Global Lipschitz Invariant Center Manifolds for ODEs with Generalized Trichotomies

In a Banach space, assuming that a linear nonautonomous differential equation $v'=A(t)v$ admits a very general type of trichotomy, we establish conditions for the existence of global Lipschitz invariant center manifold of the perturbed equation $v'=A(t)v+f(t,v)$. Our results not only improve results already existing in the literature, as well include new cases.


Introduction
Let X be a Banach space and let B(X) be the Banach algebra of all bounded linear operators acting on X. In this paper we are going to study the existence of global Lipschitz invariant center manifolds for differential equations of the type where A : R → B(X) is continuous, the perturbation f : R×X → X is a continuous function such that f (t, 0) = 0 for every t ∈ R, the function f t : X → X given by f t (v) = f (t, v) is Lipschitz for every t ∈ R and the linear differential equation Center manifolds are a powerful tool in the study of stability and in the study of bifurcations because in many cases allow the reduction of the dimension of the state space (see Carr [8], Henry [18], Guckenheimer and Holmes [15], Hale and Koçak [16] and Haragus and Iooss [17]). The first results on the existence of center manifolds were obtained by Pliss [26] in 1964 and by Kelley [19,20] in 1967. After that many authors studied the problem and proved results about center manifolds. A good expository paper for the case of autonomous differential equations in finite dimension was written by Vanderbauwhede [29] (see also Vanderbauwhede and Gils [31]) and for the case of autonomous differential equations in infinite dimension we recommend Vanderbauwhede and Iooss [30]. For more details in the finite dimensional case see Chow, Liu and Yi [11,10] and for the infinite dimensional case see Sijbrand [28], Mielke [23], Chow and Lu [12,13] and Chicone and Latushkin [9].
For nonautonomous differential equations the concept of exponential trichotomy is an important tool to obtain center manifolds theorems. This notion goes back to Sacker and Sell [27], Aulbach [1] and Elaydi and Hajek [14] and is inspired by the notion of exponential dichotomy that can be traced back to the work of Perron in [24,25]. However, as in the case of exponential dichotomies, the notion of exponential trichotomy is very demanding and several generalizations have appeared in the literature. Essentially we can find two ways of generalization: on one hand replace the exponential growth rates by nonexponential growth rates and on the other hand consider exponential trichotomies that also depend on the initial time and hence are nonuniform. Trichotomies with nonexponential growth rates have been introduced by Fenner and Pinto in [21] where the authors study the so called (h, k)−trichotomies and the nonuniform exponential trichotomies have been consider by Barreira and Valls in [2,3].
Hence, it is natural to consider trichotomies that are both nonuniform and nonexponential. This was done by Barreira and Valls in [4,5] where have been introduced the so-called ρ−nonuniform exponential trichotomies, but these trichotomies do not include as a particular case the (h, k)−trichotomies of Fenner and Pinto [21].
In this paper we are going to consider a very general type of trichotomies that includes as particular cases all the types of trichotomies mentioned above, as well new types of trichotomies. We only consider that the linear equation admits an invariant splitting in three invariant subspaces and the norms of the linear evolution operator composed with the three different projections are bounded by general functions that only depend on the initial and on the final time (see (D1), (D2) and (D3)). Despite of that we were able to obtain invariant manifolds provided that the Lipschitz constants of the perturbation are sufficiently small. Note that for dichotomies this has already been done in [6] and in [7] for differential and for difference equations, respectively.
The proof of the main theorem of this paper is based in the so-called classical Lyapunov-Perron method (see [22,24,25]) that consists in the following: − the variation of constants formula that allows to relate the solutions of the linear equation with the solutions of the perturbed equation; − the construction of a suitable space of functions that is a complete metric space; − the construction of a suitable contraction on the complete metric space mentioned above; − the application of Banach's fixed point theorem to the mentioned contraction gives a function that is the only fixed point of the contraction and whose graph is the invariant manifold. This method was used by many authors, namely [3,4,6,7]. However, in this paper we have introduced a novelty in the application of the method. In [3,4,6,7] are used two applications of the Banach's fixed point theorem, the first one to obtain the solutions of the perturbed equation along the stable/center direction and the other to obtain the solutions of the perturbed equation in the other directions. Here, with only one application of the Banach's fixed point theorem, we obtain the solutions of the perturbed equation in all directions.
As particular cases of our main result we improve the results obtained by Barreira and Valls [4,3] for the so-called ρ−nonuniform exponential trichotomies and nonuniform exponential trichotomies, respectively. Moreover, we also obtain as particular cases new results for nonuniform (a, b, c, d) −tricotomies and for µ−nonuniform polynomial trichotomies, concepts that have been introduced for the first time in this paper.
The structure of the paper is as follows. In Section 2 we introduce the notation and preliminearies. The main theorem of the paper is stated in Section 3 and in Section 4 we apply our main result to particular cases of trichotomies. Finally, in Section 5, we prove the main theorem.

