VARIABLE COEFFICIENTS IN BANACH SPACES

VARIABLE COEFFICIENTS IN BANACH SPACES

ALEXANDRU MIHAIL

The purpose of the paper is to study the convergence of a linear recurrence with positive variable coefficients with elements from a Banach space and to estimate the speed of the convergence. The conditions depend of the sum of the coeffi- cients and the minimum of the coefficients. In this paper we study the case when the sequence of the sum of the coefficients is convergent to 1. The results are first proved for recurrences of real numbers. These results are extended later for Banach spaces.

AMS 2000 Subject Classification: 40A05.

Key words: linear recurrence, homogeneous linear recurrence, Banach space.

1. INTRODUCTION AND PRELIMINARIES

The aim of the paper is the study of linear recurrences with positive variable coefficients in Banach spaces (see Definition 1.1 below). The paper extends the results obtained in [2], where the convergence or the divergence of the linear recurrence was studied. Here we want to estimate the speed of the convergence of the linear recurrence.

The Banach spaces will be supposed to be real Banach spaces although this is not necessary. The results are also valid for complex Banach spaces because every complex Banach space can be seen as a real Banach space. For a Banach spaceX, 0_X denotes the identity element ofXand for a set A⊂X, hAi denotes the Banach subspace of X generated by Aand convhAidenotes the convex closure of the set A, that is

convA=

X

i=1,n

aixi|x₁, x2, . . . , xn∈A, a1, a2, . . . , an∈[0,1], X

i=1,n

ai= 1

.

n, m, k, l, i, j, p denotes natural numbers if we do not say otherwise, δⁱ_j =

=

1 ifi=j 0 ifi6=j.

MATH. REPORTS12(62),1 (2010), 9–29

Definition 1.1. Let X be a real normed space andk ≥ 1 be fixed. Let a_n= (a¹_n, a²_n, . . . , a^k_n)

n≥k+1be a sequence of elements fromR^kand (b_n)n≥k+1

be a sequence of elements from X. The sequence (x_n)n≥1 given by x_n+k=a¹_n+kxn+k−1+a²_n+kxn+k−2+· · ·+a^k_n+kxn+b_n+k

forn≥1 is called the linear recurrence of orderkwith coefficients (an)n≥k+1, free terms (b_n)n≥k+1 and initial values x₁, x₂, . . . , x_k ∈ X. The sequence is called homogeneous if bn= 0 for everyn≥k+ 1.

For two sequences of real numbers (a_n)n≥1and (b_n)n≥1,(a_n)n≥1∼(b_n)n≥1

means that (a_n)n≥1 is convergent if and only if (b_n)n≥1 is convergent.

The infinite series and products are denoted by P

n≥1

and Q

n≥1

. Notation. P

l=m,n

a_l= 0 and Q

l=m,n

a_l = 1 if m > n.

The following lemma is well-known (see [1] or [3]).

Lemma 1.1. For a sequence of real positive numbers (a_n)n≥1 we have a) Q

n≥1

(1 +an)∼ P

n≥1

an; b) Q

n≥1

(1−an)>0⇔ P

n≥1

an<∞ if an<1for everyn∈N^∗.

We recall, from [2], the following important remark on homogenous real linear recurrences with variable coefficients:

Remark1.1. Let (a_n)n≥k+1be a fixed sequence witha_n∈R^k+and (x_n)n≥1

be the homogeneous linear recurrence of order kassociated with the sequence (an)n≥k+1 with initial values x1, x2, . . . , x_k >0 that is

xn+k =a¹_n+kxn+k−1+a²_n+kxn+k−2+· · ·+a^k_n+kxn

for n≥1. Then

a) If there exist initial values x1, x2, . . . , x_k > 0 such that lim

n→∞xn = 0,

n→∞limx_n=∞and respectively (x_n)n≥1is bounded, then for every initial values x₁, x₂, . . . , x_k (not necessary greater than 0 in the first and the third case)

n→∞limxn= 0, lim

n→∞xn=∞ and respectively (xn)n≥1is bounded.

b) If there exists initial values x₁, x₂, . . . , x_k > 0 such that lim

n→∞x_n ∈ (0,∞), then lim

n→∞

P

j=1,k

a^jn= 1.

This remark suggested us to divide the problem when x_n is real in the cases lim

n→∞xn = 0, lim

n→∞xn=∞ and (xn)n≥1 is a bounded sequence, in parti- cular when lim

n→∞x_n ∈ (0,∞). These cases depend on the behavior of the

sequence P

j=1,k

a^jn

n≥k+1. In Banach spaces we have similar cases. When P

j=1,k

a^jn

is smaller than 1, (x_n)n≥1 tends to be convergent to 0, for example when lim sup

n→∞

P

j=1,k

a^jn < 1 or when P

j=1,k

a^jn < 1 and Q

n≥k+1

P

j=1,k

a^jn

= 0. When P

j=1,k

a^jn is greater than 1, (xn)n≥1 tends to be divergent, for example when lim inf

n→∞

P

j=1,k

a^jn >1 or when P

j=1,k

a^jn>1 and Q

n≥k+1

P

j=1,k

a^jn

=∞. In particu- lar if x1, x2, . . . , xk are in a cone then (kx_nk)n≥1 tends to be divergent to ∞.

