1. Introduction

We consider nonautonomous linear difference equations

v m + 1 = A m v m + B m ( λ ) v m
(1.1)

in a Banach space, where λ is a parameter in some open subset Y of a Banach space (the parameter space), and λ → B m (λ) is of class C1 for each mJ = ℕ. Assuming that the unperturbed dynamics

v m + 1 = A m v m
(1.2)

admits a very general nonuniform dichotomy (see Section 2 for the definition), and that

sup m J , λ Y B m ( λ ) and sup m J , λ Y B m ' ( λ )

are sufficiently small, we establish the optimal C1 regularity of the stable subspaces on the parameter λ for Equation (1.1).

The classical notion of (uniform) exponential dichotomy, essentially introduced by Perron in [1], plays an important role in a large part of the theory of differential equations and dynamical systems. We refer the reader to the books [25] for details and references. Inspired both in the classical notion of exponential dichotomy and in the notion of nonuni-formly hyperbolic trajectory introduced by Pesin in [6, 7], Barreira and Valls [811] have introduced the notion of nonuniform exponential dichotomies and have developed the corresponding theory in a systematic way for the continuous and discrete dynamics during the last few years. See also the book [12] for details. As mentioned in [12], in finite-dimensional spaces essentially any linear differential equation with nonzero Lyapunov exponents admits a nonuniform exponential dichotomy. The works of Barreira and Valls can be regarded as a nice contribution to the nonuniform hyperbolicity theory [13].

There are some works concerning the smooth dependence of the stable and unstable sub-spaces on the parameter. For example, in the case of continuous time, that is, for linear differential equations

v = [ A ( t ) + B ( t , λ ) ] v ,

Johnson and Sell [14] considered exponential dichotomies in ℝ (in a finite dimensional space), and proved that for C k perturbations, if the perturbation and its derivatives in λ are bounded and equicontinuous in the parameter, then the projections are of class C k in λ. In the case of discrete time, Barreira and Valls established the optimal C1 dependence of the stable and unstable subspaces on the parameter in [15] for the uniform exponential dichotomies and in [16] for the nonuniform exponential dichotomies.

In our study, we establish the optimal C1 dependence of the stable subspaces on the parameter for very general nonuniform dichotomies (which was first introduced by Bento and Silva in [17]) for (1.1). Such dichotomies include for example the classical notion of uniform exponential dichotomies, as well as the notions of nonuniform exponential dichotomies and nonuniform polynomial dichotomies. The proof in this article follows essentially the ideas in [16], with some essential difficulties because we consider the new dichotomies. We also note that we can establish the optimal C1 dependence of the unstable subspaces on the parameter using the similar discussion as in [16], and we omit the detail for short.

2. Setup

Let ( X ) be the set of bounded linear operations in the Banach space X. Let (A m )mJbe a sequence of invertible operators in ( X ) . For each m, nJ, we set

A ( m , n ) = A m - 1 . . . A n , if m > n , Id, if m = n , A m - 1 . . . A n - 1 - 1 , if m < n .

In order to introduce the notion of nonuniform dichotomy, it is convenient to consider the notion of growth rate. We say that an increasing function μ : J → (0, +∞) is a growth rate if

μ ( 0 ) = 1 and lim n + μ ( n ) = + .

Given two growth rates μ and ν, we say that the sequence (A m )mJ(or the cocycle A ( m , n ) ) admits a nonuniform (μ, ν) dichotomy if there exist projections P m ( X ) for each mJ such that

A ( m , n ) P n = P m A ( m , n ) , m , n , J

and there exist constants α, D > 0 and ε > 0 such that

A ( m , n ) P n D μ ( m ) μ ( n ) - α ν ε ( n ) ,
(2.1)

and

A ( m , n ) - 1 Q m D μ ( m ) μ ( n ) - α ν ε ( m ) ,
(2.2)

for each mn, where Q m = Id - P m is the complementary projection of P m .

When μ(m) = ν(m) = eρ(m), we recover the notion of ρ-nonuniform exponential dichotomy, while we recover the notion of nonuniform polynomial dichotomy when μ(m) = ν(m) = 1 + m.

