Integral conditions for nonuniform $\mu$-dichotomy on the half-line

We give necessary integral conditions and sufficient ones for the existence of a general concept of $\mu$-dichotomy for evolution operators defined on the half-line which includes as particular cases the well-known concepts of nonuniform exponential dichotomy and nonuniform polynomial dichotomy, and also contains new situations. Additionally, we consider an adapted notion of Lyapunov function and use our results to obtain necessary and sufficient conditions for the existence of nonuniform $\mu$-dichotomies using these Lyapunov functions.


Introduction
The notion of exponential dichotomy is a fundamental tool in the study of stability of difference and differential equations and can be traced back to the work of Perron [17] on the stability of ordinary differential equations, and of Li [13] for discrete time systems. We also refer to the book of Chicone and Latushkin [11] for important results in infinite-dimensional spaces.
In some situations, in particular for nonautonomous systems, the concept of exponential dichotomy is too restrictive and it is important to look for more general hyperbolic behavior.
We can identify, at least, two ways to generalize this concept: allow some loss of hyperbolicity along the trajectories, a path leading to notions similar to Pesin's nonuniform hyperbolicity [18,20,19], and consider asymptotic behavior that is not necessarily exponential, an approach followed by Naulin and Pinto in [16,21], where the authors considered uniform dichotomies with asymptotic behavior given by general growth rates.
In recent years, a large number of papers study different aspects of the dynamical behavior of systems with nonuniform exponential dichotomies, a type of dichotomic behavior where some exponential loss of hyperbolicity along the trajectories is allowed (see for example the work of Barreira and Valls [4] and papers [14,22,24]). Also, several results were obtained in [1,2,3,5,6,7,8,9,10] for dichotomic behaviors that are both nonuniform and not necessarily exponential.
One of the most important results in the stability theory of evolution operators is due to Datko [12] which has given an integral characterization of uniform exponential stability. This characterization is used to obtain a necessary and sufficient condition for uniform exponential stability in terms of Lyapunov functions. Preda and Megan extend Datko theorem to uniform exponential dichotomy [23]. Generalizations of this result in the case of nonuniform exponential dichotomy are given in [15,14,22]. For more details and history about Datko theorem we refer the reader to [25].
In this paper, we consider a notion of dichotomy which is both nonuniform and not necessarily exponential in the general context of evolution operators, with the purpose of obtaining necessary conditions and sufficient ones in the spirit of Datko's results. We emphasize that this type of dichotomy includes as particular cases the notions of nonuniform exponential dichotomy and nonuniform polynomial dichotomy, respectively. We show that our results extend previous theorems and also contain new situations. Also, we note that we do not need to assume the invertibility of the evolution operators on the whole space, which allow us to apply our results to compact operators defined in infinite dimensional spaces.

Notions and preliminaries
Let X be a Banach space and let B(X) be the Banach algebra of all bounded linear operators on X. We denote by Id the identity operator of B(X). Throughout this paper, we also denote by R + 0 the set of non-negative real numbers and we consider ∆ the set defined by ∆ = (t, s) ∈ R + 0 × R + 0 : t ≥ s . We first recall the definition of an evolution operator: 1. An operator valued function U : ∆ → B(X) is said to be an evolution operator if (1) U (t, t) = Id for every t ≥ 0; (2) U (t, τ )U (τ, s) = U (t, s) for all t ≥ τ ≥ s ≥ 0; (3) (t, s) → U (t, s)x is continuous for every x ∈ X.
for all δ > 0. Setting δ → 0 in the relation above and using the fact that µ is a differentiable function, we obtain and hence (i) holds.
Definition 2.4. A strongly continuous function P : Given a projection valued function P : R + 0 → B(X), we denote by Q the complementary projection valued function, that is Q(t) = Id−P (t) for every t ≥ 0. Definition 2.5. We say that a projection valued function P : R + 0 → B(X) is compatible with an evolution operator U : ∆ → B(X) if, for all t ≥ s ≥ 0, we have (1) P (t)U (t, s) = U (t, s)P (s); (2) the restriction U (t, s)| Q(s)X : Q(s)X → Q(t)X is an isomorphism and we denote its inverse by U Q (s, t).
Remark 2.6. If P : R + 0 → B(X) is a projection valued function compatible with an evolution operator U : ∆ → B(X), then for every (t, s) ∈ ∆, it follows Definition 2.7. Given a growth rate µ : R + 0 → [1, +∞) and a projection valued function P : R + 0 → B(X) compatible with an evolution operator U : ∆ → B(X), we say that U has a nonuniform µ-dichotomy with projection valued function P if there exist constants a, b > 0, ε ≥ 0 and N 1 , N 2 ≥ 1 such that, for all t ≥ s ≥ 0, we have When ε = 0, we say that U has a uniform µ-dichotomy with projection valued function P .
In the following we consider particular cases of the notion of nonuniform µ-dichotomy: (1) if µ(t) = e t , then we recover the notion of nonuniform exponential dichotomy (in the sense of Barreira-Valls) [4] and in particular (when ε = 0) the classical notion of uniform exponential dichotomy; (2) if µ(t) = t + 1, then we recover the notion of nonuniform polynomial dichotomy [5,7,8].
be a continuous growth rate and ε ≥ 0 be a non-negative real number. On X = R 2 endowed with the norm (x 1 , x 2 ) = max{|x 1 |, |x 2 |}, we consider the projection valued function and its complementary projection valued function Obviously, we have that Given a, b > 0, we consider the evolution operator U : ∆ → B(R 2 ), Since P (t)P (s) = P (s), Q(t)Q(s) = Q(t) and Q(t)P (s) = 0, we have that P is a projection valued function compatible with U . Moreover, it follows that By (1) and using the relations above, we deduce that for t ≥ s ≥ 0, which shows that the evolution operator U has a nonuniform µ-dichotomy with projection valued function P . We will now show that for ε > 0 the nonuniform µ-dichotomy with projection valued function P is not a uniform µ-dichotomy with projection valued function P . Assume that U has a uniform µ-dichotomy with projection valued function P , then there exist ν > 0 and N ≥ 1 such that which is equivalent to Setting t = s in (2) we have and this is absurd because lim t→+∞ µ(t) = +∞. Therefore, when ε > 0 the evolution operator U does not have a uniform µ-dichotomy with projection valued function P .