Also when P

j=1,k

a^j_n is convergent to 1 quickly enough, the sequence (x_n)n≥1 is bounded and if the coefficients are not to small then it is convergent to a limit which is different from 0 in general. These cases are also valid for inhomoge- neous linear recurrences. In this paper we will study the last case when the series P

n≥k+1

P

j=1,k

a^j_n

−1

is convergent. This case seems to be difficult in the sense that the calculations are longer than in the other cases. The paper is divided into 6 sections. The first section is the introduction and the second one gives some particular cases in which is possible to calculate the general term. These cases give an idea of what happens in the general case. The third section contains the study of the real homogenous case when the sum of the coefficients is 1. The next section contains the study of the general homogenous case when the sum of the coefficients is 1. The fifth section contains the results obtained in the general inhomogeneous case when the sum of the coefficients is 1. The last section contains the general case.

2. PARTICULAR CASES

In this part we will give some particular cases when it is possible to find the general term of the recurrence. We study first the casek= 1.

Let (an)n≥1 and (bn)n≥1 be sequences of real numbers with an 6= 0 and let (xn)n≥0 be the sequence given by the linear recurrence

xn+1=anxn+bn

for n≥0.An inductive calculation shows us that

xn= Y

l=1,n

a_l

!

x0+ X

l=1,n

Y

j=l+1,n

aj

b_l

! .

We remark that (xn)n≥0 is convergent for every x0 if and only if the product Q

l≥1

a_l is convergent and the sequence P

l=1,n

b_l Q

j=l+1,n

a_j

is also conver- gent.

Let us suppose that the series P

n≥1

|a_n−1|is convergent (this imply that the product Q

l≥1

a_lis convergent and different from 0). Because the convergence of the product Q

l≥1

al and of the sequence P

l=1,n

bl Q

j=l+1,n

aj

does not change when we change a finite number of terms of the sequence (a_n)n≥1 (maintaining the condition a_n 6= 0) we can suppose also that in this case we have a_n ∈ (1−ε,1 +ε) for an ε >0. If bn ≥ 0 the sequence (xn)n≥0 is convergent for every x₀ if and only if the series P

l≥1

b_l is convergent. To see this let us remark that 0 < Q

n≥1

(1− |a_n −1|) ≤ Q

j=l+1,n

aj ≤ Q

n≥1

(1 +|a_n−1|) < +∞. If the series P

l≥1

|b_l|is convergent then (x_n)n≥0 is also convergent for everyx₀. If the series P

l≥1

|b_l|is divergent there exists a sequence (an)n≥1 such that (xn)n≥0 is divergent.

We now study the casek= 2 when the sum of the coefficients is 1.

Lemma 2.1. Let (x_n)n≥0 be the sequence of real numbers given by the linear recurrence

xn+1 = (1−an)xn+anxn−1

for n≥1 where 0< a_n<1. If x₁ 6=x₀ the sequence (x_n)n≥1 is convergent if and only if the product Q

j≥1

aj is convergent to 0.

Proof. We have

xn+1−xn=−a_n(xn−xn−1), x_n+1−x_n= (−1)ⁿ(x₁−x₀)Y

l=1,n

a_l and

x_n+1 = (x₁−x₀) 1 + X

l=1,n

(−1)^lY

j=1,l

a_j !

+x₀.

If the sequence (x_n)n≥0 is convergent then the sequence (x_n+1−x_n)n≥0

is convergent to 0 and then Q

j≥1

aj is also convergent to 0.

If Q

j≥1

ajis convergent to 0, because 0< an<1, the sequence Q

l=1,n

al

n≥1

is decreasing to 0 and the series P

l≥1

(−1)^l Q

j=1,l

a_j

is convergent (by the Leib- nitz corollary for alternate series).

Lemma 2.2. Let (xn)n≥0 be the sequence given by the linear recurrence x_n+1= (1−a_n)x_n+a_nxn−1+b_n

for n≥1, where 0< an<1 and (bn)n≥1 is a sequence of real numbers. Then a)If the sequence (x_n)n≥0 is convergent for every x₀ and every x₁ then the product Q

j≥1

aj is convergent to 0, the sequence P

l=1,n

(−1)^l

Q

j=l+1,n

aj

b_l

is convergent and the sequence

P

l=1,n

(−1)^lb_l P

j=l,n

(−1)^j Q

i=l+1,j

ai

n≥1

is also convergent.

b) If the product Q

j≥1

aj is convergent to 0 and the sequence

X

l=1,n

(−1)^lbl

X

j=l,n

(−1)^j Y

i=l+1,j

ai

!

n≥1

is convergent then the sequence (x_n)n≥0 is convergent for every x₀ and every x1.

c) If the product Q

j≥1

aj is convergent to 0, the series P

j≥1

|b_j| is conver- gent and the sequence

P

j≥n

Q

i=n+2,j+1

a_i

n≥1

is bounded then the sequence (x_n)n≥0 is also convergent for every x₀ and every x₁.

d) If there is an ε > 0 such that an ≤ 1−ε for n ≥ 1 and the series P

j≥1

|b_j| and P

j≥n+1

Q

i=n,j

ai

are convergent, then the sequence (xn)n≥1 is also convergent for every x₀ and every x₁.

Proof. Let (xn)n≥0 be the sequence given by the linear recurrence xn+1= (1−an)xn+anxn−1+bn

for n≥0 where 0< a_n<1.