For example, if μ and ν are arbitrary growth rates and ε, α > 0, consider a sequence of linear operators A n : ℝ2 → ℝ2 given by diagonal matrices

A n = a n 0 0 b n ,

where

a m = μ ( m + 1 ) μ ( m ) - α e ε 2 log ν ( m + 1 ) ( cos ( m + 1 ) - 1 ) - ε 2 log ν ( m ) ( cos m - 1 ) , b m = μ ( m + 1 ) μ ( m ) α e - ε 2 log ν ( m + 1 ) ( cos ( m + 1 ) - 1 ) + ε 2 log ν ( m ) ( cos m - 1 ) ,

for any mJ. Then (A m )mJadmits a nonuniform (μ, ν) dichotomy with the projections P m : ℝ2 → ℝ2 defined by P m (x, y) = (x, 0), and we have

A ( m , n ) P n μ ( m ) μ ( n ) - α ν ε ( n ) ,

and

A ( m , n ) - 1 Q m μ ( m ) μ ( n ) - α ν ε ( m )

for each mn.

In this article, for each nJ, we define the stable and unstable subspaces by

E n = P n ( X ) and F n = Q n ( X ) .

3. Main results

We establish the existence of stable subspaces E n λ on J for each λ ∈ Y, such that the maps λ E n λ are of class C1. As the same in [10], we look for each space E n λ as a graph over E n . More precisely, we look for linear operators Φn: E n F n such that

E n λ = graph ( I d E n + Φ n , λ ) , n J , λ Y .

Given a constant κ < 1, let X be the space of families Φ = (Φn)nJ,λ∈Yof linear operators Φn:E n F n such that

Φ : = sup Φ n , λ ν ε ( n ) : ( n , λ ) J × Y κ

and

C λ μ ( Φ ) : = sup Φ n , λ - Φ n , μ ν ε ( n ) : n J κ λ - μ

for each λ, μY. Equipping X with the distance

Φ - Ψ = sup Φ n , λ - Ψ n , λ ν ε ( n ) : ( n , λ ) J × Y ,

it becomes a complete metric space. Given ΦX and λ ∈ Y, for each nJ, we consider the vector space

E n λ = graph ( I d E n + Φ n , λ ) = ξ , Φ n , λ ξ : ξ E n .

Moreover, for each m, nJ, we set

A λ ( m , n ) = C m - 1 . . . C n , if m > n , Id, if m = n , C m - 1 . . . C n - 1 - 1 , if m < n ,

where C k = A k + B k (λ) for each kJ.

Now we state the main result of this article.

Theorem 3.1. Assume that the sequence (A m )mJadmits a nonuniform (μ, ν) dichotomy, and for eachmJ, B m :Y ( X ) are C1functions satisfying

B m ( λ ) δ ν - β ( m + 1 ) a n d B m ( λ ) δ ν - β ( m + 1 ) .
(3.1)

Suppose further that

ϑ = n = 1 μ ( n ) μ ( n + 1 ) - α ν 3 ε - β ( n + 1 ) < .
(3.2)

Then for δ sufficiently small there exists a unique ΦX such that

E m λ = A λ ( m , n ) E n λ
(3.3)

for each m, nJ. Moreover,

  1. (1)

    for each nJ, mn and λ ∈ Y we have

    A λ ( m , n ) E n λ D μ ( m ) μ ( n ) α ν ε ( n )
    (3.4)

for some constant D' > 0;

  1. (2)

    The map λ ↦ Φn is of class C 1 for each nJ.

Proof. Given nJ and (ξ, η) ∈ E n × F n , the vector

( x m , y m ) = A λ ( m , n ) ( ξ , η ) E m × F m

satisfies

x m = A ( m , n ) ξ + l = n m - 1 P m A ( m , l + 1 ) B l ( λ ) ( x l , y l )
(3.5)

and

y m = A ( m , n ) η + l = n m - 1 Q m A ( m , l + 1 ) B l ( λ ) ( x l , y l )
(3.6)

for each mn.

Due to the required invariance in (3.3), given ( x n , y n ) E n λ we must have y m = Φmx m for each m, and thus Equations (3.5)-(3.6) are equivalent to

x m = A ( m , n ) x n + l = n m - 1 P m A ( m , l + 1 ) B l ( λ ) ( I d E l + Φ l , λ ) x l
(3.7)

and

Φ m , λ x m = A ( m , n ) Φ n , λ x n + l = n m - 1 Q m A ( m , l + 1 ) B l ( λ ) ( I d E l + Φ l , λ ) x l
(3.8)

for each mn.

