, q − 1. Let K is an arbirary ring, p ∈ ℕ, q ∈ ℕ, n = p + q; for any X ∈ K^{n} let π_{1}(X) be the element in K^{p}, which have for i = 1, …, q as its i−th coordinate the i-coordinate of X and let π_{2}(X) be the element in K^{p}, which have for i = 1, …, p as its i−th coordinate the q + i-coordinate of X. Let B^{∧} be the q × q-matrix, in which ▽^{k−1}b_{i−k} stands on the intersection of i-th row and k-th columns for i = 1, …, q and k = 1, …, q.Clearly B^{∧} is the lower triangle matrix in Mat_{q}(𝔙_{m}). Let U be the column with q elements and u_{k−1} for k = 1, …, q is k-th of these elements. Let B be the column with p elements and b_{k−1} for k = 1, …, p is k-th of these elements. Let A be the column with n elements and a_{k−1} for k = 1, …, n is k-th of these elements. We see that equalities (27) may be rewritten in the form

(28)  π_{1}(A) = B^{∧}U.

Taking in (26) κ = q + μ with μ = 0, …, p − 1 we obtain the equalities

(29)  a_{q+μ} = ▽^{q}b_{μ} + ∑_{s=0}^{q−1} u_{s} ▽^{s} b_{q+μ−s}.

Therefore ▽^{q}b_{i−1} = a_{q+i−1} − ∑_{k=1}^{q} u_{k−1} ▽^{k−1} b_{q+i−k}, where i = 1, …, p. Let 𝔅 be

the p × q-matrix,in which ▽^{k−1}b_{q+i−k} stands on the intersection of i-th row and k-th columns for i = 1, …, p and k = 1, …, q. Clearly, (29) may be rewritten in the form

(30)  ▽^{q}B = π_{2}(A) − 𝔅U.

Let m ∈ ℕ−1 and let s_{m} be the map of 𝔙_{0} into 𝔙_{m}, for which (s_{m}(y))(ν) = y(ν−m) for any ν ∈ ℕ − 1 + m. Clearly, s_{m} is the ℂ-isomorphism of ℂ-algebra 𝔙_{0} onto ℂ-algebra 𝔙_{m}. Let m ∈ ℕ − 1 and let r_{m} be the map of 𝔙_{0} into 𝔙_{m}, for which r_{m}(y) is the restriction of y ∈ 𝔙_{0} onto ℕ − 1 + m. Clearly, s_{m}▽^{m} = r_{m} for each

m ∈ ℕ − 1. Let r_{m}(D) = ∑_{k=0}^{∞} 𝔪_{r_{m}(𝔠_{k}(D))} ∘ ▽^{k}, where m ∈ ℕ − 1, D ∈ 𝔙_{0} < ▽ >;

clearly, r_{m}(𝔙_{0} < ▽ >) = 𝔙_{m} < ▽ > . Let D ∈ (𝔙_{0})|_{n} < ▽ >, and a_{k} = 𝔠_{k}(D)

for k = 0, …, n. Then D = ∑_{k=0}^{∞} 𝔪_{a_{k}} ∘ ▽^{k}, r_{m}(D) = ∑_{k=0}^{∞} 𝔪_{r_{m}(a_{k})} ∘ ▽^{k}, where

m ∈ ℕ − 1, r_{m}(D) is linear operator on 𝔙_{m} and Ker (r_{m}(D)) for any m ∈ ℕ − 1 coincides with the space V_{m} of solutions of the equation (7).

Let the functin ξ(x) is defined on [0, +∞) and let m ∈ ℕ − 1; we denote by τ_{m}(ξ) the restriction of ξ on m + ℕ − 1. We denote further by l_{m,∞} the set of all the bounded functions from 𝔙_{m}, by l_{m}^{∼} - the set all the functions y = y(ν) ∈ 𝔙_{m}, for which there exists lim_{ν→∞} y(ν).

If the functin ξ = ξ(x) is defined on [0, +∞), and m ∈ ℕ − 1, let

l_{m,ξ} = τ_{m}(ξ)l_{m,∞} = {τ_{m}(ξ)y: y ∈ l_{m,∞}}.

For each m ∈ ℕ−1, the field ℂ is embeded into 𝔙_{m} in the form ℂ 1_{m} . If S is some nonempty subset of the ring 𝔙_{m}, let S < ▽ > (respectively S|_{n} < ▽ >) be the set of all the D ∈ 𝔙_{m} < ▽ > (respectively D ∈ (𝔙_{m})|_{n} < ▽ >), for which 𝔠_{k}(D) ∈ S for all the k ∈ ℕ − 1. Clearly, if S is ℂ−subalgebra of 𝔙_{m} and < ▽ > turns S into S, then S < ▽ > is ℂ−subalgebra of 𝔙_{m} < ▽ >; in particular,(ℂ 1_{m}) < ▽ > is ℂ−subalgebra of 𝔙_{m} < ▽ > . On l_{m}^{∼} there exists the ℂ-linear functional lim, for which lim(y) = lim_{ν→∞} y(ν) for any y ∈ l_{m}^{∼}; the functional lim may be uniquely continued to linear map of l_{m}^{∼} < ▽ > onto (ℂ1_{m}) < ▽ > by means of the equality

lim(D) = ∑_{k=0}^{∞} 𝔪_{(lim(𝔠_{k}(D))) 1_{m}} ∘ ▽^{k}.

For each D ∈ l_{m}^{∼} < ▽ > we name the polynomial

T_{D}(z) = ∑_{k=0}^{∞} lim(𝔠_{k}(D))z^{k} ∈ ℂ[z]

by charactristic polynomial of the operator D.

§2. Reformulation of the problem.

Reformulation of the Perron’s decomposition Lemma. Let D ∈ l_{0}^{∼}|_{n} < ▽ > and 𝔠_{n}(D) = 1_{0} . Let the polynomial T_{D}(z) coinsides with the polynomial T (z) in (8). Let, as before, the equality (8) holds wih T_{1}(z) and T_{2}(z) defined in (9). Let absolute value of each root of the polynomial T_{1}(z) is greater than the absolute value of each root of the polynomial T_{2}(z). Then there exist m ∈ ℕ − 1, D_{1} ∈ l_{m}^{∼}|_{p} < ▽ > and D ∈ l_{m}^{∼}|_{q} < ▽ > such that 𝔠_{p}(D_{1}) = 1_{m}, 𝔠_{q}(D_{2}) = 1_{m}, T_{D_{1}}(z) = T_{1}(z), T_{D_{2}}(z) = T_{2}(z) and

(31)  r_{m}(D) = D_{2} ∘ D_{1}.

We reformulate the Theorem 5 also.

Reformulation of the Theorem 5. Let ξ(x) satisfies to all the conditions of the Theorem 5. Let all the conditions of the reformulated Perron’s decomposition Lemma are fulfilled and together with (2) the more strong condition holds: D − lim(D) ∈ l_{0,ξ} < ▽ >= 𝔪_{τ_{0}(ξ)}l_{0,∞} < ▽ > Then for D_{1} and D_{2} in (31) the following conditions hold:

D_{k} − lim(D) ∈ l_{m,ξ} < ▽ >= 𝔪_{τ_{m}(ξ)}l_{m,∞} < ▽ >,

where k = 1, 2.

§3. The general plan of the proof of the Theorem 5.

For given a_{k} = a_{k}(ν) in (25) we prove that u_{k} = u_{k}(ν) and b_{k} = b_{k}(ν) in (25) are determined by means some succesive procedure together with increasing of ν. More precisely, if for given ν ∈ ℕ − 1 are known b_{i−1}(ν + s) for i = 0, …, p and s = 0, …, q − 1 and the matrix B^{∧}(ν) for this ν is invertible (we remark that B^{∧} is the map of ℕ + m − 1 into Mat_{q}(ℂ) and therefore Mat_{q}(ℂ) must contain B^{∧}(ν) for each given ν ∈ ℕ + m − 1), then U(ν) is determined by means of the equality

(32)  U(ν) = (B^{∧}(ν))^{−1}π_{1}(A(ν),

which follows from (28). But when is known U(ν), the value b_{i−1}(ν + q) for i = 1, …, p is determined by means of the equality

(33)  (▽^{q}B)(ν) = π_{2}(A(ν)) − 𝔅(ν)U(ν),

which follows from (30). We show that, if for some ν = m ∈ ℕ − 1 the values b_{i−1}(ν + s) for i = 0, …, p and s = 0, …, q − 1 are taken sufficiently near to b_{i−1}^{∼} for i = 1, …, p, then this passage from b_{i−1}(ν + s) for i = 1, …, p and s = 0, …, q − 1 to b_{i−1}(ν + 1 + s) for i = 1, …, p and s = 1, …, q is infinite. Moreover, for specially constructed norm on ℂ^{p} in this situation acts some kind of the fixed point theorem, which determines the reqired speed of convergence of b_{i−1}(ν) for i = 1, …, p to coordinates of this fixed point, i.e. to b_{i−1}^{∼}, where i = 1, …, p.

§4. The polynomials with coefficients in a linear space.

Let x be a variable and let L be a linear space over the field K. The expres-

sion P = P (x) = ∑_{k=0}^{n} x^{k}a_{k}, where n ∈ ℕ − 1, a_{k} ∈ L, k = 0, …, n, I name by

polynomial on x with coefficients in L; all the coefficients I write from the right of degrees of variable x; I name P by the zero polynomial, if a_{k} = 0_{L}, k = 0, …, n; if a_{n} ≠ 0_{L}, then I name the number n by degree of the polynomial P and I denote

the degree of the polynomial P by deg(P). If α ∈ K then ∑_{k=0}^{n} α^{k}a_{k} ∈ L; I name this

element in L by value of the polynom P at the point x = α and denote by P (α). For given P ∈ L[x] I shall say that the map

(34)  f = f_{P} : α → P (α)

of K into L is defined by P; the connected with (34) the map

(35)  P → f_{P}

of the set L[x] into L^{K} has the following property:

if deg(P) ≤ n and the field K contains at list n + 1 elements, then the layer over the image of the element P by the map (35) consists of single element P. The set L[x] has the natural structure of a left K[x]−module; moreover L[x] is a graduated K[x]−module, in which homogeneous cpomponents of nonnegative degree are isomorph to L and homogeneous cpomponents of negative degrees consit of the single element 0; in particular, there is a natural algorithm of division of P ∈ L[x] by Q ∈ K[x] with the remainder, and therefore the equality P (α) = 0 is fulfilled if and only if P (x) ∈ (x − α)L[x]. If deg(P) ≤ n, then as it was mentioned above, the coefficients of P are uniqely determined by the values of P in the n + 1 mutually distinct points x_{0}, …, x_{n} and holds the formula of Lagrange:

P (x) = ∑_{k=0}^{n} \frac{T_{k}(x)}{T_{k}(x_{k})}P (x_{k}),

where T_{k}(x) ∈ K[x] is defined by means the equality

(x − x_{k})T_{k}(x) = ∏_{i=0}^{n}(x − x_{i}).

If P [x_{k}] = 0_{L} for k = 1, …, n, then P (x) = ((T_{0}(x))/T_{0}(x_{k}))P (x_{k}). Reasonings similar to those, which done in [5], volume 2, chapter 10, section 4, show that for any f ∈ L^{K} and for any sequence x_{1}, …, x_{n}, x of mutually distinct elements holds the equality

(36)  f(x) = ∑_{k=0}^{n} ((∏_{κ=0}^{k−1}(x − x_{κ})) f_{k−1}(x_{1}, …, x_{k})) +

(∏ −i = 0^{n}(x − x_{i})) f_{n}(x_{1}, …, x_{n}, x)

with

(37)  f_{0}(x) = f(x),

where x ∈ K,

(38)  f_{1}(x_{1}, x) = (x − x_{1})^{−1}(f_{0}(x) − f_{0}(x_{1})),

where x ∈ K⧹x_{1},

(39)  f_{2}(x_{1}, x_{2}, x) = (x − x_{2})^{−1}(f_{0}(x_{1}, x) − f_{0}(x_{1}, x_{2})),

where x ∈ K⧹{x_{1}, x_{2}},

(40)  f_{k}(x_{1}, …, x_{k−1}, x_{k}, x) = (x − x_{k})^{−1}×
(f_{1}(x_{1}, …, x_{k−1}, x) − f_{1}(x_{1}, …, x_{k−1}, x_{k}))

where x ∈ K⧹{x_{1}, …, x_{k}}.

Remark 4.1. All this is almost trivial. There exists in L a Hamel’s basis e_{i}, where i runs through a some set I of indexes; L is embeded as subspace of of all the finite functions in the linear space L^{∨} = K^{I} over K. Each P ∈ L^{∨}[x] with deg(P) ≤ n may be considered as system P = {P_{i}(x): i ∈ I}, where P_{i}(x) ∈ K[x], deg P_{i}(x) ≤ n, where i ∈ I and for L = K the formula (36) is well known (see, for example, [5], volume 2, chapter 10, section 4). Clearly, L[x] may be regarded also as K[x] ⊗_{K} L.

We show below that, when f ∈ L^{K} is defined by means of the polynomial P ∈ L[x] which is a zero polynomial or have deg(P) ≤ n, the formula (36) may be spread on the arbitrary sequences

(41)  x_{1}, …, x_{n}, x

consisting of not necessary mutually distinct elements in K.