The main results
For a given evolution operator U : ∆ → B(X) and a projection valued function P : R + 0 → B(X) compatible with U , we denote the Green function associated to the evolution operator U and the projection valued function P compatible with U by We have the following result Theorem 3.1. Let p > 0 and µ : R + 0 → [1, +∞) be a differentiable growth rate. If the evolution operator U : ∆ → B(X) has a nonuniform µ-dichotomy with a dichotomy projection valued function P : R + 0 → B(X), then for every positive constant γ < min{a, b} it follows that for every (t, x) ∈ R + 0 × X, where Proof. For (t, x) ∈ R + 0 × X and 0 < γ < min{a, b}, we have and this proves the result.
The following result is a partial converse of Theorem 3.1 and it can be considered a Datko type theorem [12,14] for the existence of nonuniform µ-dichotomy.
In the particular case when µ(t) = e t , we recover Theorem 1 in [14]: If there exist p ≥ 1, γ > α, ε ≥ 0 and D > 0 such that for every (t, x) ∈ R + 0 × X, then U has a nonuniform exponential dichotomy with projection valued function P .
A similar result to the one above can be obtained in the case of nonuniform polynomial dichotomy: Corollary 3.4. We assume that U : ∆ → B(X) is an evolution operator and P : R + 0 → B(X) is a projection valued function compatible with U such that there exist constants ω > 0, α ≥ 0 and M ≥ 1 with If there exist p ≥ 1, γ > α, ε ≥ 0 and D > 0 such that for every (t, x) ∈ R + 0 × X, then U has a nonuniform polynomial dichotomy with projection valued function P .
In the following we consider an evolution operator that has a nonuniform µ-dichotomy for a given growth rate µ, different from both exponential and polynomial functions.
Furthermore, proceeding in a similar manner to the proof of Theorem 3.1, we obtain for γ ∈ (α + 1, min{a, b}) and D = 1 , which shows that U has a nonuniform µ-dichotomy with projection valued function P . such that for every t ≥ 0 and every x ∈ X.
Theorem 3.6. Let p ≥ 1 and µ : R + 0 → [1, +∞) be a differentiable growth rate. If the evolution operator U : ∆ → B(X) has a nonuniform µ-dichotomy with a dichotomy projection valued function P : R + 0 → B(X), then for every positive constant γ < min{a, b} and every H ∈ H µ γ,p (P ) there is a function L : where the constant D is given by (4).
Proof. Let We have We first compute On the other hand, using the inequality x + y p ≤ 2 p−1 x p + 2 p−1 y p , for x, y ∈ X, it follows that Now, we have, by (13) and (14), for all (t, s, x) ∈ ∆ × X. Clearly L(t, P (t)x) ≥ 0 and L(t, Q(t)x) ≤ 0. Moreover, by Theorem 3.1, we deduce that for all (t, x) ∈ R + 0 × X. This ends the proof. Theorem 3.7. Let µ : R + 0 → [1, +∞) be a differentiable growth rate that verifies (5). Assume that U : ∆ → B(X) is an evolution operator and P : R + 0 → B(X) is a projection valued function compatible with U such that (6) holds. If for some p ≥ 1, γ > α and every H ∈ H µ γ,p (P ) there is a function L : R + 0 × X → R that satisfies conditions (i)-(iii) in Theorem 3.6 for some ε ≥ 0 and D > 0, then U has a nonuniform µ-dichotomy with projection valued function P .