Now we introduce linear operators related to (3.7). Given ΦX, nJ and λ ∈ Y, we consider the linear operators W m , λ n = W m , Φ , λ n : E n E m determined recursively by the identities

W m , λ n = P m A ( m , n ) + l = n m - 1 P m A ( m , l + 1 ) B l ( λ ) ( I d E l + Φ l , λ ) W l , λ n
(3.9)

for m > n, setting W n , λ n = I d E n . We note that for x n = ξE n , the sequence

x m = W m , λ n x n = W m , λ n ξ
(3.10)

is the solution of Equation (3.5) with y l = Φlx l for each ln. Equivalently, it is a solution of Equation (3.7).

Using (3.10) we can rewrite (3.8) in the form

Φ m , λ W m , λ n = A ( m , n ) Φ n , λ + l = n m - 1 Q m A ( m , l + 1 ) B l ( λ ) ( I d E l + Φ l , λ ) W l , λ n
(3.11)

Lemma 3.2. Given δ sufficiently small, for eachΦXand λ ∈ Y, the following properties are equivalent:

  1. (1)

    (3.11) holds for every nJ and mn;

  2. (2)

    for every nJ and mn we have

    Φ n , λ = - l = n Q n A ( l + 1 , n ) - 1 B l ( λ ) ( I d E l + Φ l , λ ) W l , λ n
    (3.12)

Proof of the lemma. We first show that the series in (3.12) are well defined. Using (2.2) and (3.1), we obtain

l = n Q n A ( l + 1 , n ) - 1 B l ( λ ) ( I d E l + Φ l , λ ) W l , λ n ν ε ( n ) δ D ( 1 + κ ) l = n μ ( l + 1 ) μ ( n ) - α ν ε - β ( l + 1 ) W l , λ n ν ε ( n ) 2 δ D l = n μ ( l + 1 ) μ ( n ) - α ν ε - β ( l + 1 ) W l , λ n ν ε ( n ) .
(3.13)

By (3.9), for each mn we have

W m , λ n D μ ( m ) μ ( n ) - α ν ε ( n ) + δ D ( 1 + κ ) l = n m - 1 μ ( m ) μ ( l + 1 ) - α ν ε - β ( l + 1 ) W l , λ n .
(3.14)

Setting

ϒ = sup m n μ ( m ) μ ( n ) α W m , λ n .

Then we have

ϒ D ν ε ( n ) + δ D ( 1 + κ ) ϒ l = n m - 1 μ ( l ) μ ( l + 1 ) - α ν ε - β ( l + 1 ) D ν ε ( n ) + 2 δ D ϑ ϒ .
(3.15)

Taking δ sufficiently small such that 2δϑD < 1/2 (independently of n) we obtain

ϒ 2 D ν ε ( n ) ,

and therefore,

W m , λ n 2 D μ ( m ) μ ( n ) - α ν ε ( n ) .
(3.16)

Combined (3.13) and (3.16), we have

l = n Q n A ( l + 1 , n ) - 1 B l ( λ ) ( I d E l + Φ l , λ ) W l , λ n ν ε ( n ) 4 δ D 2 l = n μ ( l ) μ ( l + 1 ) μ 2 ( n ) - α ν ε - β ( l + 1 ) ν 2 ε ( n ) 4 δ D 2 l = n ν 3 ε - β ( l + 1 ) κ
(3.17)

provided that δ sufficiently small.

Now we assume that identity (3.11) holds. It is equivalent to

Φ n , λ = Q n A ( m , n ) - 1 Φ m , λ W m , λ n - l = n m - 1 Q n A ( l + 1 , n ) - 1 B l ( λ ) ( I d E l + Φ l , λ ) W l , λ n .
(3.18)

Using (3.16), for each mn we have

Q n A ( m , n ) - 1 Φ m , λ W m , λ n 2 κ D 2 μ ( m ) μ ( n ) - 2 α ν ε ( n )

Since α > 0, letting m → +∞ in (3.18) we obtain identity (3.12).