Let k ∈ ℕ, s ∈ ℤ. Let K_{0} be the ring of all the polynomials over K of countable set of variables

(42)  u_{l},

where l ∈ ℕ and let K^{∧} be the quotient field of the ring K_{0};let ϕ_{s}(u_{1}) = u_{1}^{s} for s ∈ ℕ − 1; let ϕ_{s}(u_{1}, …, u_{k}) for k ≥ 2 be the homogenous form of degree s of

the expansion of the fraction 1/ ∏_{κ=1}^{k}(1 − u_{κ}) in the formal multiple power series

relatively variables u_{1}, …, u_{k}; so

(43)  ϕ_{s}(u_{1}, …, u_{k}) = 0,

if k < 0;

(44)  ϕ_{s}(u_{1}, …, u_{k}) = 1,

if k = 0;

(45)  ϕ_{s}(u_{1}, …, u_{k}) = ∑_{\begin{matrix}α_{1}∈ℕ−1, ... , α_{1}∈ℕ−1 \\ ∑_{κ=1}^{k} α_{κ}=s, \end{matrix}} (∏_{κ=1}^{k} u_{κ}^{α_{κ}})

for k ∈ ℕ. In view of the equality

(u_{k} − u_{k−1}) ∏_{κ=1}^{k}(1 − u_{κ})^{−1} = ((1 − u_{k})^{−1} − (1 − u_{k−1})^{−1}) ∏_{κ=1}^{k}(1 − u_{κ}),

(46)  ϕ_{s−1}(u_{1}, u_{2})(u_{2} − u_{1}) = ϕ_{s}(u_{2}) − ϕ_{s}(u_{1}),

(47)  ϕ_{s−1}(u_{1}, u_{2}, u_{3}(u_{3} − u_{2}) = ϕ_{s}(u_{1}, u_{3}) − ϕ_{s}(u_{1}, u_{2}),

(48)

(u_{k} − u_{k−1})ϕ_{s−1}(u_{1}, …, u_{k−2}, u_{k−1}, u_{k}) = ∑_{h=0}^{1}(−1)^{h}ϕ_{s}(u_{1}, …, u_{k−2}, u_{k−h}),

where s ∈ ℕ. Let L_{0} = K_{0} ⊗_{K} L, n ∈ ℕ − 1, k ∈ ℕ, a_{κ} ∈ L, κ = 0, …, n

(49)  P = P (x) = ∑_{κ=0}^{n} x^{κ}a_{κ}.

Let

(50)  ▽(P |u_{1}, …, u_{k}) = ∑_{κ=0}^{n} ϕ_{κ−k+1}(u_{1}, …, u_{k}) ⊗_{K} a_{κ}.

Then for each specialisation u_{1} = α_{1} ∈ K, …, u_{k} = α_{k} ∈ K holds the equality

△(P |α_{1}, …, α_{k}) = ∑_{κ=0}^{n} ϕ_{κ−k+1}(α_{1}, …, α_{k}) ⊗_{K} a_{κ}.

Lemma 4.1. For the sequence (42) we have

(51)  △(P |u_{1}) = P (u_{1}), △(P |u_{1}, u_{2})(u_{2} − u_{1}) = (P |u_{2}) − (P |u_{1}),

(52)  (u_{k} − u_{k−1}) △ (P |u_{1}, …, u_{k−2}, u_{k−1}, u_{k}) =

∑_{h=0}^{1}(−1)^{h} △ (P |u_{1}, …, u_{k−2}, u_{k−h}).

Proof. The assertion of the Lemma follows from (46)-(50). ■

Corollary. If the sequence (41) consists of mutually distinct elements of the field K and P ∈ L[x], then

P_{0}(x) = △(P |x), P_{1}(x_{1}, x) = △(P |x_{1}, x),
P_{n−1}(x_{1}, …, x_{n−1}, x) = △(P |x_{1}, …, x_{n−1}, x),

where n ∈ ℕ + 1.

Proof. The assertion of the Corollary follows from (37)-(40) and Lemma 4.1. ■

Lemma 4.2. For the sequence (42) and P ∈ L[x] we have

(53)  P (u_{k}) = P (u_{1}) + ∑_{κ=2}^{k} (∏_{i=1}^{κ−1}(u_{k} − u_{i})) △ (P |u_{1}, …, u_{κ}),

where k ∈ 1 + ℕ.

Proof. For k − 1 = 1 the assertion of the Lemma follows from (51). Let k −1 ∈ 1+ℕ and assertion of the Lemma holds for k −1−1. Then for the sequence {u_{κ}: κ ∈ [1, k − 2] ∩ ℕ}, u_{k} we have the equality

(54)  P (u_{k}) = P (u_{1}) + (∑_{κ=2}^{k−2} (∏_{i=1}^{κ−1}(u_{k} − u_{i})) △ (P |u_{1}, …, u_{κ})) +

(∏_{i=1}^{k−2}(u_{k} − u_{i})) △ (P |u_{1}, …, u_{k−2}, u_{k}).

In view of (54) and (52),

P (u_{k}) = P (u_{1}) + (∑_{κ=2}^{k−2} (∏_{i=1}^{κ−1}(u_{k} − u_{i})) △ (P |u_{1}, …, u_{κ})) +

(∏_{i=1}^{k−2}(u_{k} − u_{i})) ×

(△(P |u_{1}, …, u_{k−2}, u_{k−1}) + (u_{k} − u_{k−1}) △ (P |u_{1}, …, u_{k−2}, u_{k−1}, u_{k})) .

The right part of the last equality coincides wih the right part of the equality (53). ■

So, when the function f is represented by means of P ∈ L[x], the mentioned above spreading of (36) for arbitrary sequence (41) is fulfilled. Let a^{0} = 1_{K}, where a ∈ K (in particular, a = 0_{K} .) Let n ∈ ℕ. Let 𝔨 be a field contained the field K and and contained elements

(55)  z_{1}, …, z_{n}

algebraically independent over K. For the elements (55) we define σ_{k,l} for k ∈ ℤ and l ∈ [1, n] ∩ ℕ and as follows:

σ_{k,l} = 0, if k ∈ −ℕ ∪ (l + ℕ), let σ_{0,l} = 1, and let σ_{k,l} = σ_{k,l}(z_{1}, …, z_{l}) is k-th elementar symmetric function of z_{i} with i ∈ [1, l] ∩ ℕ, if k ∈ [1, l] ∩ ℕ. Let n ∈ ℕ and for κ ∈ ℤ let a_{κ} = (−1)^{n−κ}σ_{n−κ,n}. Let us consider the polynomial

(56)  z^{n} + ∑_{κ=0}^{n} a_{κ}z^{κ}.

The sequence (55) consists of all the zeros of the polynomial (56). Let

(57)  A = A(z_{1}, …, z_{n}) = ( \begin{matrix}0 & 1 & 0 & … & 0 \\ 0 & 0 & 1 & … & 0 \\ ⋮ & ⋮ & ⋮ & … & ⋮ \\ 0 & 0 & 0 & … & 1 \\ −a_{0} & −a_{1} & −a_{2} & … & −a_{n−1} \end{matrix} ) .

For i = 1, …, n and k = 1, …, n let

(58)  c_{i,k} = ϕ_{i−k}(z_{1}, …, z_{n}),
d_{i,k} = (−1)^{i−k}σ_{i−k,i−1};

let C = C(z_{1}, …, z_{n}) and D = D(z_{1}, …, z_{n}) are n×n-matrices, in which respectively c_{i,k} and d_{i,k} stand on the intersection of i-th row and k−th column. Clearly, C and D are a lower triangle matrices, all the diagonal elements of which are equal to 1.

Lemma 4.3. The following equality holds:

(59)  D = C^{−1}.

Proof. Applying the Lemma 4.2 to the polinomial P (z) = z^{i−1} and the sequence z_{1}, …, z_{i−1}, z, we see that

z^{i−1} = z_{1}^{i−1} + ∑_{k=2}^{i−1} (∏_{κ=1}^{k−1}(z − z_{κ})) △ (z^{i−1}|z_{1}, …, z_{k})+

∏_{κ=1}^{i−1}(z − z_{κ}) △ (z^{i−1}|z_{1}, …, z_{i−1}, z).

Comparing coefficients by z^{l−1}, l ∈ [1, n] ∩ ℕ we see that assertion of the Lemma 4.3 is true. ■

Let L = K^{n}, let i = 1, …, n and e_{i} is the i−th column of the unit n × n-matrix

and let P_{0} = P_{0}(z) = ∑_{i=1}^{n} z^{i−1}e_{i}. Then P_{0}(z) ∈ L(z).

Lemma 4.4. Let g_{k} = ▽(P_{0}|z_{1}, …, z_{k}), where k = 1, …, n; then g_{k} coincides with k−th column of matrix C.

Proof. The assertion of the Lemma follows from (50) ■

Lemma 4.5. For the matrix A in (57) the following equalities hold :

(60)  Ag_{1} = z_{1}g_{1},

(61)  Ag_{k} = z_{k}g_{k} + g_{k−1},

where k ∈ ℕ ∩ [2, n].

Proof. Since z_{1} is the zero of polynomial (56), the equality (60) may be verfied directly. Since the elements (55) are mutually distinct, itfollows that

Ag_{2} = A(z_{2} − z_{1})^{−1}(P_{0}(z_{2}) − P_{0}(z_{1})) =
(z_{2} − z_{1})^{−1}(z_{2}P_{0}(z_{2}) − z_{1}P_{0}(z_{1})) =
z_{2} ▽ (P_{0}|z_{1}, z_{2}) + P_{0}(z_{1}) = z_{2}g_{2} + g_{1}.

Let k ∈ [3, n] ∩ ℕ and (61) is true for k − 1. Then

Ag_{k} = A(z_{k} − z_{k−1})^{−1}(▽(P_{0}|z_{1}, …, z_{k−2}, z_{k}) − ▽(P_{0}|z_{1}, …, z_{k−2}, z_{k−1}) =

(z_{k} − z_{k−1})^{−1}(z_{k} ▽ (P_{0}|z_{1}, …, z_{k−2}, z_{k}) − z_{k−1} ▽ (P_{0}|z_{1}, …, z_{k−2}, z_{k−1})) =

z_{k} ▽ (P_{0}|z_{1}, …, z_{k−2}, z_{k−1}, z_{k}) + P_{0}|z_{1}, …, z_{k−2}, z_{k−1}) = z_{k}g_{k} + g_{k−1}.■

Let

S = S(z_{1}, …, z_{n}) = ( \begin{matrix}z_{1} & 1 & 0 & … & 0 \\ 0 & z_{2} & 1 & … & 0 \\ ⋮ & ⋮ & ⋮ & … & ⋮ \\ 0 & 0 & z_{3} & … & 1 \\ 0 & 0 & 0 & … & z_{n} \end{matrix} ) .

Lemma 4.6. The following equality holds: C^{−1}AC = S.

Proof. The assertion of the Lemma follows from the Lemma 4.5. ■

Lemma 4.7. The folloing equalities hold:

∑_{k=0}^{l}(−1)^{k}σ_{k,n}ϕ_{l−k}(z_{1}, …, z_{n}) = 0,

where l ∈ ℕ.

Proof. The assertion of the Lemma follows from the equality

(∑_{k=0}^{∞} ϕ_{k}(z_{1}, …, z_{n})) × ∏_{1}^{n}(1 − z_{k}) = 1. ■

Corollary. Let m ∈ [1, n] ∩ ℕ, P and Q are the lower triangle m × m-matrices, in which respectively (−1)^{i−k}σ_{i−k,n} and ϕ_{i−k}(z_{1}, …, z_{n}), stand on the intersection of the i-th row and k−th column, where i = 1, …, m and k = 1, …, m. Then P^{−1} = Q.

Proof. Let

P = (p_{i,j}), Q = (q_{j,k}), R = (r_{i,k}) = P Q,

where {i, j, k} ⊂ {1, …, m}. Then

r_{i,k} = ∑_{j=1}^{m}(−1)^{i−j}σ_{i−j,n}ϕ_{j−k}(z_{1}, …, z_{n}) =

∑_{κ=0}^{i−1}(−1)^{κ}σ_{κ,n}ϕ_{i−k−κ}(z_{1}, …, z_{n}).

If i < k, then all the summands of the last sum are equal to 0, if k > i, then the last sum is equal to 0, according to the Lemma 4.7, and if i = k, then last sum have only one nonzero summand σ_{0,n}ϕ_{0}(z_{1}, …, z_{n}) = 1. ■

Remark 4.2. . Since all the elements of the matrices in (57), (58), (59) and in the Corollary of the Lemma 4.7 lie in K[z_{1} …, z_{n}], it follows that the assertions of the Lemmata 4.3 - 4.7 and of the Corollary of the Lemma 4.7 remain true for each specialisation of the point (z_{1}, …, z_{n}) into the space K^{n}. Therefore we may assume that 𝔨 = K and the sequence (55) consists of the elements of the field K. Of course, the objects, which are obtained as result of above specialization, inherit the notations of their prototypes. For example, the element, in which turns σ_{k,n} from the Lemma 4.7 by any specialization, we again denote by σ_{k,n}.

§5. A several constants.

Here we collected some constants, which are used in the further sections. We recommend to omit this section and return here during reading the other sections. Let n ∈ ℕ, m ∈ ℕ, p ∈ ℕ, q ∈ ℕ, ρ > 0,

ε > 0, γ_{0}(n, ρ) = max(1, ρ^{n−1}), γ_{1}(n, ρ) = min(1, ρ^{n−1},)

(62)  γ_{0}^{∼}(n, ρ) = ∑_{k=0}^{n−1} ρ^{k}, γ_{1}^{∼}(n, ρ, ε) = γ_{0}(n, ε/ρ)/ρ,

(63)  γ_{2}(n, ρ, ε) = max(ε, 1/ε)(1 + ρ_{0})^{2})^{n−1}, γ_{3}(n, m, ρ) = ∑_{κ=0}^{m−1} \binom{n}{κ}(ρ)^{n−κ},

γ_{4}(n, ρ) = max(1, γ_{3}(n, n, ρ)), for n ∈ ℕ + 1 and γ_{4}(1, ρ) = ρ,

(64)  γ_{5}(m, n, ρ) = ∑_{k=0}^{m−1} \binom{n + k − 1}{k}ρ^{k}.

