Open Access

C1 regularity of the stable subspaces with a general nonuniform dichotomy

Advances in Difference Equations20122012:31

https://doi.org/10.1186/1687-1847-2012-31

Received: 13 December 2011

Accepted: 14 March 2012

Published: 14 March 2012

Abstract

For nonautonomous linear difference equations, we establish C1 regularity of the stable subspaces under sufficiently C1-parameterized perturbations. We consider the general case of nonuniform dichotomies, which corresponds to the existence of what we call nonuniform (μ, ν)-dichotomies.

Mathematics Subject Classification 2000: Primary 34D09; 34D10; 37D99.

Keywords

difference equationsparameter dependencenonuniform dichotomies

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 m J = . 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 (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, n J, 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 m J 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 : 22 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 m J. Then (A m )mJadmits a nonuniform (μ, ν) dichotomy with the projections P m : 22 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 n J, 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)nJYof 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, μ Y. 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 Φ X and λ Y, for each n J, 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, n J, 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 k J.

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 each m J , 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 Φ X such that
E m λ = A λ ( m , n ) E n λ
(3.3)
for each m, n J. Moreover,
  1. (1)
    for each n J, 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 n J.

     
Proof. Given n J 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 Φ X , n J 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 = ξ E 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 each Φ X and λ Y, the following properties are equivalent:
  1. (1)

    (3.11) holds for every n J and mn;

     
  2. (2)
    for every n J 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 Φ X , n J, 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 and A ( 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)) ≤ κλ - μ. This shows that A ( X ) X .

Now we note be the space of sequences U = (Un)nYof linear operators Un: E n F n indexed by Y such that λ Unis continuous for each n J, 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 × , n J, 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, and B ( 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 2 δ D ϑ < 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, Ψ X , 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 2 δ D ϑ < 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' ν ε (n) ≤ 1. This shows that A is a contraction.

Nextly, also given Φ, Ψ X , U , V and 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 2 δ D ϑ < 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 n J, 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 ξ E 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 ξ E 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 n J 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 m J, if λ Φ n , λ m is of class C1 for each n J, and U n , λ m = d d λ Φ m , λ m for every n J 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 n J, 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 n J 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 n J 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 n J and λ Y. This completes the proof of Theorem 3.1.

Declarations

Acknowledgements

This study was supported by the National Natural Science Foundation of China (Grant No. 11171090), the Program for New Century Excellent Talents in University (Grant No. NCET-10-0325), China Postdoctoral Science Foundation funded project (Grant No. 20110491345) and the Fundamental Research Funds for the Central Universities.

Authors’ Affiliations

(1)
Department of Mathematics, College of Science, Hohai University

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© Wang; licensee Springer. 2012

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