Conversely, let us assume that identity (3.12) holds. Then

A ( m , n ) Φ n , λ + l = n m - 1 Q m A ( m , l + 1 ) B l ( λ ) ( I d E l + Φ l , λ ) W l , λ n = - l = n Q m A ( l + 1 , m ) - 1 B l ( λ ) ( I d E l + Φ l , λ ) W l , λ n + l = n m - 1 Q m A ( l + 1 , m ) - 1 B l ( λ ) ( I d E l + Φ l , λ ) W l , λ n = - l = m Q m A ( l + 1 , m ) - 1 B l ( λ ) ( I d E l + Φ l , λ ) W l , λ n

for each mn. Since W l , λ n = W l , λ m W m , λ n , it follows from (3.12) with n replace by m that (3.11) holds for each mn.

We define linear operators A(Φ)n: E n F n each ΦX, nJ, and λ ∈ Y by

A ( Φ ) n , λ = - l = n Q n A ( l + 1 , n ) - 1 B l ( λ ) ( I d E l + Φ l , λ ) W l , λ n .

Lemma 3.3. For δ sufficiently small, A is well defined andA ( X ) X.

Proof of the lemma. By (3.17) the operator A is well-defined and

A ( Φ ) κ

for δ sufficiently small. Furthermore, writing

W l , λ n = W l , λ , W l , μ n = W l , μ ,

we have

b l : = B l ( λ ) ( I d E l + Φ l , λ ) W l , λ - B l ( μ ) ( I d E l + Φ l , μ ) W l , μ B l ( λ ) - B l ( u ) W l , λ 1 + Φ l , λ + B l ( μ ) W l , λ - W l , μ 1 + Φ l , λ + B l ( μ ) W l , μ Φ l , λ - Φ l , μ 2 δ D ( 1 + κ ) λ - μ ν - β ( l + 1 ) μ ( l ) μ ( n ) - α ν ε ( n ) + δ ( 1 + κ ) ν - β ( l + 1 ) W l , λ - W l , μ + 2 δ D κ ν - β ( l + 1 ) λ - μ μ ( l ) μ ( n ) - α ν ε ( n ) ν - ε ( l ) 6 δ D λ - μ μ ( l ) μ ( n ) - α ν - β ( l + 1 ) ν ε ( n ) + 2 δ ν - β ( l + 1 ) W l , λ - W l , μ .

Therefore,

W m , λ - W m , μ l = n m - 1 P m A ( m , l + 1 ) b l 6 δ D 2 μ ( m ) μ ( n ) - α ν ε ( n ) λ - μ l = n m - 1 μ ( l ) μ ( l + 1 ) - α ν ε - β ( l + 1 ) + 2 δ D l = n m - 1 μ ( m ) μ ( l + 1 ) - α ν ε - β ( l + 1 ) W l , λ - W l , μ 6 δ D 2 ϑ μ ( m ) μ ( n ) - α ν ε ( n ) λ - μ + 2 δ D l = n m - 1 μ ( m ) μ ( l + 1 ) - α ν ε - β ( l + 1 ) W l , λ - W l , μ .

Setting

ϒ l = μ ( l ) μ ( n ) α W l , λ - W l , μ .

Then we have from the above inequality that

ϒ m 6 δ D 2 ϑ ν ε ( n ) λ - μ + 2 δ D l = n m - 1 ϒ l μ ( l ) μ ( l + 1 ) - α ν ε - β ( l ) .

Setting ϒ = sup{ϒ m : mn}, we obtain

ϒ 6 δ D 2 ϑ ν ε ( n ) λ - μ + 2 δ D ϑ ϒ .

Taking δ sufficiently small such that 2δDϑ < 1/2, we obtain

ϒ 12 δ D 2 ϑ ν ε ( n ) λ - μ ,

and therefore,

W m , λ - W m , μ 12 δ D 2 ϑ μ ( m ) μ ( n ) - α ν ε ( n ) λ - μ .
(3.19)

Therefore, it follows form (3.19) that

b l 6 δ D λ - μ μ ( l ) μ ( n ) - α ν - β ( l + 1 ) ν ε ( n ) + 2 δ ν - β ( l + 1 ) 12 δ D 2 ϑ μ ( l ) μ ( n ) - α ν ε ( n ) λ - μ = K δ λ - μ μ ( l ) μ ( n ) - α ν - β ( l + 1 ) ν ε ( n )

where K = 6D + 24δD2ϑ > 0.