ON THE DIFFERENCE EQUATION OF THE POINCARÉ TYPE 19

Clearly, if 0 < ρ < 1, then 0 < γ_{5}(n, 1, ρ) = γ_{0}^{∼}(n, ρ) = (1−ρ^{n})/(1−ρ) < 1/(1−ρ); moreover, if ε ∈ [0, ρ], then

1 − εγ_{5}(n, 1, ε/(2ρ))/(2ρ) > 1 − (ε/(2ρ))/(1 − ε/(2ρ)) =

(1 − ε/ρ)/(1 − ε/(2ρ))

and therefore

(65)  0 < γ_{6}(n, ρ, ε) = \frac{γ_{5}(n, 1, ε/(2ρ))/ρ}{1 − εγ_{5}(n, 1, ε/(2ρ))/ρ} < 1/(ρ − ε).

Let n ∈ ℕ, κ ∈ ℕ − 1 and η(n, κ) be the coefficient by x^{κ} in the formal series

1 + ∑_{k=1}^{∞}((1 + x)^{n} − 1)^{k},

(66)  γ_{7}(m, n, ρ) = ∑_{κ=0}^{m−1} η(n, κ(ρ)^{n+κ}.

Let 𝔐_{η} be the set of the zeros of the polynomial T_{η}, η = 1, 2 in (9),

ρ_{η,0} = sup{|λ|: λ ∈ 𝔐_{η}}, ρ_{η,1} = inf{|λ|: λ ∈ 𝔐_{η}},

where η = 0, 1. Clearly, ρ_{1,1} > ρ_{2,0}. Let ε_{1} ∈ (0, 1/3), ε_{2} = ε_{1}(ρ_{1,1} − ρ_{2,0}),

γ_{8} = γ_{2}(p; ρ_{1,0}, ε_{2}/2), γ_{9} = γ_{2}(q; ρ_{2,0}, ε_{2}/2), γ_{10} = γ_{6}(p, ρ_{1,1}, ε_{2})

γ_{11} = γ_{7}(q, p, γ_{10}), γ_{12} = γ_{4}(p, γ_{6}(p, ρ_{1,1}, ε_{2})), γ_{13} = γ_{4}(p, ρ_{1,0} + ε_{2}),
γ_{14} = γ_{0}(1 + q, ρ_{1,0} + ε_{2}), γ_{15} = min(γ_{8}(γ_{6}(p, ρ_{1,1}, ε_{2}))^{q}, (γ_{12})^{q}),

γ_{16} = min(γ_{8}γ_{14}, γ_{0}(q + 1, γ_{13})), γ_{17} = 2γ_{8}γ_{15}γ_{16}, γ_{18} = γ_{4}(n, ρ_{1}, 0) + γ_{8}ε_{2}/2. Let

(67)  ε_{3} = \frac{ρ_{2,0}}{ρ_{1,1} − ε_{2}} , ε_{4} ∈ (0, 1 − ε_{3}], ε_{5} ∈ (0, 1), ε_{6} = 1 − (1 − ε_{5})(ε_{4})^{q},
γ_{19} = min(γ_{8}(γ_{6}(p, ρ_{1,1}, ε_{2}))^{q}, (γ_{12})^{q}, γ_{20} = min(γ_{8}γ_{14}, γ_{0}(q + 1, γ_{13})),
γ_{21} = γ_{8}γ_{19}γ_{20}, γ_{22} = (1 + ρ_{2,0})^{q−1}γ_{21},
γ_{23} = (1 + qγ_{18}γ_{11}), γ_{24} = qγ_{11}γ_{4}(n, ρ_{1,0})(1 + qγ_{11}γ_{18}).

§6. Some special norms on ℂ^{n}.

Let now K = ℂ. According to the remark 4.2, we suppose that sequence (55) consists of the elements of the field ℂ. If 𝔮 is a norm on ℂ^{n}, the function q^{∼} on Mat_{n}(ℂ) defined by means the equality

𝔮^{∼}(A) = sup{𝔮(AX): X ∈ ℂ^{n}, 𝔮(X) ≤ 1}

is a norm on Mat_{n}(ℂ) and they name this norm 𝔮^{∼} by the norm associated with the norm 𝔮. We make use of the following norms on the space ℂ^{n} and associated with them norms on Mat_{n}(ℂ) :

q_{∞}(X) = sup{|x_{i}|: i = 1, …, n, X = (x_{i}) ∈ ℂ^{n}},

q_{∞}^{∼}(A) = sup{∑_{k=1}^{n} |a_{i,k}|: i = 1, …, n, A = (a_{i,k}) ∈ Mat_{n}(ℂ)},

q_{1}(X) = ∑_{i=1}^{n} |x_{i}|: i = 1, …, n, X = (x_{i}) ∈ ℂ^{n},

q_{1}^{∼}(A) = sup{∑_{i=1}^{n} |a_{i,k}|: k = 1, …, n, A = (a_{i,k}) ∈ Mat_{n}(ℂ)}.

Although all the norms on ℂ^{n} are equivalent, they are suitable for concrete calculations in a different degree. Below we construct some norms convenient for our purposes.

Let K = ℂ, ε > 0, n ∈ ℕ. Let T_{n,ε} be the diagonal n × n−matrix having as its k−th diagonal element the number ε^{k−1}. Let further the numbers (55) are contained in ℂ and

(68)  ρ_{0} = sup{|z_{i}|: i = 1, …, n}, ρ_{1} = inf{|z_{i}|: i = 1, …, n}.

For the sequence (55) and any ε ∈ (0, 1) we define the norm q_{z_{1}, ... , z_{n};ε} as follows:

(69)  q_{z_{1}, ... , z_{n};ε}(X) = q_{∞}((C(z_{1}, …, z_{n})T_{n,ε})^{−1}(X)),

where X ∈ ℂ^{n}. Clearly, if B ∈ Mat_{n}(ℂ), then

(70)  q_{z_{1}, ... , z_{n};ε}^{∼}(B) = q_{∞}((C(z_{1}, …, z_{n})T_{n,ε})^{−1}F C(z_{1}, …, z_{n})T_{n,ε}).

In particular, according to the Lemma 4.5, for the matrix A in (57) holds the equality

(71)  q_{z_{1}, ... , z_{n};ε}^{∼}(A) = ρ_{0} + ε min(n − 1, 0).

If ρ_{1} > 0, then, clearly, there exists A^{−1} and the following inequaliy holds:

(72)  q_{z_{1}, ... , z_{n};ε}^{∼}(A^{−1}) ≤ γ_{1}^{∼}(n, ρ_{1}, ε).

Let us establish some supplementary connections between the norms q_{∞} and q_{z_{1}, ... , z_{n};ε} (respectively between the norms q_{∞}^{∼} and q_{z_{1}, ... , z_{n};ε}^{∼}). Lemma 6.1. The following inequaliy holds:

q_{∞}^{∼}(C(z_{1}, …, z_{n})) ≤ (1 + ρ_{0})^{n−1}.

Proof. Clearly, q_{∞}^{∼}(C) = sup{ ∑_{k=1}^{n} |c_{i,k}|: i = 1, …, n. In view of (58), (43) - (45),

|c_{i,k}| ≤ \binom{i−1}{k−1}ρ_{0})^{i−k}, where k = 1, …, i, and c_{i,k} = 0 for i < k. Therefore

∑_{k=1}^{n} |c_{i,k}| ≤ ∑_{k=1}^{i} \binom{i − 1}{k − 1}ρ_{0})^{i−k} = (1 + ρ_{0})^{i−1}.■

Lemma 6.2. The following inequality holds:

q_{∞}^{∼}((C(z_{1}, …, z_{n}))^{−1}) ≤ (1 + ρ_{0})^{n−1}.

Proof. The assertion of the Lemma follows directly from the Lemma 4.6. ■ Lemma 6.3. The following inequalities hold:

q_{z_{1}, ... , z_{n};ε} ≤ (max(1, 1/ε)(1 + ρ_{0}))^{n−1}q_{∞},
q_{∞} ≤ (max(1, ε)(1 + ρ_{0}))^{n−1}q_{z_{1}, ... , z_{n};ε},
(73)  q_{z_{1}, ... , z_{n};ε}^{∼} ≤ γ_{2}(n, ρ, ε)q_{∞}^{∼},

(74)  q_{∞}^{∼} ≤ γ_{2}(n, ρ, ε)q_{∞}^{∼}vertz_{1}, …, z_{n}; ε,

whith γ_{2}(n, ρ, ε) from (63).

Proof. In view of (69) and (70),

(75)  q_{∞}(X) = q_{z_{1}, ... , z_{n};ε}(C(z_{1}, …, z_{n})T_{n,ε}X),

(76)  q_{∞}^{∼}(F) = q_{z_{1}, ... , z_{n};ε}((C(z_{1}, …, z_{n})T_{n,ε})F (C(z_{1}, …, z_{n})T_{n,ε})^{−1}).

where X ∈ ℂ^{n} and F ∈ Mat_{n}(ℂ). The assertions of the Lemma follow directly from (69), (70), (75), (76), Lemma 4.6 and Lemma 6.2. ■

We introduce on ℂ the order relation ⪯ as follows: we say that z_{1} ⪯ z_{2} if |z_{1}| < |z_{2}| or if 0 < |z_{1}| = |z_{2}| and arg(z_{1}) < arg(z_{2}), where arg(z) for any z ∈ ℂ is taken in (−π, π]. Let the elements (55) are contained in ℂ and z_{k+1} ⪯ z_{k}, for k ∈ [1, n − 1] ∩ ℕ. We suppose that the sequence (55) consists of all the roots of the polynomial (56) and one meets each root in (55) as many times as the multiplicity of this root is. Then the sequence (55) is uniquely defined by means of the matrix A from (57). Therefore the norms q_{z_{1}, ... , z_{n};ε} and q_{z_{1}, ... , z_{n};ε}^{∼} further will be denoted respectively by q_{A,ε} and q_{A,ε}^{∼}

Lemma 6.4.Let η(t) = 0 for t < 0 and η(t) = 1 for t ≥ 0 The following inequalities hold for P and Q defined (according to the remark 4.2) in the Corollary 1 of the Lemma 4.7 :

q_{1}^{∼}(P^{−1}) ≤ γ_{5}(m, n, ρ_{0}), q_{∞}^{∼}(P^{−1}) ≤ γ_{5}(m, n, ρ_{0}),
q_{1}^{∼}(P) ≤ (1 + ρ_{0})^{n} − η(n − m)(ρ_{0})^{n}, q_{∞}^{∼}(P) ≤ (1 + ρ_{0})^{n} − η(n − m)(ρ_{0})^{n}

with ρ_{0} from (68) and γ_{5}(m, n, ρ_{0}) from (64).

Proof. Clearly,

∑_{i=1}^{m} |ϕ_{i−k,}(z_{1}, …, z_{n})| ≤ ∑_{i=k}^{m} \binom{i − k + n − 1}{i − k}(ρ_{0})^{i−k} =

∑_{κ=0}^{m−k} \binom{κ + n − 1}{κ}(ρ_{0})^{κ} ≤ ∑_{κ=0}^{m−1} \binom{κ + n − 1}{κ}(ρ_{0})^{κ} = γ_{5}(m, n, ρ_{0}),

∑_{k=1}^{m} |ϕ_{i−k,}(z_{1}, …, z_{n})| ≤ ∑_{k=1}^{i} \binom{i − k + n − 1}{i − k}(ρ_{0})^{i−k} =

∑_{κ=0}^{i−1} \binom{κ + n − 1}{κ}(ρ_{0})^{κ} ≤ ∑_{κ=0}^{m−1} \binom{κ + n − 1}{κ}(ρ_{0})^{κ} = γ_{5}(m, n, ρ_{0}),

∑_{i=1}^{m} |σ_{i−k,n}| ≤ ∑_{i=k}^{m} \binom{n}{i − k}(ρ_{0})^{i−k} =

∑_{κ=0}^{m−k} \binom{n}{κ}(ρ_{0})^{κ} ≤ (1 + ρ_{0})^{n} − η(n − m)(ρ_{0})^{n},

∑_{k=1}^{m} |σ_{i−k,n}| ≤ ∑_{k=1}^{i} \binom{n}{i − k}(ρ_{0})^{i−k} = ∑_{κ=0}^{i−1} \binom{n}{κ}(ρ_{0})^{κ} ≤ (1 + ρ_{0})^{n} − η(n − m)(ρ_{0})^{n}.■

Lemma 6.5. Let n ∈ ℕ, m ∈ ℕ Let, as before, the matrix (57) corresponds to the sequence (55), ρ_{0} and ρ_{1} are defined in (68). Let A_{1} be the lower triangle m × m-matrix, in which a_{i−k}, stands on the intersection of the i-th row and k−th column, where k = 1, …, m and i = k, …, m, but all other elements are equal to 0. Then

(77) q_{∞}^{∼}(A) ≤ γ_{4}(n, ρ_{0}), q_{∞}^{∼}(A_{1}) ≤ γ_{3}(n, m, ρ_{0}), q_{1}^{∼}(A_{1}) ≤ γ_{3}(n, m, ρ_{0}).