Thenxn+1−xn=an(xn−1−xn) +bn=−a_n(xn−xn−1) +bn.

If we denote x_n+1−x_n by y_n then the sequence (y_n)n≥0 is defined by yn=−a_nyn−1+bn and y0 =x1−x0.

We have y_n= (−1)ⁿ

Q

l=1,n

a_l

(x₁−x₀) + P

l=1,n

(−1)^l

Q

j=l+1,n

a_j b_l

. Then

xn=x0+ X

j=1,n

(xj−xj−1) =

= X

j=1,n

(−1)^j

"

Y

l=1,j

a_l

(x₁−x₀) + X

l=1,j

(−1)^l

Y

i=l+1,j

a_i

b_l #

+x₀ =

= (x₁−x₀) X

j=1,n

(−1)^j

Y

l=1,j

a_l

+ X

j=1,n

X

l=1,j

(−1)^l+j

Y

i=l+1,j

a_i

b_l !

+x₀=

= (x1−x0)X

j=1,n

(−1)^j

Y

l=1,j

a_l

+ X

l=1,n

(−1)^lb_l X

j=l,n

(−1)^j

Y

i=l+1,j

a_i !

+x0. a) If the sequence (xn)n≥0 is convergent for every x0 and every x1 then the sequence (y_n)n≥0 is convergent for every y₀ and from the case k = 1 it follows that the series P

l=1,n

(−1)^l

Q

j=l+1,n

aj

bl

is convergent and the product Q

j≥1

(−a_j) is also convergent. Because 0< a_n<1 we have Q

j≥1

a_j= 0. Takingx₀= x₁= 0 we obtain that the sequence

P

l=1,n

(−1)^lb_l

P

j=l,n

(−1)^j Q

i=l+1,j

a_i

n≥1

is also convergent.

b) If the product Q

j≥1

a_j is convergent to 0, then the sequence X

j=1,n

(−1)^j

Y

l=1,j

a_l !

n≥1

is also convergent. Taking account that the sequence X

l=1,n

(−1)^lb_l

X

j=l,n

Y

i=l+1,j

a_i !

n≥1

is convergent, we obtain that the sequence (x_n)n≥0 is convergent for every x₀ and every x1.

c) Let the sequence

P

j≥n

Q

i=n+2,j+1

a_i

n≥1

be bounded byM >0. We have to prove that the sequence

P

l=1,n

(−1)^lb_l(−1)^j

P

j=l,n

Q

i=l,j

ai

n≥1

is

convergent. Let u_n= P

l=1,n

(−1)^lb_l

P

j=l,n

(−1)^j Q

i=l+1,j

a_i . Then

|u_n+1−u_n|=

(−1)ⁿ⁺¹ X

l=1,n+1

(−1)^l Y

i=l+1,n+1

a_i b_l

≤ X

l=1,n+1

Y

i=l+1,n+1

a_i

|b_l|

and

X

n≥1

|u_n+1−u_n| ≤X

n≥1

X

l=1,n+1

Y

i=l+1,n+1

a_i

|b_l|=

=X

l≥1

|b_l|

X

n≥l−1

Y

i=l+1,n+1

ai

≤MX

l≥1

|b_l|.

This imply that the sequence

P

l=1,n

(−1)^lb_l

P

j=l,n

Q

i=l+1,j

a_i

n≥1

is convergent.

d) results from c).

3. THE REAL HOMOGENOUS CASE WHEN THE SUM OF THE COEFFICIENTS IS 1

In this part we study real homogenous linear recurrence with positive coefficients when the sum of the coefficients is one.

For a real linear recurrence of orderkas in Definition 1.1 let y_n= min{x_n, xn−1, . . . , xn−k+1}, n≥k,

z_n= max{x_n, xn−1, . . . , x_n−k+1}, n≥k, dn=zn−yn,

m_n= min{a¹_n, a²_n, . . . , a^k_n}, n≥k+ 1,

m^l_n= min{m_n, mn−1, . . . , mn−l+1}, n≥k+l and

m_n=m^k−1_n , n≥2k−1.

Lemma 3.1. Let (an)n≥k+1 be a fixed sequence with an ∈R^k+ such that c_n= P

j=1,k

a^j_n= 1. Let (x_n)n≥1 be the homogeneous linear recurrence of order kassociated to the sequence (an)n≥k+1 with initial values x1, x2, . . . , x_k, that is x_n+k =a¹_n+kxn+k−1+a²_n+kxn+k−2+· · ·+a^k_n+kx_n. Then dn+k−1 ≤d_n(1− m_n+k−1) for every n≥k.

Proof. Letyn= min{x_n, xn−1, . . . , xn−k+1} and zn= max{x_n, xn−1, . . . , xn−k+1} forn≥k. It is clear thaty_n≤x_n+1≤z_n.It follows thatz_n+1≤z_n, yn+1 ≥yn and dn+1≤dn.

Ifm_n+k−1 = 0 the result is obvious. So, we can supposem_n+k−1 >0.

