Smooth linearization of nonautonomous differential equations with a nonuniform dichotomy

Abstract

In this paper we give a smooth linearization theorem for nonautonomous differential equations with a nonuniform strong exponential dichotomy. In terms of a discretized evolution operator with hyperbolic fixed point 0, we formulate its spectrum and then give a spectral bound condition for the linearization of such equations to be simultaneously differentiable at 0 and Hölder continuous near 0. Restricted to the autonomous case, our result is the first one that gives a rigorous proof for simultaneously differentiable and Hölder linearization of hyperbolic systems without any nonresonant conditions.

Higher regularity of homeomorphisms in the Hartman-Grobman theorem for semilinear evolution equations

“…It is expansive, and the expansive estimate leads us to prove that the homeomorphism is Hölder continuous, not Lipschitzian. Therefore, most of the previous works on the regularity of conjugacy [13,20,41,42,46,52,54] is Hölder continuous.…”

“…As pointed out by Backes et al [1], even if the diffeomorphism F is C ∞ , the conjugacy (homeomorphism) can fail to be locally Lipschitz. The homeomorphisms are in general only (locally) Hölder continuous (see [1,2,3,4,5,8,13,20,42,54]). However, we prove that the conjugacy (homeomorphism) is Lipchitzian, but the inverse is Hölder continuous.…”

Higher regularity of homeomorphisms in the Hartman-Grobman theorem and a conjecture on its sharpness

“…An important and interesting problem is the regularity of the linearization, which have greatly attracted many mathematicians' attentions. Among the works on the linearization mentioned above, a lot of papers were devoted to proving the Hölder continuity of the homeomorphisms in the linearization theorem (see Backes et al [13], Barreira and Valls [14,15,16,17], Dragičević et al [36,37], Huerta et al [18,19], Hein and Prüss [11], Jiang [20,21], Pötzche [25], Shi and Zhang [28], Rodrigues and Solà-Morales [39,40], Xia et al [23,29], Zhang et al [42,43,44], Tan [46], Shi and Xiong [47]). For the sake of easier illustration, we restate the Palmer's linearization theorem [12] which has extended the classical Hartman-Grobman theorem ( [4,5]) to the nonautonomous case.…”

“…Sternberg [30,31] initially studied the smooth linearization problem. Recently, the smooth linearization for C k (1 ≤ k ≤ ∞) diffeomorphisms are well improved by Sell [32], Belitskill et al [33,34], Cuong et al [35], Dragičević et al [36,37], Elbialy [38], Rodrigues and Solà-Morales [39,40,41], Zhang et al [42,43,44,45]. In particular, a set of nice results on the sharp regularity of linearization for hyperbolic diffeomorphisms were established in Zhang et al [42,43,44].…”

A note on the differentiability of Palmer's topological equivalence for discrete systems

“…Other authors such as D. Dragičević et al [11,10] have also been interested in these properties of differentiability in a continuous and discrete context respectively . Recently, faced with the problem in its discrete context, the authors on [5] have studied the differentiability of said homeomorphism under the assumption that the linear system satisfies a nonuniform contraction.…”