If

(78)  a_{0} ≠ 0,

then

(79)  q_{∞}^{∼}(A^{−1}) ≤ γ_{4}(n, 1/ρ_{1}), q_{∞}^{∼}((A_{1})^{−1}) ≤
γ_{5}(m, n, 1/ρ_{1})(ρ_{1})^{−n}, q_{1}^{∼}((A_{1})^{−1}) ≤ γ_{5}(m, n, 1/ρ_{1})(ρ_{1})^{−n}.

Proof. The inequalities (77) are obvious. We show that, if (78) is fulfilled, then

(79) hold. Clearly, A^{−1} = ( \begin{matrix}\frac{−a_{1}}{a_{0}} & \frac{−a_{2}}{a_{0}} & … & \frac{−1}{a_{0}} & \frac{−1}{a_{n}} \\ 1 & 0 & … & 0 & 0 \\ 0 & 1 & … & 0 & 0 \\ ⋮ & ⋮ & … & ⋮ & ⋮ \\ 0 & 0 & … & 1 & 0 \end{matrix} ) . From (78) follows the

first of the inequalities (79). Let G = (a_{n})^{−1}A_{1}, w_{k} = 1/z_{k}, where k = 1, …, n. We see that for P all the conditons of the Lemma 6.4 are fulfilled with w_{k} instead z_{k} for k = 1, …, n. The second of the inequalities (79) follows from this Lemma. ■

Since ∥A∥_{sp} := ρ_{0}, ∥A^{−1}∥_{sp} := 1/ρ_{1}, it follow that (77) and (78) may be rewritten respectively in the form

q_{∞}^{∼}(A) ≤ γ_{4}(n, ∥A∥_{sp}), q_{∞}^{∼}(A_{1}) ≤ γ_{3}(n, m, ∥A∥_{sp}), q_{1}^{∼}(A_{1}) ≤ γ_{3}(n, m, ∥A∥_{sp}),

q_{∞}^{∼}(A^{−1}) ≤ γ_{4}(n, ∥A^{−1}∥_{sp}), q_{∞}^{∼}((A_{1})^{−1}) ≤ γ_{5}(m, n, ∥A^{−1}∥_{sp})(∥A^{−1}∥_{sp})^{n},

q_{1}^{∼}((A_{1})^{−1}) ≤ γ_{5}(m, n, ∥A^{−1}∥_{sp})(∥A^{−1}∥_{sp})^{n}.

Lemma 6.6. Let A be the matrix from (57), let ε > 0, let B ∈ Mat_{n} ℂ and let

(80)  q_{A,ε/2}^{∼}(B − A) ≤ ε/2.

Then (a) ∥B∥_{sp} ≤ q_{A,ε}^{∼}(B) ≤ ε + ∥A∥_{sp};

(b) if for some b_{κ} ∈ ℂ, where κ ∈ ℤ, b_{n} = 1 and b_{κ} = 0 for κ ∈ (−ℕ) ∪ (n + ℕ) the following equality holds:

(81)  B = ( \begin{matrix}0 & 1 & 0 & … & 0 \\ 0 & 0 & 1 & … & 0 \\ ⋮ & ⋮ & ⋮ & … & ⋮ \\ 0 & 0 & 0 & … & 1 \\ −b_{0} & −b_{1} & −b_{2} & … & −b_{n−1} \end{matrix} ) ,

B_{1} be the lower triangle m × m-matrix, in which b_{i−k}, stands on the intersection of the i-th row and k−th column, where k = 1, …, m and i = k, …, m, but all other elements are equal to 0, then

q_{∞}^{∼}(B) ≤ γ_{4}(n, ∥A∥_{sp} + ε), q_{∞}^{∼}(B_{1}) ≤ γ_{3}(n, m, ∥A∥_{sp} + ε),
q_{1}^{∼}(B_{1}) ≤ γ_{3}(n, m, ∥A∥_{sp} + ε).

(c) if (78) is fulfilled and 0 < ε < (∥A^{−1}∥_{sp})^{−1}, then

(82)  |b_{0}| ≥ ((∥A^{−1}∥_{sp})^{−1} − ε)^{n} > 0,

(83)  q_{A^{;}ε_{2}/2}(B^{−1}) ≤ γ_{6}(n, (∥A^{−1}∥_{sp})^{−1}, ε),

(84)  q_{∞}^{∼}(B^{−1}) ≤ γ_{4}(n, γ_{6}(n, (∥A^{−1}∥_{sp})^{−1}, ε)),

(85)  q_{∞}^{∼}((B_{1})^{−1}) ≤ γ_{5}(m, n, γ_{6}(n, ρ, ε))γ_{6}(n, ρ, ε))^{n},
q_{1}^{∼}((B_{1})^{−1}) ≤ γ_{5}(m, n, γ_{6}(n, ρ, ε))γ_{6}(n, ρ, ε))^{n}.

Proof. Since q_{A,ε/2}^{∼}(B) ≤ q_{A,ε/2}^{∼}(A) + ε/2, the inequality ∥B∥_{sp} ≤ 𝔮^{∼}(B) holds for any norm 𝔮 on ℂ^{n} and, in fiew of (71),the inequality q_{A,ε/2}(A) ≤ ∥A∥_{sp} + ε/2 holds, it follows that the assertion (a) of the Lemma is proved. The assertion (b) follows from the assertion (a) and Lemma 6.5. Let us to prove the assertion (c). If (78) is fulfilled, there exisists A^{−1} and, in view of (72),

(86)  q_{A;ε/2}^{∼}(A^{−1}) ≤ γ_{1}^{∼}(n, ρ_{1}, ε/2).

Clearly, B = A(E − A^{−1}(A − B)). In view of (80) and (86), q_{A,ε/2}^{∼}(A^{−1}(A − B)) ≤

(γ_{1}^{∼}(n, ρ_{1}, ε/2))ε/2 ≤ \frac{ε/(2ρ_{1}))}{1−ε/(2ρ_{1}))} < 1 Therefore (E − A^{−1}(A − B))^{−1} exists and

(87)  q_{A,ε/2}^{∼}(E − A^{−1}(A − B))^{−1} ≤ 1/(1 − (γ_{1}(n, ρ_{1}, ε/2))ε/2).

Consequently, B^{−1} = (E−A^{−1}(A−B))^{−1}A^{−1} exists also; according to the equality γ_{5}(n, 1, ρ) = γ_{0}^{∼}(n, ρ) and in view of (86), (87), and (65),

(88)  ∥B^{−1}∥_{sp} ≤ q_{A,ε}^{∼}(B−1) ≤ \frac{γ_{1}^{∼}(n, ρ_{1}, ε/2)}{1 − (γ_{1}^{∼}(n, ρ_{1}, ε/2))ε/2} =

\frac{γ_{5}(n, 1ρ_{1}, ε/(2ρ_{1}))/ρ_{1}}{1 − (ε/2)γ_{5}(n, 1ρ_{1}, ε/(2ρ_{1}))/ρ_{1}} = γ_{6}(n, ρ_{1}, ε) < \frac{1}{ρ_{1} − ε}.

In view of (88) and (81), |b_{0}| ≥ (ρ_{1} − ε)^{n} > 0 and therefore (82) is fulfilled. The inequalities (84) and (85) follow from (88) and (79). ■

Lemma 6.7. Let m ∈ ℕ, n ∈ ℕ. Let A be the matrix of the Lemma 6.6, b_{i,κ} ∈ ℂ, where i = 1, …, m, κ ∈ ℤ, b_{i,n} = 1 and b_{i,κ} = 0 for i = 1, …, m, κ ∈ (−ℕ) ∩ (n + ℕ) and

B_{i} = ( \begin{matrix}0 & 1 & 0 & … & 0 \\ 0 & 0 & 1 & … & 0 \\ ⋮ & ⋮ & ⋮ & … & ⋮ \\ 0 & 0 & 0 & … & 1 \\ −b_{i,0} & −b_{i,1} & −b_{i,2} & … & −b_{i,n−1} \end{matrix} ) .

Let q_{A,ε/2}^{∼}(B_{i} − A) ≤ ε/2, where i = 1, …, m, let B^{∧} be the lower triangle m × mmatrix, in which b_{k,i−k}, stands on the intersection of the i-th row and k−th column, where k = 1, …, m and i = k, …, m, but all other elements are equal to 0. Then

(89)  q_{∞}^{∼}(B^{∧}) ≤ γ_{3}(n, m, ∥A∥_{sp} + ε), q_{1}^{∼}(B^{∧}) ≤ γ_{3}(n, m, ∥A∥_{sp} + ε).

If (78) is fulfilled and 0 < ε < (∥A^{−1}∥_{sp})^{−1}, then (B^{∧})^{−1} exists and q_{∞}^{∼}((B^{∧})^{−1}) ≤ γ_{7}(m, n, γ_{6}(n, (∥A^{−1}∥_{sp})^{−1}, ε)),

q_{1}^{∼}((B^{∧})^{−1}) ≤ γ_{7}(m, n, γ_{6}(n, (∥A^{−1}∥_{sp})^{−1}, ε)).

Proof. In view of the lemma 6.6, ∥B_{i}∥_{sp} < ε + ∥A∥_{sp}, where i = 1, …, m. Consequently, |b_{i,k}| < \binom{n}{k}(ε + ∥A∥_{sp})^{n−k}, where i = 1, …, m, k = 0, …, n. Therefore (69) holds. If (78) is fulfilled and 0 < ε < (∥A^{−1}∥_{sp})^{−1}, then it follows from the lemma 6.6 that b_{i,0} ≠ 0 for i = 1, …, m and

(90)  1/|b_{i,0}| ≤ γ_{6}(n, (∥A^{−1}∥_{sp})^{−1}, ε)^{n},

where i = 1, …, m. Let D_{1} be the diagonal m × m-matrix, in which b_{i,0}, stands on the intersection of the i-th row and i−th column, where i = 1, …, m, and let B_{1}^{∧} = (D_{1})^{−1}B_{1}. Then B_{1}^{∧} is the lower triangle m × m-matrix, in which all the diagonal elements are equal to 1 and B_{1}^{∧} = E_{m} − F, where F^{m} is the zero m × mmatrix and E_{m} is the unit m × m-matrix. Let 𝔇_{m} and ℜ_{m} are the rings of all the diagonal and respectively the lower triangle m × m-matrices in Mat_{m}(ℂ); let

N_{m} be the m × m-matrix having the form ( \begin{matrix}0 & 0 \\ E_{m−1} & 0 \end{matrix} ) , where E_{m−1} is the unit

(m − 1) × (m − 1)-matrix; let 𝔇_{m,0} = 𝔇_{m}, 𝔇_{m,l} = 𝔇_{m}(N_{m})^{l} = (N_{m})^{l}𝔇_{m} where l ∈ ℕ. Clearly, ℜ_{m} splits into direct sum ℜ_{m} = ⊕\begin{matrix}∞ \\ l=0 \end{matrix}𝔇_{m,l} = ⊕\begin{matrix}m−1 \\ l=0 \end{matrix}𝔇_{m,l} and is the graduated ring, for which 𝔇_{m,l} is the additive group of homgeneous elements of

degree l. Clearly, F = ∑_{l=1}^{m−1} F_{l}, where F_{l} ∈ 𝔇_{m,l} for l = 1, …, m − 1. According to the Lemma 6.6, (B_{i})^{−1} exists and ∥B^{−1}∥_{sp} ≤ γ_{6}(n, ρ_{1}, ε). The absolute value of the each element of the matrix F_{l} is not greater, than \binom{n}{l}(γ_{6}(n, (∥A^{−1}∥_{sp})^{−1}, ε))^{l}. On the other hand

(91) (B_{1}^{∧})^{−1} = E_{m} + ∑_{l=0}^{m−1} F^{l} = E_{m} + ∑_{k=1}^{m−1} ( ∑_{l=1}^{m−1} ( ∑_{\begin{matrix}l_{1}∈ℕ, ... , l_{s}∈ℕ \\ ∑_{s=1}^{k} l_{s}=l \end{matrix}} ∏_{s=1}^{k} F_{l_{s}} )) .

The absolute value of the each element of the matrix

∑_{\begin{matrix}l_{1}∈ℕ, ... , l_{s}∈ℕ \\ ∑_{s=1}^{k} l_{s}=l \end{matrix}} ∏_{s=1}^{k} F_{l_{s}}

is not greater than

(γ_{6}(n, (∥A^{−1}∥_{sp})^{−1}, ε))^{l} ∑_{\begin{matrix}l_{1}∈ℕ, ... , l_{s}∈ℕ \\ ∑_{s=1}^{k} l_{s}=l \end{matrix}} ∏_{s=1}^{k} \binom{n}{l_{s}}

The last sum is equal to the coefficient by x^{l} in the polynomial ((1 + x)^{n} − 1)^{l}; we denote this coefficient by η^{∼}(n, k, l). Then, in view of (91), (66) for q = q_{1} and for q = q_{∞}, we have

q^{∼}((B_{1}^{∧})^{−1}) ≤ 1 + ∑_{k=1}^{m−1} ∑_{l=1}^{m−1} η^{∼}(n, k, l)(γ_{6}(n, (∥A^{−1}∥_{sp})^{−1}, ε))^{l} =

1 + ∑_{l=1}^{m−1}(γ_{6}(n, (∥A^{−1}∥_{sp})^{−1}, ε))^{l} ∑_{k=1}^{m−1} η^{∼}(n, k, l) =

1 + ∑_{l=1}^{m−1}(γ_{6}(n, (∥A^{−1}∥_{sp})^{−1}, ε))^{l}η(n, l) =

γ_{7}(m, n, γ_{6}(n, ∥A^{−1}∥_{sp})^{−1}, ε))(γ_{6}(n, (∥A^{−1}∥_{sp})^{−1}, ε))^{−n}.