We want to estimate dn+k−1 =zn+k−1−yn+k−1 = max

i,j=n,n+k−1

|x_i−x_j|, where

xn+i=a¹_n+ixn+i−1+a²_n+ixn+i−2+· · ·+a^k_n+ixn+i−k, i= 1, k−1, under the conditions

a¹_n+i+a²_n+i+· · ·+a^k_n+i = 1, i= 1, k−1,

mn+i≤a¹_n+i, mn+i≤a²_n+i, . . . , mn+i ≤a^k_n+i, i= 1, k−1, and

yn≤xn+1−i ≤zn, i= 1, k.

Let us consider the function ˜dn+k−1 : ×^k

i=1[y_n, z_n]→Rdefined by d˜n+k−1(t₁, . . . , tk−1, t_k) = max

i,j=n,n+k−1

|˜x_i(t₁, . . . , tk−1, t_k)−x˜_j(t₁, . . . , tk−1, t_k)|, where ˜x_i : ×^k

i=1[y_n, z_n] → R for i ≥ n−k+ 1 are the functions defined such that ˜xi(t1, . . . , tk−1, tk) is the value of the ith term of the linear recurrence if xn+1−i=ti fori= 1, kthat is ˜xi are defined inductively by

˜

x_i(t₁, . . . , tk−1, t_k) =a¹_ix˜i−1(t₁, . . . , tk−1, t_k)+

+a²_ix˜i−2(t₁, . . . tk−1, t_k) +· · ·+a^k_ix˜i−k(t₁, . . . , tk−1, t_k), i≥n+ 1, and by

˜

x_i(t₁, . . . , tk−1, t_k) =tn+1−i, i=n−k+ 1, n.

It is easy to see that ˜x_i are linear functions. Because |x|= max(x,−x), d˜n+k−1 is the maximum of a finite family of linear functions defined on a compact convex set. After the linear programming theory it results that the maximum is taken into an extreme point of the convex set. So, we can suppose that xn+1−i =z_n orxn+1−i=y_n fori= 1, k.

If m_n+k−1 > 0 then m_n > 0 and yn < xn+1 < zn. Also we have yn ≤ yn+i−1 < x_n+i < zn+i−1 ≤ z_n for i = 2, k−1. It follows that dn+k−1 < d_n. Let l be such that dn+l < dn and dn+l−1 = dn. This means that zn+l−1 = zn+l−2 = · · · = z_n and yn+l−1 = yn+l−2 = · · · = y_n. Then z_n+l < z_n or y_n+l> y_n. Let us suppose thatz_n+l < z_n. In this case, forj = 1, l

xn+j =a¹_n+jxn+j−1+a²_n+jxn+j−2+· · ·+a^k_n+jxn+j−k≤

≤(1−mn+j)zn+j−1+mn+jyn+j−1= (1−mn+j)zn+mn+jyn=

=z_n−m_n+jd_n≤z_n−m_n+k−1d_n.

Also from the assumption thatxn+1−i=znorxn+1−i =ynfori= 1, kit follows thatxn+1−i=y_nfori= 1, k−l. If there is ani∈ {1,2, . . . , k−l}such that xn+1−i = zn we should have z_n+l = zn. Then z_n+l ≤ zn−m_n+k−1dn. Finally,

dn+k−1≤dn+l=zn+l−yn+l≤zn+l−yn≤

≤zn−m_n+k−1dn−yn=dn(1−m_n+k−1).

Proposition 3.1. Let (an)n≥k+1 be a fixed sequence with an∈R^k+ and (x_n)n≥1 be the homogeneous linear recurrence of order kassociated to the se- quence (an)n≥k+1with initial values x1, x2, . . . , x_k, that isx_n+k=a¹_n+kxn+k−1

+a²_n+kxn+k−2+· · ·+a^k_n+kxn such that cn = P

j=1,k

a^jn = 1. If P

n≥k+1

m_n = ∞ then the sequence (x_n)n≥1 is convergent to a finite limit l and

|x_n−l| ≤d_k Y

l=2,[^n−pk−1]

1−m_l(k−1)+p ,

where p∈N^∗ is fixed and n≥p+ 2k−2.

Proof. It is clear that yn ≤ xn+1 = a¹_n+kxn+k−1+a²_n+kxn+k−2+· · ·+ a^k_n+kx_n≤ z_n.It follows that z_n+1 ≤z_n, y_n+1 ≥y_n and d_n+1 ≤d_n. To prove the convergence it is enough to prove that dn = zn−yn → 0 when n → ∞ because [y_n+1, z_n+1]⊂[y_n, z_n].

From Lemma 3.1, dn+k−1 ≤dn(1−m_n+k−1) for every n≥k. It follows for n≥2 that

d_n(k−1)+p ≤dk+p−1

Y

l=2,n

1−m_l(k−1)+p

≤d_k Y

l=2,n

1−m_l(k−1)+p .

Because +∞ = P

n≥1

m_n =

k

P

p=1

P

l≥1

m_l(k−1)+p

there is a p ∈ {1,2, . . . , k}

such that P

l≥1

m_l(k−1)+p = +∞. It follows thatd_n=z_n−y_n→0 whenn→ ∞.

Letl= lim

n→∞xn. Then |x_n−l| ≤dn≤d[^n−pk−1]^(k−1)+p and

|x_n−l| ≤d_k Y

l=2,[^n−pk−1]

1−m_l(k−1)+p .

4. THE GENERAL HOMOGENOUS CASE WHEN THE SUM OF THE COEFFICIENTS IS 1

In this part we study real homogenous linear recurrence in Banach space with positive coefficients when the sum of the coefficients is one.