Notation and preliminaries
Let X be a Banach space, let B(X) be the Banach algebra of all bounded linear operators acting on X and let A : R → B(X) be a continuous map. Consider the linear differential equation with s ∈ R and v s ∈ X. We are going to assume that (1) has a global solution and denote by T t,s the linear evolution operator associated to equation (1), i.e., for every t, s ∈ R.
We say that (1) admits an invariant splitting if, for every t ∈ R, there exist bounded projections P t , Q + t , Q − t ∈ B(X) such that (S1) P t + Q + t + Q − t = Id for every t ∈ R; (S2) P t Q + t = 0 for every t ∈ R; (S3) P t T t,s = T t,s P s for every t, s ∈ R; (S4) Q + t T t,s = T t,s Q + s for every t, s ∈ R. From (S1) and (S2) we have for every t ∈ R and from (S1), (S3) and (S4) it follows immediately that For each t ∈ R, we define the linear subspaces E t = P t (X), Given functions α : and denoting α(t, s), β + (t, s) and β − (t, s) by α t,s , β + t,s and β − t,s , respectively, we say that equation (1) admits a generalized trichotomy with bounds α = (α t,s ) (t,s)∈R 2 , β + = β + t,s (t,s)∈R 2 and β − = β − t,s (t,s)∈R 2 , or simply with bounds α t,s , β + t,s and β − t,s , if it admits an invariant splitting such that (D1) T t,s P s α t,s for every (t, s) ∈ R 2 ; (D2) T t,s Q + s β + t,s for every (t, s) ∈ R 2 ; (D3) T t,s Q − s β − t,s for every (t, s) ∈ R 2 . Example 2.1. Let a, b, c, d : R → ]0, +∞[ be C 1 functions and let ε a , ε b , ε c , ε d : R → [1, +∞[ be C 1 functions in R \ {0} and with derivatives from the left and from the right at t = 0. In R 4 , equipped with the maximum norm, consider the differential equation where The evolution operator of this equation is given by Morover, assuming that we have Therefore, if (3) is satisfied, then equation (2) has a generalized trichotomy with bounds We call the trichotomies with bounds of this type nonuniform (a, b, c, d) −trichotomies.
In (4), making and with a, b, c, d, D, ε ∈ R such that D 1 and ε 0, we get This bounds for the trichotomy were consider by Barreira and Valls in [5,4] and are called ρ−nonuniform exponential trichotomies. Note that in this case condition (3) is equivalent to a + c 0. When ρ(t) = t we obtain the nonuniform exponential trichotomies consider by Barreira and Valls in [3] and [2] with the bounds of the form