In view of (90), q^{∼}(D_{1}) < (γ_{6}(n, (∥A^{−1}∥_{sp})^{−1}, ε))^{n}. Therefore

q^{∼}((B^{∼})^{−1}) ≤ q^{∼}((B_{1}^{∼})^{−1})q^{∼}((D_{1})^{−1}) ≤ γ_{7}(m, n, γ_{6}(n, (∥A^{−1}∥_{sp})^{−1}, ε)). ■

§7. Some remarks on contracted maps and fixed point Theorem.

If (Y, ρ) is a bounded metric space (i.e. d(Y) = sup{ρ(x, y): (x, y) ∈ Y^{2}} < +∞), then following function ρ^{∼} is defined on X^{2}, where X = Y^{Y}: ρ^{∼}(A, B) = sup{ρ(A(y), B(y)): y ∈ Y), where A ∈ X = Y^{Y} and B ∈ 𝕏. Clearly, (X, ρ^{∼}) is bounded metric space and d(X) ≤ d(Y); (X, ρ^{∼}) is complete if (Y, ρ) is complete. In this section we consider only bounded metric spaces. Let γ_{0} ∈ (0, 1). If A is a contacted map of (Y, ρ) into (Y, ρ) with coefficient of contraction equal to γ_{0} then, clearly, ρ^{∼}(AB, AC) ≤ γ_{0}ρ(B, C) for any B ∈ Y^{Y} and C ∈ Y^{Y} . To avoid the superfluous notations, if Z ⊂ Y, then the restriction of ρ on Z^{2} we shall again denote by ρ.

Lemma 7.1. Let (Y, ρ) is the bounded metric space, r ∈ ℕ. Let A ∈ X = Y^{Y} , γ_{0} ∈ (0, 1), A^{r} is the contracted map (Y, ρ) into (Y, rho) with coefficient of contraction equal to γ_{0} and fixed point y^{∗}. Let y_{0} ∈ Y, B_{n} ∈ X = Y^{Y} , where n ∈ ℕ−1, let y(n+1) = B_{n}(y_{n}), where n ∈ ℕ − 1, let δ_{r}(y) = sup{ρ(y^{∗}, A^{s}(y)): s = 0, …, r − 1}, where y ∈ Y and let δ_{r}^{∼}(B) = sup{ρ^{∼}(A^{κ}, A^{κ−1}B): κ = 1, …, r}. Then

(92)  ρ(y_{n}, y∗) ≤ (γ_{0})^{[n/r]}δ_{r}(y_{0}) + ∑_{κ=1}^{n}(γ_{0})^{[(κ−1)/r]}δ_{r}^{∼}(B_{n−κ}) =

(γ_{0})^{n}δ_{r}(y_{0}) + ∑_{κ=1}^{n}(γ_{0})^{[(n−κ)/r]}δ_{r}^{∼}(B_{κ−1}).

If lim_{ν→+∞} δ_{r}^{∼}(B_{n}) = 0 then lim_{n→+∞} ρ(y_{n}, y^{∗}) = 0. Moreover, if the functin ξ(x) is

defined on [0, +∞), ξ(x) > 0 for x ∈ [0, +∞), ξ(x) decreases together with increasing of x ∈ [0, +∞), lim_{x→∞}(ξ(x)) = 0, (12), (13) are fulfilled and δ_{r}^{∼}(B_{n}) < ξ(n) for n ∈ ℕ − 1, then ρ(y_{n}, y^{∗}) = O(ξ(n)).

Proof. Clearly,

(93)  ρ(y_{n}, y^{∗}) = ρ(B_{n−1}(y_{n−1}), A(y^{∗})) ≤

ρ(A(y_{n−1}), A(y^{∗})) + ρ(B_{n−1}(y_{n−1}), A(y_{n−1}) ≤ ρ(A(y_{n−1}), A(y^{∗})) + ρ^{∼}(B_{n−1}, A)

Iterating of (93) we obtain the inequality

(94)  ρ(y_{n}, y^{∗}) ≤ ρ(A^{k}(y_{n−k}), A^{k}(y^{∗})) + ∑_{κ=1}^{k} ρ^{∼}(A^{κ−1}B_{n−κ}, A^{κ}),

where k ∈ ℕ, n ∈ k + ℕ − 1. In fact, (94) coincides with (93) for k = 1. Let n ∈ ℕ + 1, k ∈ [2, n] ∩ ℕ and (94) is true for k − 1. Then

ρ(y_{n}, y^{∗}) ≤ ρ(A^{k−1}(y_{n−k+1}), A^{k−1}(y^{∗})) + ∑_{κ=1}^{k−1} ρ^{∼}(A^{κ−1}B_{n−κ}, A^{κ}),

ρ(y_{n}, y^{∗}) ≤ ρ(A^{k−1}(B_{n−k}y_{n−k}), A^{k−1}(Ay^{∗})) + ∑_{κ=1}^{k−1} ρ^{∼}(A^{κ−1}B_{n−κ}, A^{κ}) ≤

ρ(A^{k−1}(Ay_{n−k}), A^{k−1}(Ay^{∗})) + ρ(A^{k−1}(B_{n−k}y_{n−k}), A^{k−1}(Ay_{n−k}))+

∑_{κ=1}^{k−1} ρ^{∼}(A^{κ−1}B_{n−κ}, A^{κ}) ≤

ρ(A^{k}y_{n−k}), A^{k}y^{∗})) + ρ^{∼}(A^{k−1}B_{n−k}, A^{k}) + ∑_{κ=1}^{k−1} ρ^{∼}(A^{κ−1}B_{n−κ}, A^{κ}).

We take now k = n = rl + s, κ = rλ + σ, where s = 0, …, r − 1, l ∈ ℕ, σ = 1, …, r, λ ∈ ℕ − 1 and obtain the inequalities

(95)  ρ(y_{n}, y^{∗}) ≤ ρ(A^{k}(y_{n−k}), A^{k}(y^{∗})) ≤ (γ_{0})^{l}ρ(A^{s}(y_{n−k}), y^{∗}) ≤
(γ_{0})^{l} sup{ρ^{∼}(A^{s}(y_{0}), y^{∗}): s = 0, …, q − 1} = (γ_{0})^{l}δ_{r}(y_{0}),
(96)  ρ^{∼}(A^{κ−1}B_{n−κ}, A^{κ}) =
ρ^{∼}(A^{rλ+σ−1}B_{n−κ}, A^{rλ+σ}) ≤ (γ_{0})^{λ}ρ^{∼}(A^{σ−1}B_{n−κ}, A^{σ}) ≤
(γ_{0})^{λ} sup{ρ^{∼}(A^{σ−1}B_{n−κ}, A^{σ}): σ = 1, …, r, } = (γ_{0})^{[(κ−1)/r]}δ_{r}^{∼}(B_{n−κ}).

The inequality (92) follows from (95) and (96). Let us rewrite (92) in the form

(97)  ρ(y_{n}, y∗) ≤ (γ_{0})^{n}δ_{r}(y_{0}) + ∑_{κ=1}^{[n/2]}(γ_{0})^{[(κ−1)/q]}δ_{r}^{∼}(B_{n−κ})+

∑_{κ=[n/2]+1}^{n}(γ_{0})^{[(κ−1)/q]}δ_{r}^{∼}(B_{n−κ}).

Other assertions of the Lemma follow from (97). ■

Let (Y, ρ) is the bounded metric space, A ∈ X = Y^{Y} . We name A by centripetal map relatively ρ if there exists γ_{0} ∈ (0, 1) and fixed point y^{∗} ∈ Y of the map A such that ρ(A(y), y^{∗}) ≤ γ_{0}ρ(A(y), y^{∗}) for any y ∈ Y. If 𝔮 is an norm on linear space L then we denote by O_{𝔮,δ}(Z^{∗}) the open ball with the center Z^{∗} and radius r i.e., {Z ∈ L: 𝔮(Z − Z^{∗}) < δ}.

Lemma 7.2. Let 𝔮 be a norm on the linear space L over ℂ (or ℝ) and L^{∼} be te space of the all linear continuous operatos on L. Let further 𝔮^{∼} be the norm on L^{∼} associated with the norm q. Let Z^{∗} ∈ L and D is a bounded open neighborhood of the zero element in L. Let further Φ(Z) a differentiable map in the sence of Frech’et of the set Z^{∗} + D in L and let the Frechét derivative (\frac{∂}{∂Z} Φ) (Z) is continuous in Z^{∗} + D. Let A = (\frac{∂}{∂Z} Φ) (Z^{∗}), let 𝔮^{∼}(A) < 1 and let ε ∈ (0, 1 − 𝔮(A)). Then there exist δ ∈ (0, 1) such that for any Z^{∨} and Z^{∧} in Z^{∗} + δD the following inequality holds: 𝔮(ϕ(Z^{∨}) − ϕ(Z^{∧})) ≤ (𝔮^{∼}(A) + ε)𝔮(Z^{∨} − Z^{∧}).

Proof. Since D is open, it follows that there exists δ_{1} > 0, such that O_{𝔮,δ_{1}}(Z^{∗}) ∈ D. Since (\frac{∂}{∂Z} Φ) (Z) is continuous, it follows that for any ε > 0 there exists δ_{2} ∈ (0, δ_{1}) such that

𝔮^{∼} ((\frac{∂}{∂Z} Φ) (Z) − A) < ε

for Z ∈ O_{𝔮,δ_{2}}(Z^{∗}). Then 𝔮^{∼} ((\frac{∂}{∂Z} Φ) (Z)) < 𝔮^{∼}(A) + ε for Z ∈ O_{𝔮,δ_{2}}(Z^{∗}). Let

Z^{∨} ∈ O_{𝔮,δ_{2}}(Z^{∗}), Z^{∧} ∈ O_{𝔮,δ_{2}}(Z^{∗}).

Since O_{𝔮,δ_{2}}(Z^{∗}) is convex,it folows that ϕ(t) = Φ((1 − t)Z^{∨} + tZ^{∧}), is defined [0, 1] map of the segment [0, 1] in L and, moreover, this map has a continuous derivative (\frac{∂}{∂t}ϕ) (t) = (\frac{∂}{∂Z} Φ) ((1 − t)Z^{∨} + tZ^{∧})(Z^{∧} − Z^{∨}). Therefore

𝔮(Φ(Z^{∧}) − Φ(Z^{∨})) = 𝔮(ϕ(1) − ϕ(0)) = 𝔮 (∫_{0}^{1} (\frac{∂}{∂t}ϕ) (t)dt) =

𝔮 ((∫_{0}^{1} ((\frac{∂}{∂Z} Φ) ((1 − t)Z^{∨} + tZ^{∧})) dt) (Z^{∧} − Z^{∨})) ≤

𝔮^{∼} (∫_{0}^{1} (\frac{∂}{∂Z} Φ) ((1 − t)Z^{∨} + tZ^{∧}))dt) 𝔮(Z^{∧} − Z^{∨}) ≤

(∫_{0}^{1} (𝔮^{∼} (\frac{∂}{∂Z} Φ) ((1 − t)Z^{∨} + tZ^{∧})) dt) 𝔮(Z^{∧} − Z^{∨})

≤ (𝔮^{∼}(A) + ε)𝔮(Z^{∧} − Z^{∨}).

Since D is bounded, it follows that there exist δ > 0 such that Z^{∗} + δD ⊂ O_{𝔮,δ_{2}}(Z^{∗}). ■

Lemma 7.3. Let L be a linear normed space over ℂ (or ℝ) with norm 𝔮. Let further 𝔮^{∼} be the norm on L^{∼} associated with the norm q. Let Z^{∗} ∈ L, D is bounded open neighborhood of the zero element in L. Let further Φ(Z) is a map of the set Z^{∗} +D in ℂ^{n}, which is differentiable in the sence of Frechét at the point Z^{∗} and let A = (\frac{∂}{∂Z} Φ)(Z^{∗}). Let there exists γ_{0} ∈ (0, 1) such that 𝔮(Φ(Z) − Z^{∗}) ≤ γ_{0}𝔮(Z − Z^{∗}) for any Z ∈ D. Then 𝔮^{∼}(A) < 1.

Proof. Since D is open, it follows that there exists δ_{1} > 0 such that O_{𝔮,δ_{1}}(Z^{∗}) ∈ D. Since Φ(Z) is differentiable at Z^{∗} in the sence of Frechét, it follows that 𝔮(Φ(Z^{∗}+ H) − P (Z^{∗}) − AH) = ω(H)𝔮(H), where H ∈ O_{𝔮,δ_{1}}(Z^{∗}) and ω(H) is a function defined on O_{𝔮,δ_{1}}(Z^{∗}) such that lim_{H→0} ω(H) = ω(0_{L}) = 0. Therefore for ε ∈ (0, 1−γ_{0})

there exists δ_{2} ∈ (0, δ_{1}) such that 0 ≤ ω(H) < ε. for any H ∈ O_{𝔮,δ_{1}}(Z^{∗}). Then

𝔮(AH) ≤ 𝔮(Φ(Z^{∗} + H) − P (Z^{∗}) − AH) + 𝔮(Φ(Z^{∗} + H) − P (Z^{∗})) ≤
(γ_{0} + ω(H))𝔮(H) ≤ (γ_{0} + ε)𝔮(H)

for any H ∈ O_{𝔮,δ_{1}}(Z^{∗}). Therefore 𝔮^{∼}(A) ≤ ε + γ_{0} < 1. ■

Lemma 7.4. Let 𝔮 is a norm on the linear space L over ℂ (or ℝ) and L^{∼} be te space of the all linear continuous operatos on L. Let further 𝔮^{∼} be the norm on L^{∼} associated with the norm q. Let Z^{∗} ∈ L, D is bounded open neighborhood of the zero element in L Let further Φ(Z) a differentiable map in the sence of Frechét of the set Z^{∗}+D in Z^{∗}+D, let Z^{∗} is the fixed point of Φ and let the Frechét derivative (\frac{∂}{∂Z} Φ) (Z) is continuous in Z^{∗} + D. Let Φ is centripetal. Then there exists r ∈ ℕ such that Φ is contracted map of set Φ^{r}(Z^{∗} + D) into set Φ^{r}(Z^{∗} + D).

