C1 regularity of the stable subspaces with a general nonuniform dichotomy
© Wang; licensee Springer. 2012
Received: 13 December 2011
Accepted: 14 March 2012
Published: 14 March 2012
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.
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 , plays an important role in a large part of the theory of differential equations and dynamical systems. We refer the reader to the books [2–5] 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 [8–11] 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  for details. As mentioned in , 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 .
Johnson and Sell  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  for the uniform exponential dichotomies and in  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 ) 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 , 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 , and we omit the detail for short.
for each m ≥ n, 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 each m ≥ n.
3. Main results
where C k = A k + B k (λ) for each k ∈ J.
Now we state the main result of this article.
- (1)for each n ∈ J, m ≥ n and λ ∈ Y we have(3.4)
The map λ ↦ Φn,λ is of class C 1 for each n ∈ J.
for each m ≥ n.
for each m ≥ n.
(3.11) holds for every n ∈ J and m ≥ n;
- (2)for every n ∈ J and m ≥ n we have(3.12)
provided that δ sufficiently small.
Since α > 0, letting m → +∞ in (3.18) we obtain identity (3.12).
for each m ≥ n. Since , it follows from (3.12) with n replace by m that (3.11) holds for each m ≥ n.
Lemma 3.3. For δ sufficiently small, A is well defined and.
where K = 6D + 24δD2ϑ > 0.
and provided that δ is sufficiently small, we obtain Cλμ(A(Φ)) ≤ κ∥λ - μ∥. This shows that .
for m > n, setting Zn,λ= 0. One observe that by the continuity of the functions Φl,λand Ul,λon λ the functions λ ↦ Wl,λand λ ↦ Zl,λare also continuous.
Lemma 3.4. For δ sufficiently small, the operator B is well defined, and.
provided that δ is sufficiently small. This shows that B is well defined for each n, and that ∥B(Φ, U) ∥ ≤ 1. Therefore, .
By Lemmas 3.3 and 3.4, it is clearly that the maps S is well defined and .
Lemma 3.5. For δ sufficiently small, the operator S is a contraction.
for K' = 8δD3ϑ2 + 2D2ϑ > 0, provided that δK' ν ε (n) ≤ 1. This shows that A is a contraction.
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.
is a solution of (3.7) and (3.8). This means (3.3) holds.
Thus, if (x m , y m ) is the solution of Equation (1.1) with x n = ξ and y n = Φ n,λξ, then y m = Φm,λx m for m ≥ n. This means (3.7) and (3.8) hold. Furthermore, the sequence satisfies (3.9) and (3.11) holds. So .
which implies that (3.4) holds with .
for each n ∈ J and λ ∈ Y.
Moveover, if is the unique fixed point for the contraction map S. then the sequence converge uniformly to and the sequence converge uniformly to for each n ∈ J and λ ∈ Y.
for each n ∈ J and λ ∈ Y. This completes the proof of Theorem 3.1.
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.
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