Theorem 4.1.Let (X,k k) be a normed space. Let (an)n≥k+1 be a fixed sequence with a_n ∈ R^k+ and (x_n)n≥1 ⊂ X be the homogeneous linear re- currence of order k associated to the sequence (a_n)_n≥k+1 with initial values x1, x2, . . . , xk ∈X, such that cn= P

j=1,k

a^jn= 1.If P

n≥1

m_n=∞ then(xn)n≥1 is convergent to a limit l∈X and kx_n−lk ≤ dk Q

l=2,[^n−pk−1]

1−m_l(k−1)+p

, where p∈N^∗ is fixed, n≥p+ 2k−2 and d_k= max

i,j=1,k

kx_i−xjk.

Proof. Let us suppose first that X is finite dimensional. Let ϕ : X → R be a linear function. Then ϕ(xn)

n≥1 is the homogeneous linear recur- rence of order k associated to the sequence (a_n)n≥k+1 with initial values ϕ(x₁), ϕ(x₂), . . . , ϕ(x_k) ∈ R. From Proposition 3.1 ϕ(x_n)

n≥1 is convergent to a finite limit l_ϕ and

|ϕ(x_n)−lϕ| ≤ max

i,j=1,k

|ϕ(x_i)−ϕ(xj)| Y

l=2,[^n−pk−1]

1−m_l(k−1)+p .

Since max

i,j=1,k

|ϕ(x_i)−ϕ(xj)| ≤ kϕk max

i,j=1,k

kx_i−xjk=kϕkd_k we have

|ϕ(x_n)−l_ϕ| ≤ kϕkd_k Y

l=2,[^n−pk−1]

1−m_l(k−1)+p .

SinceX is finite dimensional, (x_n)n≥1 is convergent to a limitl∈X and ϕ(l) =lϕ for every linear function ϕ:X→R. It follows that

kx_n−lk ≤d_k Y

l=2,[^n−pk−1]

1−m_l(k−1)+p .

This ends the proof in the case of finite dimensional spaces. The case whenXis not finite dimensional can be reduced to the case whenX is finite dimensional because x_n∈ hx₁, x₂, . . . , x_ki for everyn≥1.

Corollary 4.1. Let (X,k k) be a normed space. Let (an)n≥k+1 be a fixed sequence with an ∈ R^k+ and (xn)n≥1 ⊂ X be the homogeneous linear recurrence of order k associated to the sequence (a_n)n≥k+1 with initial values x₁, x₂, . . . , x_k ∈X, such that c_n= P

j=1,k

a^j_n= 1.Let K_n=convh{x_n, xn−1, . . . , xn−k+1}ifor n≥k and l be the limit of the sequence (xn)n≥1. Then Kn+1⊂ K_n and T

n≥1

K_n={l}. In particular, we have

klk ≤max{kx₁k,kx₂k, . . . ,kx_kk}.

Proof. We havexn+1 ∈Kn and soKn+1⊂Kn. The rest result from the convergence of the sequence (x_n)n≥1.

5. THE GENERAL INHOMOGENEOUS CASE WHEN THE SUM OF THE COEFFICIENTS IS 1

The following two lemmas are technical results that give us the possibility to reduce the inhomogeneous case to the homogenous one.

Lemma 5.1.Let (X,k k)be a Banach space. Let f :N→Nbe an increas- ing function such that lim

n→∞f(n) =∞.Let (x(m)n)n≥1 ⊂X, (lm)m≥1 ⊂X for m∈Nbe sequences of vectors and (d_n,m)n≥1 for m∈N, (t_m)m≥0 be sequences of positive numbers such that

1) x(m)_n

n≥1 is convergent to l_m; 2) P

m≥0

kl_mk is convergent;

3)kx(m)_n−lmk ≤dn,mtm for m≤f(n);

4) P

m≥0

tm is convergent;

5)d_n,m →0 when n→ ∞ for every m∈N;

6) there exists a M > 0 such that dn,m < M for every n, m ∈ N with m≤f(n).

Then the sequence y_n= P

l=0,f(n)

x(l)_n

n≥1 is convergent to P

m≥0

l_m. Proof. Let ε > 0 be fixed. From 2) and 4) there exists m₀ ∈ N and m1 ∈ N such that m0 ≤ m1, P

m≥m0

tm < _3M^ε and P

m≥m1

kl_mk < ^ε₃. There also exists nε ≥ m1 + 1 such that f(nε) ≥ m1 + 1 and for every n ≥ nε, d_n,m ≤ ₃P ^ε

m≥0tm form= 0, m₀. Then for n≥n_ε we have ky_n−X

m≥0

l_mk ≤ X

m≥f(n)+1

kl_mk+ X

m=0,f(n)

kx(m)_n−l_mk ≤

≤ X

m≥f(n)+1

kl_mk+ X

m=0,f(n)

dn,mtm≤

≤ X

m≥f(n)+1

kl_mk+ X

m=0,m0

d_n,mt_m+ X

m=m0+1,f(n)

d_n,mt_m≤

≤ X

m≥m1

kl_mk+ X

m=0,m0

dn,m

X

m=0,m0

tm

+M X

m=m0+1,f(n)

tm≤

≤ X

m≥m1

kl_mk+ ε 3 P

m≥0

t_m X

m=0,m0

tm+M X

m≥m0+1

tm≤ ε 3 +ε

3 +ε 3 =ε.