Proof. Let A = (\frac{∂}{∂Z} Φ)(Z^{∗}). According to conditions of the Lemma, there exists γ_{0} ∈ (0, 1) and such that 𝔮(Φ(Z) − Φ(Z^{∗})) ≤ γ_{0}𝔮(Z − Z^{∗})) for any Z ∈ Z^{∗} + D. According to the Lemma 6.3, 𝔮^{∼}(A) < 1. Let ε ∈ (0, 1 − 𝔮(A)). According to the Lemma 6.2, there exist δ ∈ (0, 1) such that for any Z^{∨} and Z^{∧} in Z^{∗} + δD the following inequality holds: 𝔮(ϕ(Z^{∨}) − ϕ(Z^{∧})) ≤ (𝔮^{∼}(A) + ε)𝔮(Z^{∨} − Z^{∧}). Since 0 < 𝔮^{∼}(A) + ε < 1 there exists r ∈ ℕ such that (𝔮^{∼}(A) + ε)^{r} < δ. Clearly Φ maps of set Φ^{r}(Z^{∗} + D) into set Φ^{r+1}(Z^{∗} + D) ⊂ Φ^{r}(Z^{∗} + D) and for any Z^{∨} and Z^{∧} in Z^{∗} + δD the following inequality holds:

𝔮(Φ(Φ^{r}(Z^{∨})) − Φ(Φ^{r}(Z^{∧}))) ≤ (𝔮^{∼}(A) + ε)𝔮(Φ^{r}(Z^{∨}) − Φ^{r}(Z^{∧})). ■

§8. End of the proof of the Theorem 5.

We can now enter upon the realization of the plan marked in the section 3. For any matrix X we denote by X^{𝔱} the matrix transposed to the matrix X. We introduce on ℂ^{q} the norm 𝔮_{3} as follows:

with each column f ∈ ℂ^{p} we connect the matrix ϑ(f) ∈ Mat_{p}(ℂ), such that all the elements of its first p − 1 rows are equal to 0 and last row i equal to f^{𝔱}; let

B_{1}^{∼} = ( \begin{matrix}0 & 1 & 0 & … & 0 \\ 0 & 0 & 1 & … & 0 \\ ⋮ & ⋮ & ⋮ & … & ⋮ \\ 0 & 0 & 0 & … & 1 \\ −b_{0}^{∼} & −b_{1}^{∼} & −b_{2}^{∼} & … & −b_{p−1}^{∼} \end{matrix} ) ,

q_{3}^{∼} = q_{B_{1}^{∼};ε_{2}/2}^{∼}, 𝔮(f) = q_{3}^{∼}(ϑ(f)).

If 𝔮(f) ≤ ε_{2}/2, then acording to the Lemma 6.6, the matrix B_{1}^{∼} − ϑ(f) is invertible and, in view of (88),

(98)  q_{3}^{∼}((B_{1}^{∼} − ϑ(f))^{−1}) ≤ γ_{6}(n, ρ_{1,1}, ε_{2}) < \frac{1}{ρ_{1,1} − ε_{2}} .

We introduce on the space A_{p,q}(ℂ) of all the p × q matrices the norm 𝔮^{∗} as follows: if F ∈ A_{p,q}(ℂ), then let F_{∗,k−1}, k = 1, …, q denotes the k-th column of the matrix F and let 𝔮^{∗}(F) = sup{𝔮(F_{∗,k−1}): k = 1, …, q}. For any F ∈ A_{p,q}(ℂ), i = 1, …, p, k = 1, …, q let F_{i−1,k−1} denotes the number, which stands in i-th row and k-th column of the matrix F, F_{p,k−1} = 1 for k = 1, …, q, and let further F_{i−1,k−1} = 0 for k = 1, …, q, i ∈ ℤ⧹[0, p].

Remark 8.1. Actually F_{i−1,k−1} play further the role of b_{i−1}(ν+k−1)−b_{i−1}^{∼} for i = 1, …, p, k = 1, …, q, where b_{i−1}(ν + k − 1) and b_{i−1}^{∼} are considered at the end of the section 3.

Let 𝔒(ε) = {F ∈ A_{p,q}ℂ: 𝔮^{∗}(F) < ε, }, 𝔒_{0} = 𝔒(ε_{2}/2). Let B_{1}^{∗∗}(F, k − 1) = B_{1}^{∼} − ϑ(F_{∗,k−1}), where F ∈ A_{q,p}ℂ. If F ∈ 𝔒_{0}, and k = 1, …, q, then according to Lemma 6.6, the matrix B_{1}^{∗∗}(F, k − 1) is invertible and

q_{3}^{∼}((B_{1}^{∗∗}(F, k − 1))^{−1}) ≤ γ_{6}(p, ρ_{1,1}, ε_{2}) < \frac{1}{ρ_{1,1} − ε_{2}} .

For any F ∈ A_{q,p}ℂ, let B_{2}(F) denotes the lower triangle q × q-matrix, in which for i = 1, …, q, k = 1, …, q, the number b_{i−k}^{∼} + F_{i−k,k−1} stands on the intersection of the i-th row and k-th column and B_{3}(F) denotes the p × q-matrix, in which for i = 1, …, p, k = 1, …, q, the number b_{q+i−k}^{∼} + F_{q+i−k,k−1} stands on the intersection of the i-th row and k-th column, B_{3}^{∼} = B_{3}(0).

Remark 8.2. Actually B_{2}(F) will play further the role of the matrix B^{∧}(ν) in (32) and B_{3}(F) will play further the role of the matrix 𝔅(ν) in (33).

Applying the Lemma 6.7 to the case, when the role of n plays p, the role of m plays q, the role of A plays B_{1}^{∼}, the role of B^{∧} plays B_{2}(F) with F ∈ 𝔒_{0}, we see that the following inequalities holds

q_{∞}^{∼}(B_{2}(F)) ≤ γ_{3}(p, q, ρ_{1,0} + ε_{2}), q_{1}^{∼}(B_{2}(F)) ≤ γ_{3}(p, q, ρ_{1,0} + ε_{2});

moreover, (B_{2}(F))^{−1} exists and

(99)  q_{∞}^{∼}((B_{2}(F))^{−1}) ≤ γ_{11}, q_{1}^{∼}((B_{2}(F))^{−1}) ≤ γ_{11},

where γ_{11} is defined in the section 5. Let A^{∨} be the column in ℂ^{n}, in which the i-th element coincides with a_{i−1}^{∼} for i = 1, …, n. Let B^{∨} be the column in ℂ^{p}, in which the i-th element coincides with b_{i−1}^{∼} for i = 1, …, p. Let U^{∨} be the column in ℂ^{q}, in which the i-th element coincides with u_{i−1}^{∼} for i = 1, …, q. For any column H ∈ ℂ^{n}, the i-th element of H will be denoted by H_{i−1,∗} for i = 1, …, n. Let

(100)  U^{∗}(F, H) = (B_{2}(F))^{−1}π_{1}(A^{∨} + H),

(101)  Φ_{H}(F) = π_{2}(A^{∨} + H) − B_{3}(F)U^{∗}(F, H) − B_{1}^{∨} =
π_{2}(A^{∨} + H) − B_{3}(F)(B_{2}(F))^{−1}π_{1}(A^{∨} + H) − B_{1}^{∨}

for any column H ∈ ℂ^{n} and any F ∈ 𝔒_{0}. Let further u_{i−1}^{∗}(F, H) for i = 1, …, q be the i-th element of the column U^{∗}(F, H), u_{q}^{∗}(F, H) = 1 and u_{i−1}^{∗}(F, H) = 0 for any column H ∈ ℂ^{n}, any F ∈ 𝔒_{0} and any i ∈ ℤ⧹[0, q].

Remark 8.3. Actually A^{∨} + H will play further the role of the column A(ν) in (32) and (33), that U^{∗}(F, H) in (101) will play the role of U(ν) in (32), and Φ_{H}(F) + B^{∨} will play the role of (▽^{q}B)(ν) in (33).

Clearly,

(102)  U^{∗}(0, 0) = U^{∨}, Φ_{0}(0) = 0.

In view of (45),

(103)  Φ_{H}(F) − Φ_{0}(F) = π_{2}(H) − B_{3}(F)(B_{2}(F))^{−1}π_{1}(H).

Let y = y(κ), κ ∈ [0, n] ∩ ℤ be an arbitrary sequence. Let

Y_{p,y}(κ) = ( y(κ) ; ⋮ ; y(κ + p − 1) ) ,

where κ ∈ [0, q + 1] ∩ ℤ. Let E_{p} be the unit p × p-matrix. Let

(104)  B_{1}^{∗}(F, H, k − 1) = B_{1}^{∗∗}(F, k − 1),

where k = 1, …, q, and B_{1}^{∗}(F, H, q) = B_{1}^{∼} − ϑ(Φ_{H}(F)), where H ∈ ℂ^{n} and F ∈ 𝔒_{0}. Making use of these notations we have

(105)  ∑_{β=0}^{q}(u_{β}^{∗}(F, H))(Y_{p,y}(β + 1) − B_{1}^{∗}(F, H, β)Y_{p,y}(β)) =

Y_{p,y}(q + 1) + ∑_{β=0}^{q−1}(u_{β−1}^{∗}(F, H)E_{q} − u_{β}^{∗}(F, H)B_{1}^{∗}(F, H, β))Y_{p,y}(β)).

ON THE DIFFERENCE EQUATION OF THE POINCARÉ TYPE 33

Let

(106)  R_{β−1}^{∗}(F, H) = u_{β−1}^{∗}(F, H)E_{q} − u_{β}^{∗}(F, H)B_{1}^{∗}(F, H, β),

where β ∈ ℤ. Then

(107)  R_{q+1}^{∗}(F, H) = E_{q}.

Iterating (106) for β = 0, , …, q. we have u_{β}^{∗}(F, H)E_{q} = −R_{β−1}^{∗}(F, H)(B_{1}^{∗}(F, H, β))^{−1} + u_{β−1}^{∗}(F, H)(B_{1}^{∗}(F, H, β))^{−1} =

− ∑_{κ=0}^{γ} R_{β−κ−1}^{∗}(F, H) ∏_{μ=β−κ}^{β}(B_{1}^{∗}(F, H, μ))^{−1}+

u_{β−γ−1}^{∗}(F, H) ∏_{μ=β−γ}^{β}(B_{1}^{∗}(F, H, μ))^{−1},

where β = 0, , …, q, and γ = 0, , …, β; in particular, for γ = β we have

u_{β}^{∗}(F, H)E_{q} = − ∑_{κ=0}^{β} R_{β−κ−1}^{∗}(F, H) ∏_{μ=β−κ}^{β}(B_{1}^{∗}(F, H, μ))^{−1}.

If β = q − 1, then

(108)  u_{q−1}^{∗}(F, H)E_{q} =

− ∑_{κ=0}^{q−1} R_{q−1−κ−1}^{∗}(F, H) ( ∏_{μ=q−1−κ}^{q−1}(B_{1}^{∗}(F, H, μ))^{−1}) (B_{1}^{∗}(F, H, q))^{−1}.

It follows from (106) for β = q and from (108) that

(109)  B_{1}^{∗}(F, H, q) = −R_{q−1}^{∗}(F, H) + u_{q−1}^{∗}(F, H) =

−R_{q−1}^{∗}(F, H) − ∑_{κ=0}^{q−1} R_{q−1−κ−1}^{∗}(F, H) ∏_{μ=q−1−κ}^{q−1}(B_{1}^{∗}(F, H, μ))^{−1} =

−R_{q−1}^{∗}(F, H) − ∑_{κ=1}^{q} R_{q−κ−1}^{∗}(F, H) ∏_{μ=q−κ}^{q−1}(B_{1}^{∗}(F, H, μ))^{−1} =

− ∑_{κ=0}^{q} R_{q−κ−1}^{∗}(F, H)Π_{q−κ,q}(F),

where Π_{ω,ω}(F) = E_{q} for ω ∈ [0, q] ∩ ℤ and Π_{α,ω}(F) = ∏_{μ=α}^{ω−1}(B_{1}^{∗∗}(F, μ))^{−1} for

ω ∈ [1, q] ∩ ℤ, α ∈ [0, ω] ∩ ℤ.