Therefore, we obtain

A ( Φ ) n , λ - A ( Φ ) n , μ ν ε ( n ) l = n Q n A ( l + 1 , n ) - 1 b l ν ε ( n ) δ K D λ - μ l = n μ ( l ) μ ( l + 1 ) μ 2 ( n ) - α ν ε - β ( l + 1 ) ν 2 ε ( n ) δ K D λ - μ l = n ν 3 ε - β ( l + 1 ) δ K D ϑ λ - μ

and provided that δ is sufficiently small, we obtain Cλμ(A(Φ)) ≤ κ∥λ - μ∥. This shows that A ( X ) X.

Now we note be the space of sequences U = (Un)n∈ℕ,λ∈Yof linear operators Un: E n F n indexed by Y such that λ ↦ Unis continuous for each nJ, and

U = sup ( n , λ ) J × Y U n , λ 1
(3.20)

Equipping with this norm, it becomes a complete metric space, For each ( Φ , U ) X×, nJ, and λ ∈ Y, we also define linear operators B (Φ, U)nby

B ( Φ , U ) n , λ = - l = n Q n A ( l + 1 , n ) - 1 G l , λ ,
(3.21)

where

G l , λ = B l ( λ ) ( Z l , λ + Φ l , λ Z l , λ + U l , λ W l , λ n ) + B l ( λ ) ( I d E l + Φ l , λ ) W l , λ
(3.22)

and the linear operators Zm= Zm,Φ,U: E n E m are determined recursively by the identities

Z m , λ = l = n m - 1 P m A ( m , l + 1 ) G l , λ
(3.23)

for m > n, setting Zn= 0. One observe that by the continuity of the functions Φland Ulon λ the functions λ ↦ Wland λ ↦ Zlare also continuous.

Lemma 3.4. For δ sufficiently small, the operator B is well defined, andB ( X × ) .

Proof of the lemma. By (3.16) and (3.20) we have

Z m , λ l = n m - 1 P m A ( m , l + 1 ) B l ( λ ) 1 + Φ l , λ Z l , λ + B l ( λ ) U l , λ + B l ( λ ) 1 + Φ l , λ W l , λ n 2 δ D l = n m - 1 μ ( m ) μ ( l + 1 ) - α ν ε - β ( l + 1 ) Z l , λ + 6 δ D 2 l = n m - 1 μ ( m ) μ ( l + 1 ) - α ν ε - β ( l + 1 ) μ ( l ) μ ( n ) - α ν ε ( n ) 2 δ D l = n m - 1 μ ( m ) μ ( l + 1 ) - α ν ε - β ( l + 1 ) Z l , λ + 6 δ D 2 ϑ μ ( m ) μ ( n ) - α

Setting ϒ m = μ ( m ) μ ( n ) α Z m , λ , we obtain

ϒ m 2 δ D l = n m - 1 μ ( l + 1 ) μ ( n ) α ν ε - β ( l + 1 ) Z l , λ + 6 δ D 2 ϑ 2 δ D ϑ ϒ l + 6 δ D 2 ϑ

and setting ϒ = sup{ϒ m : mn},

ϒ 2 δ D ϑ ϒ + 6 δ D 2 ϑ .

Thus, taking δ sufficiently small such that 2δDϑ< 1 2 , we have

ϒ 12 δ D 2 ϑ ,

and therefore

Z m , λ 12 δ D 2 ϑ μ ( m ) μ ( n ) - α
(3.24)

Setting

G = l = n Q n A ( l + 1 , n ) - 1 G l , λ ,
(3.25)

it follows from (3.16) and (3.20) that

G D l = n μ ( l + 1 ) μ ( n ) - α ν ε ( l + 1 ) . 2 δ ν - β ( l + 1 ) Z l , λ + 6 δ D ν - β ( l + 1 ) μ ( l ) μ ( n ) - α ν ε ( n ) D l = n μ ( l + 1 ) μ ( n ) - α ν ε - β ( l + 1 ) . 24 δ 2 D 2 ϑ μ ( l ) μ ( n ) - α + 6 δ D μ ( l ) μ ( n ) - α ν ε ( n ) 24 δ 2 D 3 ϑ l = n μ ( l + 1 ) μ ( l ) μ 2 ( n ) - α ν ε - β ( l + 1 ) + 6 δ D 2 l = n μ ( l + 1 ) μ ( l ) μ 2 ( n ) - α ν 2 ε - β ( l + 1 ) 24 δ 2 D 3 ϑ 3 + 6 δ D 2 ϑ 1 ,
(3.26)

provided that δ is sufficiently small. This shows that B is well defined for each n, and that ∥B(Φ, U) ∥ ≤ 1. Therefore, B ( X × ) .