Main theorem
Suppose that equation (1) admits a generalized trichotomy. Consider the initial value problem where f : R × X → X is a continuous function such that and, for every t ∈ R, i.e., the function f t : for every x, y ∈ X and every t ∈ R and taking y = 0 in the last inequality, and by (8), we have for every x ∈ X and every t ∈ R. When (1) admits a generalized trichotomy, we can write the only solution of (7) in the form where v s = (ξ, η + , η − ) ∈ E s × F + s × F − s , and then solving problem (7) is equivalent to solve the following problem for every t ∈ R.
For each τ ∈ R, we define the flow by We are going to study the existence of invariant center manifolds for equation (7) when (1) admits a generalized trichotomy. The invariant center manifolds that we are going to obtain are given as the graph of a function belonging to a certain function space that we define now. Making and choosing N ∈ ]0, +∞[, we denote by A N the space of all continuous functions ϕ : G → X such that sup Note that from (18) it follows immediately that and making ξ = 0 in (19), we have By (17), and identifying The global Lipschitz invariant center manifolds that we intend to obtain are given as the graph of suitable functions ϕ belonging to some space A N .
Before state the main theorem we need to define the following quantities: and that are supposed to be finite.
Theorem 3.1. Let X be a Banach space. Suppose that (1) admits a generalized trichotomy with bounds α t,s , β + t,s and β − t,s and let f : R × X → X be a continuous function such that (8) and (9) are satisfied. If and 2σ + 2ω < 1, where σ and ω are given by (22) and (23), respectively, then there is N ∈ ]0, 1[ and a unique ϕ ∈ A N such that for every τ ∈ R, where Ψ τ is given by (15) and V ϕ is given by (21). Moreover, The proof of last theorem will be given in Section 5.
b) Note that the function γ in last theorem can be chosen as for t = 0 and where s is a fixed real number. In fact, from (28) Theorem 4.3. Let X be a Banach space and suppose that equation (1) admits a trichotomy with bounds of the form (5). Suppose that f : R×X → X is a continuous function that satisfies (8) and (9) and If then (7) admits a global Lipschtiz invariant center manifold, i.e., there is N ∈ ]0, 1[ and a unique ϕ ∈ A N such that for all (s, ξ), (s, ξ) ∈ G and where p s,ξ = (s, ξ, ϕ(s, ξ)) and p s,ξ = s, ξ, ϕ(s, ξ) .
Note that Theorem 4.3 improves the result of Barreira and Valls [4] because our result has a better asymptotic behavior. In fact, with our notation, in (32) where we have a and c, Barreira and Valls [4] have a + 2δD and c + 2δD, respectively.
Making ρ(t) = t in last theorem we have the following result.
Again, as in last theorem, we improve the asymptotic behavior of the result obtained by Barreira and Valls in [3]. Now we are going to assume that equation (1) admits a generalized trichotomy with bounds of the form for all (s, ξ), (s, ξ) ∈ G and where p s,ξ = (s, ξ, ϕ(s, ξ)) and p s,ξ = s, ξ, ϕ(s, ξ) .
To prove this theorem we need the following lemma.
Since for every t r s and every t r s we have ε for t r s. Then, since γ > ε + 1, it follows that Here we made the substitution τ = µ(r).
Making the substitution τ = µ(s) − µ(r) we have and with the substitution τ = µ(r) − µ(s) we obtain Hence, if δ is sufficiently small we have 2σ + 2ω < 1 and the theorem is proved.
In the next corollary we will consider µ(t) = t, i.e., for all (t, s) ∈ R 2 , and this type of trichotomies are called nonuniform polynomial trichotomies.

Proof of the main theorem
Before doing the proof of Theorem 3.1 we need to prove some lemmas. .
Proof. Clearly, equalities (33) are equivalent to Hence making and we obtain immediately the first equality in (34). Taking into account that Moreover, using the definition of N and M we can put and this proves the second equality in (34). To finish the proof we note it is clear that N > 0 and since M = 1 + σN/ω we have M > 1.
From now on the numbers M and N will be given by (35) and (36). Moreover, the number N mentioned in Theorem 3.1 is also given by (36).
It is easy to see that A N is a complete metric space with the metric for all ϕ, ψ ∈ A N . Making let B M be the set of all continuous functions x : G ′ → X such that x(s, s, ξ) = ξ for all (s, ξ) ∈ G, (39) Defining, for every x, y ∈ B M , it is easy to see that (B M , d ′ ) is a complete metric space.
Now we define an operator on C M,N and we will prove that it is a contraction and this will be essential in the proof of Theorem 3. and, since σ + ω < 1/2, N < 1 and M < 2, we obtain (σ + ω) max {1 + N, M } < 1 and this implies that T is a contraction. Now we are going to prove Theorem 3.1.