Let L be a linear space over ℂ; on L^{ℤ} acts a ℂ-linear operator ▽, for which (▽X)(ν) = X(ν + 1), where X ∈ L^{ℤ}, ν ∈ ℤ. Let 𝔐^{∨}(L) be the set of all the ℂ-linear maps of the space L into L; then the equality (AX)(ν) = A(ν)X(ν), where A ∈ (𝔐^{∨}(L))^{ℤ}, X ∈ L^{ℤ}, ν ∈ ℤ determines the natural embeding of the space (𝔐^{∨}(L))^{ℤ} into the space 𝔐^{∨}(L^{ℤ}) of all the ℂ-linear maps of the space L^{ℤ} into L^{ℤ}. The identity operator on L^{ℤ} will be denoted by I and the operator, for wich (▽A)(ν) = A(ν + 1) for A ∈ (𝔐^{∨}(L))^{ℤ}, ν ∈ ℤ, will be (also) denoted by ▽. Clearly, for the elements (55) the following equality holds

(110)  ∏_{k=1}^{l}(I − z_{k}A) = I + ∑_{k=1}^{l}(−1)^{k}σ_{k,l}(z_{1}, …, z_{l})A^{k},

where A ∈ 𝔐^{∨}(L^{ℤ}) and l ∈ ℕ. Let z_{i} for i = 1, …, p are zeros of the polynomial T_{1}(z) and z_{i} for i = p + 1, …, n are zeros of the polynomial T_{2}(z) and each zero of the polynomial T (z) is counted according its multiplicity. Clearly, b_{k}^{∼} = (−1)^{p−k}σ_{p−k,p}(z_{1}, …, z_{p}) for k = 0, …, p, and also u_{k}^{∼} = (−1)^{q−k}σ_{q−k,q}(z_{p} + 1, …, z_{p+k}) for k = 0, …, q. We define b_{k}^{∼} = 0 for k ∈ ℤ⧹[0, p] and also u_{k}^{∼} = 0 for k ∈ ℤ⧹[0, q]. Let L = Mat_{p}(ℂ),

(111)  x(κ) = Δ^{(0)}(F, H, κ) = Δ(F, H, κ) = B_{1}^{∗}(F, H, κ) − B_{1}^{∼}

for κ ∈ [0, q] ∩ ℤ, but x(κ) = Δ^{(0)}(F, H, κ) = Δ^{(0)}(F, H, κ) = 0 for κℤ⧹[0, q]. In view of (102) and (98), if 𝔮^{∗}(F) ≤ ε_{2}/2 then there exists (B_{1}^{∗∗}(F, κ − 1))^{−1} for κ ∈ [1, q] ∩ ℤ, and q_{3}^{∼}((B_{1}^{∗∗}(F, κ − 1))^{−1}) < \frac{1}{ρ_{1,1}−ε_{2}} , where ε_{2} ∈ (0, ρ_{1,1}); let D ∈ L^{ℤ} and D^{∨} ∈ 𝔐(L)^{ℤ} is defined as follows:

(112)  D(κ) = (B_{1}^{∗∗}(F, κ))^{−1}

for κ ∈ [0, q −1]∩ℤ and D(κ) = 0 for κ ∈ ℤ⧹[0, q −1]. Let further D^{∨}(κ) for κ ∈ ℤ be the operator, which turns any matrix V ∈ L into V D(κ). Let

(113)  Δ^{(i)}(F, H, κ) = Δ^{(i−1)}(F, H, κ) − z_{i+p}Δ^{(i−1)}(F, H, κ − 1)D(κ − 1),

where i ∈ [1, q] ∩ ℤ, κ ∈ ℤ. Taking in (110) A = ▽^{−1}D^{∨}, we obtain the equality

(114)  Δ^{(i)}(F, H, κ) = Δ(F, H, κ)+

∑_{k=1}^{i}(−1)^{k}σ_{k,i}(z_{p+1}, …, z_{p+i})Δ(F, H, κ − k) ∏_{μ=κ−k}^{κ−1} D(μ) =

Δ(F, H, κ) + ∑_{k=1}^{i}(−1)^{k}σ_{k,i}(z_{p+1}, …, _{p+i})Δ(F, H, κ − k) ∏_{μ=κ−k}^{κ−1}(B_{1}^{∗∗}(F, μ))^{−1} =

∑_{k=0}^{i}(−1)^{k}σ_{k,i}(z_{p+1}, …, z_{q+i})Δ(F, H, κ − k)Π_{κ−k,κ},

where i ∈ [1, q] ∩ ℤ, κ ∈ [i, q] ∩ ℤ. Clearly, (114) holds also for i = 0. In view of (114),

(115)  Δ^{(q)}(F, H, q) = ∑_{k=0}^{q} u_{k}^{∼}Δ(F, H, q − k)Π_{q−k,q} =

∑_{κ=0}^{q} u_{q−κ}^{∼}Δ(F, H, k)Π_{k,q}.

Let in (106) Ω_{β−1}^{∗}(F, H) = R_{β−1}^{∗}(F, H) − R_{β−1}^{∗}(0, 0), where β ∈ [1, q] ∩ ℤ. Then, in view of (109) and (102)

(116)  B_{1}^{∗}(F, H, q) = − (∑_{κ=0}^{q} R_{q−κ−1}^{∼}(0, 0)Π_{q−κ,q}(F)) −

∑_{κ=0}^{q} Ω_{q−κ−1}^{∗}(F, H)Π_{q−κ,q}(F) = − (∑_{κ=0}^{q} u_{q−κ−1}^{∼}Π_{q−κ,q}(F)) +

(∑_{κ=0}^{q} u_{q−κ}^{∼}B_{1}^{∼}Π_{q−κ,q}(F)) − ∑_{κ=0}^{q} Ω_{q−κ−1}^{∗}(F, H)Π_{q−κ,q}(F) =

B_{1}^{∗}(F, H, q) = − (∑_{κ=1}^{q} u_{q−κ}^{∼}B_{1}^{∗∗}(F, q − κ)Π_{q−κ,q}(F)) +

(∑_{κ=0}^{q} u_{q−κ}^{∼}B_{1}^{∼}Π_{q−κ,q}(F)) − ∑_{κ=0}^{q} Ω_{q−κ−1}^{∗}(F, H)Π_{q−κ,q}(F).

In view of (116),

(117)  0 = − (∑_{κ=0}^{q} u_{q−κ}^{∼}Δ(F, H, q − κ)Π_{q−κ,q}(F)) −

∑_{κ=0}^{q} Ω_{q−κ−1}^{∗}(F, H)Π_{q−κ,q}(F).

We add (117) with (115) and obtain the equality

Δ^{(q)}(F, H, q) = − ∑_{κ=0}^{q} Ω_{q−κ−1}^{∗}(F, H)Π_{q−κ,q}(F).

In view of (113) and (112),

Δ^{(q)}(F, H, q) = Δ^{(q−k)}(F, H, q) − ∑_{j=1+q−k}^{k} z_{q+1−j+p}Δ^{(q−j)}(F, H, q − 1)D(q − 1) =

Δ^{(q−k)}(F, H, q − k) − ∑_{j=q+1−k}^{q} z_{j+p}Δ^{(j−1)}(F, H, q − 1)(B_{1}^{∗∗}(F, q − 1))^{−1},

where k ∈ [0, q] ∩ ℤ. Therefore

(118)  Δ^{(i)}(F, H, q) = Δ^{(q)}(F, H, q)+

∑_{j=i+1}^{q} z_{j+p}Δ^{(j−1)}(F, H, q − 1)(B_{1}^{∗∗}(F, q − 1))^{−1},

where i ∈ [0, q] ∩ ℤ. In view of (111), (104) and (114),

(119)  Δ^{(i)}(F, H, κ) = Δ^{(i)}(F, 0, κ) =

∑_{k=0}^{i}(−1)^{k}σ_{k,i}(z_{q+1}, …, z_{q+i})ϑ(F_{∗,κ−k})Π_{κ−k,κ},

where i ∈ [1, q − 1] ∩ ℤ, κ ∈ [i, q − 1] ∩ ℤ.

For any F ∈ 𝔒_{0}, H ∈ ℂ^{q} we denote by Ψ_{H}(F) the matrix in A_{p,q}(ℂ), in which the last column coincides with Φ_{H}(F) and on the first q − 1 columns stands the matrix obtained by the remowing of the first column in F.

Remark 8.4. The passage F → Ψ_{H}(F) corresponds to the described in the section 3 passage from b_{i−1}(ν + s) for i = 1, …, p and s = 0, …, q − 1 to b_{i−1}(ν + 1 + s) for i = 1, …, p and s = 1, …, q.

We denote below the distance determined by a norm on linar space in the same way as this norm. Let

ρ_{0}(F) = sup{q_{B_{1};ε_{2}/2}^{∼}(ε_{4})^{1−i}Δ^{(i−1)}(F, 0, k − 1): i = 1, …, q, k = 1, …, q}, where ε_{4} is defined in (67) and Y ⊂ 𝔒_{0}. Clearly, ρ^{∼}(F) is continuous function on 𝔒_{0}. Let Y = {F ∈ 𝔒_{0}: ρ_{0}(F) < ε_{2}/2.} I view of (102), 0 ∈ Y and Y is nonempty open subset in 𝔒_{0}. We study the map ψ_{H}(F) on the open set Y for H ∈ ℂ^{p}. Let

(120)  0 < γ ≤ \frac{ε_{2}}{2} , F ∈ Y, ρ_{0}(F) < γ, q_{3}^{∼}(Δ^{(μ)}(F, 0, β) < γ(ε_{4})^{μ},

where μ = 0, …, q − 1 and β = μ, …, q − 1. Let

(121)  θ^{(i)}(γ) = γ(ε_{4})^{i},

where i = 0, …, q. Clearly,

(122)  θ^{(i)}(γ) = θ^{(q)}(γ) + (1 − ε_{4}) ∑_{μ=i}^{q−1} θ^{(μ)}(γ).

In view of (120), (121) and (98),

(123)  q_{3}^{∼}(Δ^{(μ)}(F, 0, β)) ≤ θ^{(μ)}(γ),

where μ = 0, …, q − 1 and β = μ, …, q − 1,

(124)  q_{3}^{∼}(B_{1}^{∗∗}(F, 0, β)) ≤ \frac{1}{ρ_{1,1} − ε_{2}} ,

where β = 0 …, q − 1. In view of (118), (119), (122), (123), (124), (67),

(125)  q_{3}^{∼}(Δ^{(i)}(F, H, q)) ≤ q_{3}^{∼}(Δ^{(q)}(F, H, q))+

q_{3}^{∼}((B_{1}^{∗∗}(F, q − 1))^{−1}) ∑_{j=i+1}^{q} |z_{j+p}|q_{3}^{∼}(Δ^{(j−1)}(F, 0, q − 1)) <

q_{3}^{∼}(Δ^{(q)}(F, H, q)) + (1 − ε_{4}) ∑_{j=i+1}^{q} θ^{(j−1)}(γ) = q_{3}^{∼}(Δ^{(q)}(F, H, q)) + θ^{(i)}(γ) − θ^{(q)}(γ),

where i ∈ [0, q] ∩ ℤ. Let b_{α}^{∼} = 0, u_{β}^{∼} = 0 for α ∈ ℤ⧹[0, p] and β ∈ ℤ⧹[0, q]. In

view of (3), (8) and (9), a_{τ}^{∼} = ∑_{α=0}^{τ} b_{α}^{∼}u_{τ−α}^{∼}, where τ ∈ ℤ. Let b_{k−1}^{∗}(F, H, i)) be the

product of −1 and the value, which stands in the matrix B_{1}^{∗}(F, H, i) in the last row and the k − th column, where i = 0, …, q and k = 1, …, p. Then

(126)  Δ_{y}^{∗}(F, H, i) = Y_{p,y}(i + 1) − B_{1}^{∗}(F, H, i)Y_{p,y}(i) =

( 0 ; 0 ; ⋮ ; 0 ; ∑_{α=0}^{p} b_{α}^{∗}(F, H, i)y(i + α), ) ,

where i = 0, …, q. In view of (105), (106) and (107), we have

∑_{β=0}^{q+1}(u_{β}^{∗}(F, H))Δ_{y}^{∗}(F, H, β) = ∑_{β=0}^{q} R_{β−1}Y_{p,y}(β).

It follows from the definition of the matrices B_{1}^{∗}(F, H, β) for β = 0, …, q, from the dfinition of the matrices B_{2}(F), B_{3}(F) and from the definition of the numbers b_{k}^{∗}(F, H, i) for i = 0, …, q and k = 1, …, p that B_{2}(F) is the lower triangle q ×q-matrix, in which for i = 1, …, q and k = 1, …, q the number b_{i−k}^{∗}(F, H, k − 1) = b_{i−k}^{∼} + F_{k−1,i−k}, stands on the intersection of the i-th row and k-th column, B_{3}(F) is the p × q-matrix, in which for i = 1, …, p, k = 1, …, q, the number b_{q+i−k}^{∗}(F, H, k − 1) = b_{q+i−k}^{∼} + F_{q+i−k,k−1} stands on the intersection of the i-th row and k-th column. In view of (100),

(127)  a_{i−1}^{∼} + H_{i−1,∗} = ∑_{k=1}^{q} u_{k−1}^{∗}(F, H)b_{i−k}^{∗}(F, H, k − 1),

where i = 1, …, q. In view of (101), for i = 1, …, p the i − th element of the column π_{2}(A^{∨} + H) − B_{3}(F)(B_{2}(F))^{−1}π_{1}(A^{∨} + H) coincides with b_{i−1}^{∗}(F, H, q). Therefore

a_{q+i−1}^{∼} + H_{i−1,∗} = b_{i−1}^{∗}(F, H, q) + ∑_{k=1}^{q} u_{k−1}^{∗}(F, H)b_{q+i−k}^{∗}(F, H, k − 1),

where i = 1, …, p. Consequently, in view of (126) - (127),

∑_{β=0}^{q} u_{β}^{∗}(F, H) ∑_{α=0}^{p} b_{α}^{∗}(F, H, β)y(β + α) =

∑_{τ=0}^{n} (∑_{β=0}^{q} u_{β}^{∗}(F, H)b_{τ−β}^{∗}(F, H, β)) y(τ) = ∑_{τ=0}^{n} y(τ)(a_{τ}^{∼} + H_{τ,∗}^{∼})

and therefore

(128)  ∑_{β=0}^{q+1} R_{β−1}(F, H)Y_{p,y}(β) = ∑_{β=0}^{q}(u_{β}^{∗}(F, H))Δ_{y}^{∗}(F, H, β) =

( 0 ; 0 ; ⋮ ; 0 ; ∑_{τ=0}^{n} y(τ)(a_{τ}^{∼} + H_{τ,∗}^{∼}) ) ,

(129)  ∑_{β=0}^{q+1} R_{β−1}(0, 0)Y_{p,y}(β) = ( 0 ; 0 ; ⋮ ; 0 ; ∑_{τ=0}^{n} y(τ)a_{τ}^{∼} ) .