Now we define another map S:X×X× by

S ( Φ , U ) = ( A ( Φ ) , B ( Φ , U ) ) .

By Lemmas 3.3 and 3.4, it is clearly that the maps S is well defined and S ( X × ) X×.

Lemma 3.5. For δ sufficiently small, the operator S is a contraction.

Proof of the lemma. Given Φ, ΨX, and set W l , Φ = W l , Φ , λ n , W l , Ψ = W l , Ψ , λ n , we obtain

A ( Φ ) n , λ - A ( Ψ ) n , λ ν ε ( n ) D l = n μ ( l + 1 ) μ ( n ) - α ν ε ( l + 1 ) δ ν - β ( l + 1 ) . W l , Φ - W l , Ψ + Φ l , λ W l , Φ - Ψ l , λ W l , Ψ ν ε ( n ) δ D l = n μ ( l + 1 ) μ ( n ) - α ν ε - β ( l + 1 ) . 2 W l , Φ - W l , Ψ + Φ - Ψ W l , Ψ ν - ε ( l ) ν ε ( n ) 2 δ D ϑ W l , Φ - W l , Ψ ν ε ( n ) + 2 δ D 2 ϑ Φ - Ψ μ ( l ) μ ( n ) - α ν - ε ( l ) ν 2 ε ( n ) .
(3.27)

By (3.16) we obtain

W m , Φ - W m , Ψ δ D l = n m - 1 μ ( m ) μ ( l + 1 ) - α ν ε ( l + 1 ) ν - β ( l + 1 ) . 2 W l , Φ - W l , Ψ + Φ - Ψ W l , Ψ ν - ε ( l ) 2 δ D l = n m - 1 μ ( m ) μ ( l + 1 ) - α ν ε - β ( l + 1 ) W l , Φ - W l , Ψ + 2 δ D 2 Φ - Ψ μ ( m ) μ ( n ) - α l = n m - 1 μ ( l ) μ ( l + 1 ) - α ν ε - β ( l + 1 ) ν - ε ( l ) ν ε ( n ) 2 δ D ϑ μ ( m ) μ ( n ) - α μ ( l ) μ ( n ) α W l , Φ - W l , Ψ + 2 δ D 2 ϑ Φ - Ψ μ ( m ) μ ( n ) - α .

Setting ϒ m = μ ( m ) μ ( n ) α W m , Φ - W m , Ψ , then

ϒ m 2 δ D ϑ ϒ l + 2 δ D 2 ϑ Φ - Ψ

and setting ϒ = sup{ϒ m : mn},

ϒ 2 δ D ϑ ϒ + 2 δ D 2 ϒ Φ - Ψ .

Thus, taking δ sufficiently small so that 2δDϑ< 1 2 we have

ϒ 4 δ D 2 ϑ Φ - Ψ

and therefore,

W m , Φ - W m , Ψ 4 δ D 2 ϑ Φ - Ψ μ ( m ) μ ( n ) - α
(3.28)

Using (3.16) and (3.28) in (3.27) we obtain

A ( Φ ) n , λ - A ( Ψ ) n , λ 2 δ D ϑ 4 δ D 2 ϑ Φ - Ψ μ ( l ) μ ( n ) - α ν ε ( n ) + 2 δ D 2 ϑ Φ - Ψ μ ( l ) μ ( n ) - α - ε ν ε ( n ) δ K Φ - Ψ ν ε ( n )
(3.29)

for K' = 8δD3ϑ2 + 2D2ϑ > 0, provided that δK' νε(n) ≤ 1. This shows that A is a contraction.