Notation. P

l=m,n

a_l= 0 and Q

l=m,n

a_l = 1 if m > n.

Lemma 5.2.Let (X,k k)be a Banach space. Let f :N→Nbe an increa- sing function such that lim

n→∞f(n) = ∞ and f(n) ≤ n. Let (x(m)n)n≥1 ⊂X for m ∈ N, (lm)m≥0 ⊂X be sequences of vectors and (dn,m)n≥1 for m ∈N, (t_m)m≥0, (c_m)m≥1 and (s_m)m≥0 be sequences of positive numbers such that

1) x(m)_n

n≥1 is convergent to l_m; 2) P

m≥0

sm is convergent;

3)kl_mk ≤sm and tm≤sm;

4)kx(m)_n−l_mk ≤d_n,mt_m for m≤f(n);

5)dn,m = Q

l=m+1,f(n)

cl for m≤f(n)−1;

6)c_n∈(0,1].

Let (yn)n≥1 be the sequence P

l=0,f(n)

x(l)n

n≥1. Then the series P

m≥0

lm

is convergent and

ky_n−X

m≥0

l_mk ≤ min

m0=0,f(n)

X

m≥m0+1

s_m+ Y

l=m0+1,f(n)

c_l X

m=0,m0

t_m Y c_l

l=m+1,m0

≤

≤ min

m0=0,f(n)

X

m≥m0+1

sm+

Y

l=m0+1,f(n)

cl

X

m=0,m0

tm

≤

≤ min

m0=0,f(n)

X

m≥m0+1

sm+

Y

l=m0+1,f(n)

cl

X

m≥0

tm

.

Proof. Since P

m≥0

kl_mk ≤ P

m≥0

sm and the series P

m≥0

sm is convergent it results that the series P

m≥0

lm is convergent.

Letnbe fixed and m₀ be a natural number such that m₀ ≤f(n). As in the proof of Lemma 5.1, we have

ky_n−X

m≥0

lmk ≤ X

m≥f(n)+1

kl_mk+ X

m=0,m0

dn,mtm+ X

m=m0+1,f(n)

dn,mtm.

Then ky_n− X

m≥0

lmk ≤ X

m≥f(n)+1

sm+ X

m=0,m0

Y

l=m+1,f(n)

c_l

tm+ X

m=m0+1,f(n)

sm ≤

≤ X

m≥m0+1

sm+

Y

l=m0+1,f(n)

c_l

X

m=0,m0

tm

Yc_l

l=m+1,m0

≤

≤ X

m≥m0+1

sm+

Y

l=m0+1,f(n)

c_l

X

m=0,m0

tm ≤

≤ X

m≥m₀+1

s_m+

Y

l=m0+1,f(n)

c_l

X

m≥0

t_m.

Because m0 is an arbitrary natural number such that m0 ≤ f(n) we obtain the desired conclusion.

Remark 5.1. If the conditions from Lemma 5.2 are fulfilled and Q

l≥1

cl= 0 then the conditions from Lemma 5.1 are also fulfilled.

The following lemma is obvious.

Lemma 5.3.Let (an)n≥k+1be a fixed sequence with an∈R^k+and (xn)n≥1

be the homogeneous linear recurrence of order k of real numbers associated to the sequence (an)n≥k+1 with initial values x1, x2, . . . , x_k ∈R, that is x_n+k = a¹_n+kxn+k−1+a²_n+kxn+k−2+· · ·+a^k_n+kxn such that cn = P

j=1,k

a^jn = 1. Then x_n∈[min{x₁, x₂, . . . , x_k},max{x₁, x₂, . . . , x_k}].

Lemma 5.4. Let (X,k k) be a Banach space. Let (an)n≥k+1 be a fixed sequence with a_n∈R^k+ and (b_n)n≥k+1 be a sequence of elements from X and (xn)n≥1 be the linear recurrence of order kassociated to the sequence (an)n≥1

with initial values x₁ = x₂ = · · · = x_k = 0, that is x_n+k = a¹_n+kxn+k−1+ a²_n+kxn+k−2+· · ·+a^k_n+kxn+bn.We suppose that cn= P

j=1,k

a^jn= 1 and b_n= 0 for n≥k+ 1. Then

sup

i,j≥1

kx_i−x_jk ≤ X

j=1,k

kb_jk.

Proof. Let us consider the following linear recurrences (x(l)_n)n≥1 forl∈ {1,2, . . . , k}defined by x(l)1 = 0,x(l)2= 0, . . . , x(l)k= 0 forl∈ {1,2, . . . , k}

and x(l)n+k=a¹_n+kx(l)n+k−1+a²_n+kx(l)n+k−2+· · ·+a^k_n+kx(l)n+bn+kδ_n+k^l . Thenxn=x(1)n+x(2)n+· · ·+x(k)n.

Alsox(l)n=y(l)nbl, where y(l)n

n≥1 are the real linear recurrences de- fined by y(l)1 = 0,y(l)2 = 0, . . . , y(l)_k= 0 for l∈ {1,2, . . . , k}and y(l)_n+k= a¹_n+ky(l)n+k−1+a²_n+ky(l)n+k−2+· · ·+a^k_n+ky(l)n+δ^l_n+k.