Clearly, H_{n,∗}^{∼} = 0 and, in view of (107), Ω_{q+1}^{∗}(F, H) = 0. After the substraction of the equality (129) from the equality (128) we have

(130)  ∑_{β=0}^{q} Ω_{β−1}(F, H)Y_{p,y}(β) = ( 0 ; 0 ; ⋮ ; 0 ; ∑_{τ=0}^{n−1} y(τ)H_{τ,∗}^{∼} ) .

Let Z(F, H.0) = E_{p},

(131)  Z(F, H.μ) = ∏_{κ=1}^{μ}(B_{1}^{∗}(F, H, μ − κ))^{−1},

where μ ∈ [0, q + 1] ∩ ℤ. Clearly,

(132)  Π_{μ,q}(F) = Z(F, H.μ)Π_{0,q}(F),

where μ ∈ [0, q] ∩ ℤ. Therefore (128) may be rewritten in the form

(133)  Δ^{(q)}(F, H, q) = − ∑_{μ=0}^{q} Ω_{μ−1}^{∗}(F, H)Z(F, H.μ)Π_{0,q}(F).

Let y_{k}^{∗}(F, H, k − 1) = 1 for k = 1, …, p, and let y_{k}^{∗}(F, H, μ) = 0 for k = 1, …, p and μ ∈ [0, p − 1] ∩ ℤ⧹{k − 1}. Let further

y_{k}^{∗}(F, H, μ) = − ∑_{α=0}^{p−1} b_{α}^{∗}(F, H, μ − p)Zy_{α}^{∗}(F, H, μ),

for k = 1, …, p and μ ∈ [p, n] ∩ ℤ. Let

(134)  Y_{k}^{∗}(F, H, μ) = Y_{p,y_{k}^{∗}(F,H,μ)}) = ( y_{k}^{∗}(F, H, μ) ; ⋮ ; y_{k}^{∗}(F, H, μ + p − 1) ) ,

for k = 1, …, p and μ ∈ [0, q + 1] ∩ ℤ. Clearly,

(135)  Z(F, H.μ) = (Y_{k}^{∗}(1, H, μ)), …, Y_{p}^{∗}(F, H, μ)).

Let

(136)  α_{k}(F, H) = ∑_{τ=0}^{n−1} H_{τ,∗}^{∼}y_{k}^{∗}(F, H, τ),

where k = 1, …, p. In view of (131), (134), (135) , (130) and (136),

(137)  ∑_{β=0}^{q} Ω_{β−1}(F, H)Z(F, H, β) = ( \begin{matrix}0 & … & 0 \\ 0 & … & 0 \\ . & … & . \\ 0 & … & 0 \\ α_{1}(F, H) & … & α_{p}(F, H) \end{matrix} ) .

In view of (123),

(138)  q_{3}^{∼}(Δ(F, H, β)) = q_{3}^{∼}(Δ(F, 0, β)) < ε_{2}/2,

where β = 0, …, q − 1. In view of (84), (83), (74),

(139)  q_{∞}^{∼}(Π_{0,q}) ≤ γ_{19}

In view of (136) and (137),

(140)  q_{∞}^{∼}(∑_{β=0}^{q} Ω_{β−1}(F, H)Z(F, H, β) =

∑_{τ=0}^{n−1} |H_{τ,∗}^{∼}| ∑_{k=1}^{p} |y_{k}^{∗}(F, H, τ)|.

In view of (134) and (135),

∑_{k=1}^{p} |y_{k}^{∗}(F, H, τ)| ≤ sup{q_{∞}^{∼}(Z(F, H, β)): β ∈ [0, q] ∩ ℤ.}

In view of (138),

(141)  ∥B_{1}^{∗}(F, H, β)∥_{sp} ≤ q_{3}^{∼}(B_{1}^{∗}(F, H, β)) ≤ ρ_{1,0} + ε_{2},

where β = 0, …, q − 1. In view of (77),

q_{∞}(B_{1}^{∗}(F, H, β)) ≤ γ_{13},

where β = 0, …, q − 1,

(142)  sup{q_{∞}^{∼}(Z(F, H, β)): β ∈ [0, q] ∩ ℤ} ≤ γ_{0}(q + 1, γ_{13}).

In view of (141) and (74),

(143)  sup{q_{∞}^{∼}(Z(F, H, β)): β ∈ [0, q] ∩ ℤ} ≤ γ_{8}γ_{14}.

In view of (142) and (143),

(144)  sup{q_{∞}^{∼}(Z(F, H, β)): β ∈ [0, q] ∩ ℤ.} ≤ γ_{20}.

In view of to (73), (133), (139), (140), (142),

(145)  q_{3}^{∼}(Δ^{(q)}(F, H, q)) ≤ γ_{8}q_{∞}(Δ^{(q)}(F, H, q)) ≤
γ_{8}γ_{19}γ_{20}q_{∞}(H) = γ_{21}q_{∞}(H).

Let now

(146)  q_{∞}(H) ≤ γε_{5}(ε_{4})^{q}/γ_{21}.

It follows then from (144), (145) that

(147)  q_{3}^{∼}(Δ^{(q)}(F, H, q)) ≤ γε_{5}(ε_{4})^{q}.

In view of (125), (147), (123) and (122),

(148)  q_{3}^{∼}(Δ^{(i)}(F, H, q)) < γε_{5}(ε_{4})^{q}) + θ^{(i)}(γ) − θ^{(q)}(γ) =
γ(ε_{4})^{i}(1 − ε_{5}((ε_{4})^{q−i}) ≤ γ(ε_{4})^{i}ε_{6} =
θ^{(i)}(γ)ε_{6} < θ^{(i)}(γ).

Taking in (146) γ = ε_{2}/2, we see that, if (146) is fulfilled, then Ψ_{H} maps the open set Y into Y. Taking in (146) γ = ρ^{∼}(F), we see that, if (146) is fulfilled, then ρ^{∼}(Ψ_{0}^{q}(F)) ≤ ε_{6}ρ^{∼}(F) for any F ∈ Y, moreover Ψ_{0}(0) = 0. In view of (132), (139),

sup{q_{∞}^{∼}(Π_{μ,q}(F): μ ∈ [0, q] ∩ ℤ} ≤ γ_{19}γ_{20}.

Let q_{∞}^{∗}(F) = sup{q_{∞}(F_{∗,k−1}): k = 1, …, q}, where F ∈ A_{p,q}(ℂ). In view of (114),

q_{3}^{∼}Δ^{(i)}(F, H, κ) ≤ (1 + ρ_{2,0})^{i}γ_{8}γ_{19}γ_{20}q_{∞}^{∗}(F)

for i ∈ [0, q − 1] ∩ ℤ. Therefore ρ^{∼}(F) ≤ γ_{22}q_{∞}^{∗}(F) for any F ∈ Y. Clearly,

q_{∞}^{∗}(F) ≤ γ_{8}ρ^{∼}(F)

for any F ∈ Y. So,

q_{∞}^{∗}(Ψ_{0}^{qs}(F)) ≤ γ_{8}ρ^{∼}(Ψ_{0}^{qs}(F)) ≤ γ_{8}(ε_{6})^{s}ρ^{∼}(F, 0) ≤ γ_{8}γ_{22}(ε_{6})^{s}q_{∞}(F),

where F ∈ Y, s ∈ ℕ. Let is chosen s ∈ ℕ such that γ_{8}γ_{22}(ε_{6})^{s} < 1. Then Ψ_{0}^{qs} is centripetal relaively q_{∞}^{∗}. Since q_{∞}^{∗} is a norm on F ∈ A_{p,q}(ℂ), it follows that we may apply the Lemma 6.4. Therefore there exists m ∈ ℕ such that Ψ_{0}^{qsm} is contracted map of the metric space (Y, q_{∞}^{∗}). In view of (101), Φ_{0}(F^{∨}) − Φ_{0}(F^{∧}) = W_{1} + W_{2}, where

W_{1} = (B_{3}(F^{∧}) − B_{3}(F^{∨}))(B_{2}(F)^{∧})^{−1}π_{1}(A^{∨}),
W_{2} = B_{3}(F^{∨})(B_{2}(F)^{∨})^{−1}(B_{2}(F)^{∨} − B_{2}(F)^{∧})(B_{2}(F)^{∧})^{−1}π_{1}(A^{∨}).

Therefore

(149)  q_{∞}(Φ_{0}(F^{∨}) − Φ_{0}(F^{∧})) ≤ q_{∞}(W_{1}) + q_{∞}(W_{2}).

In view of (77), if F ∈ 𝔒_{0}, then the absolute value of the each element of the the matrix B_{3}(F)^{∨} is not greater than γ_{4}(n, ρ_{1}, 0) + γ_{8}ε_{2}/2 = γ_{18}. The absolute value of the each element of the matrix B_{2}(F^{∧}) − B_{2}(F^{∨}) and the matrix B_{3}(F^{∧}) − B_{3}(F^{∨}) is not greater than q_{∞}^{∗}(F^{∧} − F^{∨}). According to the inequality q_{∞}(π_{1}(A^{∨})) ≤ γ_{4}(n, ρ_{1}, 0), and, in view of (103), (100), (77), (101) and (149),

(150,)  q_{∞}(B_{3}(F)(B_{2}(F))^{−1}H) ≤ qγ_{18}γ_{11}q_{∞}(H),

(151)  q_{∞}(Φ_{H}(F) − Φ_{0}(F)) ≤ γ_{23}q_{∞}(H)

for any F ∈ O_{0}; moreover

(152)  q_{∞}^{∗}(Ψ_{0}(F^{∧}) − Ψ_{0}(F^{∨})) ≤ γ_{24}q_{∞}^{∗}(F^{∧} − F^{∨}),

where

γ_{24} = qγ_{11}γ_{4}(n, ρ_{1,0})(1 + qγ_{11}γ_{18}), {F^{∧}, F^{∨}} ⊂ Y.

In view of (151) and (152),

q_{∞}((Φ_{0}(F))^{k−1}Φ_{H}(F) − (Φ_{0}(F))^{k}) ≤ γ_{23}(γ_{24})^{k}q_{∞}(H),

where k = 1, …, qsm and F ∈ 𝔒_{0} and therefore

(153)  q_{∞}^{∗}((ψ_{0}(F))^{k−1}ψ_{H}(F) − (ψ_{0}(F))^{k}) ≤ γ_{23}γ_{0}(qsm, γ_{24})q_{∞}(H),

where k = 1, …, qsm and F ∈ 𝔒_{0}.

For the bounded metric space (Y, q_{∞}^{∗}) are fulfilled all the conditions of the Lemma 7.1, where Ψ_{0} plays the role of the map A, the map ψ_{H(ν)} with H(ν) = A(ν) − A^{∨} plays the role of the map B_{n}, the number qsm plays the role of the number r and 0 ∈ Y plays the role of the fixed point Z^{∗}. Let

(154)  lim_{ν→∞} q_{∞}(H_{ν}) = 0,

then there exists m ∈ ℕ such that q_{∞}(H_{ν}) ≤ γε_{5}(ε_{4})^{q}/γ_{21} for ν ∈ m + ℕ − 1. Then Ψ_{H_{ν}} maps Y in Y, if ν ∈ m + ℕ − 1. Let F_{m} ∈ Y, F_{ν+1} = Ψ_{H_{ν}} (F_{ν}) for all the ν ∈ m + ℕ − 1. In view of (154),

lim_{ν→∞} δ_{qsm}(Ψ_{H_{ν}}) = 0;

in view of the Lemma 7.1, lim_{ν→∞} q_{∞}^{∗}(F_{ν}) = 0. In view of (100), (99) and (101),

lim_{ν→∞} q_{∞}(U^{∗}(F_{ν}, H_{ν}) − U^{∨}) = 0, lim_{ν→∞} q_{∞}Ψ_{H_{ν}} (F_{ν}) = 0.

So, we obtain the proof of the Perron’s decomposition Lemma. If q_{∞}(H_{ν}) = O(ξ(ν)), then, in view of (152) and (153),

δ_{qsm}(Ψ_{H_{ν}}) = O(ξ(ν)).

In view of the Lemma 7.1, (100) and (99),

q_{∞}^{∗}(F_{ν}) = O(ξ(ν)), q_{∞}(U^{∗}(F_{ν}, H_{ν}) − U^{∨}) = O(ξ(ν))

and

q_{∞}(Ψ_{H_{ν}} (F_{ν})) = O(ξ(ν)). ■.

Theorem 5 is proved.

References

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[2] A.O.Gelfond and I.M.Kubenskaya, On the theorem of Perron in the theory of differrence equations (in Russian), IAN USSR, math. ser. 17 (1953), 83 – 86..

[3] M.A.Evgrafov, New proof of the theorem of Perron (in Russian) jour IAN USSR, math. ser. 17 (1953), 77 – 82.

[4] G.A.Frejman, On theorems of of Poincaré and Perron (in Russian), UMN 12, 3 (75) (1957), 243 – 245.

[5] CH.-J. de la Vallée Poussin, Course d’analyse infinitésimale (Rus. tr. by G.M.Fikhtengolts), GT-TI, 1933.

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Moscow State Institute of Electronics and Mathematics Bolshoi Trekhsvyatitelskij pereulok, 1-3/12, str.8 Moscow 109028 Russian Federation E-mail: gutnik@gutnik.mccme.ru