Nextly, also given Φ, ΨX,U,Vand λ ∈ Y, set Zl,Φ,U= Zl, Φ,Uand Zl, Ψ, U= Zl,Ψ,U, we obtain

B ( Φ , U ) n , λ - B ( Φ , V ) n , λ δ D l = n μ ( l + 1 ) μ ( n ) - α ν ε - β ( l + 1 ) . 2 Z l , Φ , U - Z l , Ψ , V + Φ l , λ - Ψ l , λ Z l , Φ , U + W l , Φ + U l , λ - V l , λ W l , Φ + W l , Φ - W l , Ψ 1 + V l , λ + Ψ l , λ .
(3.30)

By (3.16), (3.26), and (3.30)

Z m , Φ , U - Z m , Ψ , V δ D l = n m - 1 μ ( m ) μ ( l + 1 ) - α ν ε - β ( l + 1 ) . 2 Z l , Φ , U - Z l , Ψ , V + Φ l , λ - Ψ l , λ Z l , Φ , U + W l , Φ + U l , λ - V l , λ W l , Φ + W l , Φ - W l , Ψ 1 + V l , λ + Ψ l , λ 2 δ D l = n m - 1 μ ( m ) μ ( l + 1 ) - α ν ε - β ( l + 1 ) Z l , Φ , U - Z l , Ψ , V + 12 δ 2 D 3 ϑ + 2 δ D 2 + 12 δ 2 D 3 ϑ Φ - Ψ . l = n m - 1 μ ( m ) μ ( l + 1 ) - α μ ( l ) μ ( m ) - α ν ε - β ( l + 1 ) ν ε ( n ) + 2 δ D 2 U - V l = n m - 1 μ ( m ) μ ( l + 1 ) - α μ ( l ) μ ( n ) - α ν ε - β ( l + 1 ) ν ε ( n ) 2 δ D l = n m - 1 μ ( m ) μ ( l + 1 ) - α ν ε - β ( l + 1 ) Z l , Φ , U - Z l , Ψ , V + ( 12 δ 2 D 3 ϑ + 2 δ D 2 + 12 δ 2 D 3 ϑ ) Φ - Ψ ϑ μ ( m ) μ ( n ) - α + 2 δ D 2 U - V ϑ μ ( m ) μ ( n ) - α 2 δ D μ ( m ) μ ( n ) - α l = n m - 1 μ ( n ) μ ( l + 1 ) - α ν ε - β ( l + 1 ) Z l , Φ , U - Z l , Ψ , V + δ K 0 Φ - Ψ + U - V μ ( m ) μ ( n ) - α
(3.31)

for some positive constant K0 = 12δD3ϑ2 + 2D2ϑ + 12δD3ϑ2 > 0, provided that δ ≤ 1. Setting ϒ m = μ ( m ) μ ( n ) α Z m , Φ , U n - Z m , Ψ , V n , we obtain

ϒ m 2 δ D ϑ ϒ l + δ K 0 Φ - Ψ + U - V

and setting ϒ = sup{ϒ m : mn} we obtain

ϒ 2 δ D ϑ ϒ + δ K 0 Φ - Ψ + U - V

Taking δ sufficiently small so that 2δDϑ< 1 2 we obtain

ϒ 2 δ K 0 Φ - Ψ + U - V

and therefore,

Z m , Φ , U - Z m , Ψ , V 2 δ K 0 Φ - Ψ + U - V μ ( m ) μ ( n ) - α .
(3.32)

Using (3.32) in (3.30) we obtain

B ( Φ , U ) n , λ - B ( Φ , V ) n , λ δ D l = n μ ( l + 1 ) μ ( n ) - α ν ε - β ( l + 1 ) 4 δ K 0 Φ - Ψ + U - V μ ( m ) μ ( n ) - α + Φ - Ψ ( 12 δ D 2 ϑ μ ( l ) μ ( n ) - α + 2 D μ ( l ) μ ( n ) - α ν ε ( n ) ) + 2 D U - V μ ( l ) μ ( n ) - α ν ε ( n ) + 12 δ D 2 ϑ μ ( l ) μ ( n ) - α Φ - Ψ 4 δ 2 D K 0 Φ - Ψ + U - V l = n μ ( l + 1 ) μ ( m ) μ 2 ( n ) - α ν ε - β ( l + 1 ) + δ D l = n μ ( l + 1 ) μ ( n ) - α ν ε - β ( l + 1 ) ( 12 δ 2 D 3 ϑ + 2 δ D 2 + 12 δ 2 D 3 ϑ ) Φ - Ψ μ ( l ) μ ( n ) - α ν ε ( n ) + 2 δ D 2 U - V μ ( l ) μ ( n ) - α ν ε ( n ) δ L Φ - Ψ + U - V

for some positive constant L = 4δDK0ϑ + δDK0 > 0, provided that δ L ≤ 1. It follows from (3.29) and the above inequality that for δ sufficiently small the operator S is a contraction.