It follows that xn= P

l=1,k

y(l)nbl. From Lemma 5.3,y(l)n∈[0,1].

The rest is obvious.

Definition 5.1. We denote by A the set of all sequences (an)n≥1 with an∈R^k₊ and cn= P

j=1,k

a^jn= 1 for n≥k+ 1.

Definition 5.2. Let (X,k k) be a Banach space. Letb₁, b₂, . . . , b_k be ele- ments fromX. Let (an)n≥1be an arbitrary sequence withan∈R^k+and (xn)n≥1

be the linear recurrence of order k associated to the sequence (a_n)n≥1 with initial values x1 = x2 = · · · = x_k = 0, defined by x_n+k = a¹_n+kxn+k−1 + a²_n+kxn+k−2+· · ·+a^k_n+kxn+(δ_n¹b1+δ_n²b2+· · ·+δ_n^kbk) such thatcn= P

j=1,k

a^jn= 1 forn≥k. xndepends on the sequence (an)n so we can writexn=xn (an)n

. Let

v_m(b₁, b₂, . . . , b_k) = sup

(an)n∈A

max

i,j=k+1,m

x_i (a_n)_n

−x_j (a_n)_n

for m∈N^∗ and m≥k+ 1.

Lemma 5.5. With the notations from the above Definition(5.2)we have 1)

vm(b1, b2, . . . , bk) = sup

(an)n∈A

max

i,j=k+1,m

X

l=1,k

yi l,(an)n

−yj l,(an)n

bl

, where y_m(l,(a_n)_n)

m≥1 are the real linear recurrences defined by y₁ l,(a_m

m) =y₂ l,(a_m)_m

=· · ·=y_k l,(a_m)_m

= 0 for l∈ {1,2, . . . , k} and

y_n+k l,(a_m)_m

=a¹_n+kyn+k−1 l,(a_m)_m + +a²_n+kyn+k−2 l,(a_m)_m

+· · ·+a^k_n+ky_n l,(a_m)_m

+δ^l_n+k; 2)v_m(b₁, b₂, . . . , b_k)≤ P

j=1,k

kb_jk;

3)vm(b1+e1, b2+e2, . . . , b_k+e_k)≤vm(b1, b2, . . . , b_k) + P

j=1,k

ke_jk;

4)v_m(b₁, b₂, . . . , b_k)≤v_2k(b₁, b₂, . . . , b_k) for m≥2k.

Proof. 1) It is easy to see thatxi (an)n

= P

l=1,k

yi l,(an)n

b_l.

2) and 3) results from the fact thatyi l,(an)n

∈[0,1] (see Lemma 5.3).

4) Form≥2k the recurrence is homogenous and we have

xm+p ∈conv{x_m, xm−1, . . . , xm−k+1} ⊂conv{x_2k, x2k−1, . . . , xk}ifp∈N^∗. Lemma 5.6. Let (X,k k) be a Banach space. Let (an)n≥k+1 be a fixed sequence with a_n ∈ R^k+ and (b_n)n≥k+1 be a sequence of elements from X.

Let (xn)n≥1 be the linear recurrence of order k associated to the sequence (a_n)n≥k+1 with initial values x₁, x₂, . . . , x_k, that is x_n+k = a¹_n+kxn+k−1 +

· · ·+a^k_n+kxn+b_n+k such that cn = P

j=1,k

a^jn = 1. Let dn = sup

i,j=1,n

kx_i−xjk.

Then d_k+ P

l=k+1,k+p

kb_lk ≥d_k+p, where p∈N^∗.

Proof. It is enough to give the proof in the casep= 1. The general case results by induction. We have

d_k+1 = max

d_k, sup

i=1,n

kx_n+1−xik and

kx_n+1−x_ik ≤a¹_n+1kx_n−x_ik+a²_n+1kx_n−1−x_ik+· · ·+ +a^k_n+1kx₁−x_ik+kb_n+1k ≤d_k+kb_n+1k.

It follows that dk+1 ≤dk+kb_n+1k.

The following theorem extends the results of the homogenous case to the inhomogeneous case when the sum of the coefficients of the linear recurrence is 1 and it is the main result of this section.

Theorem 5.1. Let (X,k k) be a Banach space. Let (a_n)n≥k+1 be a fixed sequence with an ∈ R^k+, (bn)n≥k+1 be a sequence of elements from X and (x_n)n≥1 be the linear recurrence of order k associated to the sequences (an)n≥k+1 and (b_n)n≥k+1 with initial values x1, x2, . . . , xk, that is xn+k = a¹_n+kxn+k−1+a²_n+kxn+k−2+· · ·+a^k_n+kxn+bn+k,such that cn= P

j=1,k

a^jn= 1.

If P

n≥2k−1

m_n =∞ and P

n≥k+1

kb_nk<∞ then (x_n)n≥1 is convergent to a limit a∈X and

kx_n−ak ≤ [^n−pk−1]

mmin0=0

X

l≥m0(k−1)+p+1

kb_lk+

dk+X

l≥1

rl,p

[^n−pk−1] Y

l=m0+1

1−m_l(k−1)+p

!