Now we proceed with the proof of Theorem 3.1. By Lemma 3.5 and its proof, there exists a unique pair ( Φ ¯ , U ¯ ) X× such that S ( Φ ¯ , U ¯ ) = ( Φ ¯ , U ¯ ) and Φ ¯ is the unique sequence in X such that A ( Φ ¯ ) n , λ = Φ ¯ n , λ for each nJ, λ ∈ Y. Namely, Φ ¯ is the unique solution of Equation (3.12) as well as Equation (3.11). Together with (3.9) this implies that if ξE n , then

m W m , λ n ξ , Φ ¯ m , λ W m , λ n ξ

is a solution of (3.7) and (3.8). This means (3.3) holds.

Let Φ be another sequence for (3.3). If ξE n , then

( ξ , Φ n , λ ξ ) E n λ and A λ ( m , n ) ( ξ , Φ n , λ ξ ) E m λ

Thus, if (x m , y m ) is the solution of Equation (1.1) with x n = ξ and y n = Φ nξ, then y m = Φmx m for mn. This means (3.7) and (3.8) hold. Furthermore, the sequence x m = W m , λ n ξ satisfies (3.9) and (3.11) holds. So Φ= Φ ¯ .

Let ( x n , y n ) E n λ , then for each mn we have

( x m , y m ) = A λ ( m , n ) ( x n , y n )

and

x m = W m , λ x n and y m = Φ m , λ x m .

Therefore

( x m , y m ) = ( I d E m + Φ m , λ ) W m , λ x n .

By (3.16)

( x m , y m ) 4 D μ ( m ) μ ( n ) - α ν ε ( n ) x n .

On the other hand,

( x n , y n ) = x n + y n x n - y n = x n - Φ n , λ x n ( 1 - κ ) x n .

Thus

( x m , y m ) 4 D 1 - κ μ ( m ) μ ( n ) - α ν ε ( n ) x n , y n ,

which implies that (3.4) holds with D = 4 D 1 - κ >0.

For the C1 regularity of the maps λ Φ n , λ we consider the pair ( Φ 1 , U 1 ) = ( 0 , 0 ) X×. Clearly,

U n , λ 1 = d d λ Φ n , λ 1

for each nJ and λ ∈ Y. We define a sequence ( Φ m , U m ) X× by

( Φ m + 1 , U m + 1 ) = S ( Φ m , U m ) = ( A ( Φ m ) , B ( Φ m , U m ) )

For a given mJ, if λ Φ n , λ m is of class C1 for each nJ, and U n , λ m = d d λ Φ m , λ m for every nJ and λ ∈ Y, then the linear operators Wmand Zmsatisfy Z m , λ = d d λ W m , λ for mn and λ ∈ Y. Therefore we can apply Leibniz's rule to conclude that λ Φ n , λ m + 1 is of class C1 for every nJ, with

U n , λ m + 1 = B ( Φ m , U m ) n , λ = - l = n λ Q n A ( l + 1 , n ) - 1 B l ( λ ) ( W l , λ + Φ l , λ m W l , λ ) = d d λ A ( Φ m ) n , λ = d d λ Φ n , λ m + 1
(3.33)

for each nJ and λ ∈ Y.

Moveover, if ( Φ ¯ , U ¯ ) is the unique fixed point for the contraction map S. then the sequence Φ n , λ m m converge uniformly to Φ ¯ n , λ and the sequence U n , λ m m converge uniformly to U ¯ n , λ for each nJ and λ ∈ Y.

We know that if a sequence f m of C1 functions converges uniformly, and its derivatives f m also converges uniformly, then the limit of f m is of class C1, and its derivative is the limit of f m . Therefore, by (3.33) each function λ Φ ¯ n , λ is of class C1, and

d d λ Φ ¯ n , λ = U ¯ n , λ

for each nJ and λ ∈ Y. This completes the proof of Theorem 3.1.