≤ [^n−pk−1]

mmin0=0

X

l≥m₀(k−1)+p+1

kb_lk+

d_k+ X

l≥k+1

kb_lk

[^n−p_k−1] Y

l=m0+1

1−m_l(k−1)+p

! ,

where rl,p = v2k 0X, bl(k−1)+p+1, bl(k−1)+p+2, . . . , bl(k−1)+p+k−1

, p ∈ N^∗ is fixed, n≥p+ 2k−2 and dk= max

i,j=1,k

kx_i−xjk.

Proof. The idea of the proof is to decompose the inhomogeneous linear recurrence (xn)n≥1 into a countable sum of homogenous linear recurrences.

Then using Theorem 4.1 we can estimate the speed of the convergence of these recurrences. From Lemma 5.1 we obtain the convergence and from Lemma 5.2 we can estimate the speed of the convergence of the inhomogeneous linear recurrence (x_n)n≥1.

It is enough to make the proof only in the casep= 1.

For the general case let us consider the linear recurrence of order k, (xn)n≥1, associated to the sequences (an)n≥k+1 and (bn)n≥k+1 with initial valuesx₁, x₂, . . . , x_k, that isx_n+k =a¹_n+kxn+k−1+a²_n+kxn+k−2+· · ·+a^k_n+kx_n+ bn+k, where xn = xn+p−1, a^jn = a^j_n+p−1 and bn = bn+p−1. We have cn =

P

j=1,k

a^jn = cn+p−1 = 1. Applying the results of the Theorem to the linear recurrence (xn)n≥1 forp= 1 and taking account that

d_k+ X

l=k+1,k+p−1

kb_lk ≥dk+p−1 = sup

i,j=1,k+p−1

kx_i−x_jk

(from Lemma 5.6) we can obtain the results of the Theorem for the linear recurrence (xn)n≥1 for a generalp.

Let us consider the sequences (x(l)_n)n≥1 forl∈Ndefined by x(0)_n+k=a¹_n+kx(0)n+k−1+a²_n+kx(0)n+k−2+· · ·+a^k_n+kx(0)n,

x(0)₁ =x₁, x(0)₂ =x₂, . . . , x(0)_k =x_k forl= 0 and

x(l)_n+k=a¹_n+kx(l)n+k−1+a²_n+kx(l)n+k−2+· · ·+a^k_n+kx(l)_n+ +bn+k

δ_n+k^l(k−1)+2+δ^l(k−1)+3_n+k +· · ·+δ(l+1)(k−1)+1 n+k

, x(l)₁ = 0, x(l)₂= 0, . . . , x(l)_k= 0 for l >0.

Thenx_n=x(0)_n+x(1)_n+· · ·+x _n−2

k−1

n.

From Theorem 4.1 and Corollary 4.1 the sequences x(l)_n

n≥1 forl∈N are convergent to the limits a(l) such that

ka(0)k ≤ X

i=1,k

kx_jk,

ka(l)k ≤ X

j=l(k−1)+2,(l+1)(k−1)+1

kb_jk forl >0, kx(0)_n−a(0)k ≤d_k Y

j=2,[ⁿ⁻¹k−1]

1−m_j(k−1)+1

forl= 0

and

kx(l)_n−a(l)k ≤v_2k 0X, b_l(k−1)+2, . . . , b_l(k−1)+k Y

j=l+1,[ⁿ⁻¹k−1]

1−m_j(k−1)+1

for l >0 andn≥(l+ 1)(k−1) + 1.

We can apply Lemma 5.2 withx_nasy_n, x(m)_n

n≥1as x(m)_n

n≥1,a(m) as l_m, r_l,1 =v_2k 0_X, b_l(k−1)+2, b_l(k−1)+3, . . . , b(l+1)(k−1)+1

ast_l forl≥1, 1− m_l(k−1)+1

asc_l,s_l = P

j=l(k−1)+2,(l+1)(k−1)+1

kb_jkass_lforl≥1, P

i=1,k

kx_jkass0, dkast0and_n−1

k−1

forf(n). It results that the series P

n≥0

a(n) is convergent and

kx_n−ak ≤ min

m0=0,[ⁿ⁻¹k−1]

X

l≥m0+1

s_l+

d_k+X

l≥1

r_l,1

Y

l=m0+1,[ⁿ⁻¹k−1]

1−m_l(k−1)+1

! ,

where a= P

n≥0

a(n).

The last inequality follows from the fact that (see Lemma 5.5.2)) v_2k 0_X, b_l(k−1)+2, b_l(k−1)+3, . . . , b(l+1)(k−1)+1

≤ X

j=l(k−1)+2,(l+1)(k−1)+1

kb_jk.

It remains us to prove the convergence of the sequence (xn)n≥1. Since

+∞ = P

n≥k

m_n =

k

P

p=1

P

l≥1

m_l(k−1)+p

there is a p ∈ {1,2, . . . , k} such that P

l≥1

m_l(k−1)+p = +∞, and so Q

l≥m0+1

1−m_l(k−1)+p

= 0 for every m0. From Remark 5.1 conditions from Lemma 5.1 are also fulfilled. Therefore the con- vergence results from Lemma 5.1.

6. THE GENERAL INHOMOGENEOUS CASE

The next result is obvious.

Lemma 6.1. Let n∈N^∗,ε >0and a₁, a₂, . . . , a_n, abe such that,nε≤1, ai ≥ε for i∈ 1, n, a6= 0 and a1+a2+· · ·+an = 1 +a. Then there exists