Center Manifolds for Non-instantaneous Impulsive Equations Under Nonuniform Hyperbolicity

In this paper, we establish the existence of smooth center manifolds for a class of nonautonomous differential equations with non-instantaneous impulses under sufficiently small perturbations of the linear homogeneous part which has a nonuniform exponential trichotomy. In addition, we show the C 1 smoothness of center manifolds outside the jumping times. Funding. This work is partially supported by the National Natural Science Foundation of China (11661016), Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), Major Research Project of Innovative Group in Guizhou Education Department ([2018]012), Guizhou Data Driven Modeling Learning and Optimization Innovation Team ([2020]5016), the Slovak Research and Development Agency under the contract No. APVV-18-0308, and the Slovak Grant Agency VEGA No. 1/0358/20 and No. 2/0127/20. Manuscript received 28th September 2019, revised 24th March 2020, accepted 30th March 2020.


Introduction
Instantaneous impulsive effects arise naturally in physics, biology and control theory [1,2,23].Non-instantaneous impulsive effects start at an arbitrary fixed point and remain active on a finite time interval and this effect models certain dynamics of evolution processes in pharmaceutics.Noninstantaneous impulsive differential equations was introduced by Hernández and O'Regan [17] and is an extension of classical instantaneous impulsive differential equations [27,30]; we refer the reader to [12,18,22,[24][25][26]29] and the reference therein for results on qualitative and stability theory.
It is well known that the notion of (uniform) exponential dichotomy and exponential trichotomy play an important role in stability theory for differential equations and dynamical systems, both with continuous and discrete time.The theory of exponential dichotomies and exponential trichotomy and its applications are widely developed and we refer the reader to [8,15,16] for details and further references.Analogously, the more general notions of nonuniform exponential dichotomy and nonuniform exponential trichotomy play a similar role although under much weaker assumptions, and thus also for much larger classes of dynamics.In particular, the notion of nonuniform exponential dichotomy and nonuniform exponential trichotomy are essentially as weak as the (uniform) exponential dichotomy and exponential trichotomy.
A significant result in the theory of ordinary differential equations is the stable manifold theorem.The concept of the invariant manifold for rest points arises from the study of linear systems.Recall that if A is a linear operator on R n , then the spectrum of A splits naturally (from the point of view of the stability theorem) into three subsets: the eigenvalues with negative, zero, or positive real parts.With a linear change of coordinates that transforms A to its real Jordan normal form, we find that the differential equation u = A u decouples into an equivalent system x = Qx, y = W y, z = Rz, where (x, y, z) ∈ R k × R l × R m with k + l + m = n, and Q, W and R are linear operator whose eigenvalues have all negative, zero, and positive real parts, respectively.The subspace R k is called the stable manifold of the rest point of the original system u = A u, the subspace R l is called the center manifold, and the subspace R m is called the unstable manifold; we refer the reader to [7,Chapter 4] for details and further references.If a center manifold has dimension less than the dimension of the phase space, then the most important dynamics can be studied by considering the restriction of the original system to a center manifold.We refer the reader to [3,5,6,9,10,19,20] for more details and further references.
It should also be noted that the study of invariant manifolds has a long history.Fenner and Pinto [13] introduced the notion of (h, k) manifolds and gave conditions under which the property of being a manifold with asymptotic phase holds.In [14], Fenner and Pinto studied discrete nonautonomous nonlinear systems possessing (h, k)-trichotomies and (h, k)-hyperbolic.Li et al. [21,28] studied Lyapunov regularity and the existence of stable invariant manifolds and stable invariant manifolds of C 1 regularity for non-instantaneous impulsive equations.
In this paper we consider the following non-instantaneous linear impulsive differential equation: in R n , where we consider n × n matrices A(t ) and B ±i (t ) varying continuously for t ∈ R and impulsive point t ±i and junction point s ±i satisfying the relation t −(i +1) < s −i < t −i and s i < t i +1 < s i +1 , i ∈ N. The symbols y( + ±i ) and y( − ±i ) represent the right and left limits of y(t ) at t = ±i , respectively and set y( − ±i ) = y( ±i ).We consider the perturbed equation: C. R. Mathématique, 2020, 358, n 3, 341-364 where f : R×R n → R n and g i : R×R n → R n satisfy f (t , 0) = 0 and g i (t , 0) = 0 for each t ∈ R, i ∈ Z.We assume f and g i are piecewise continuous in t with at most discontinuities of the first kind at t i .The rest of the paper is organized as follows.In Section 2, we recall the notion of nonuniform exponential trichotomy and use Example 2 to present nonuniform exponential trichotomy for non-instantaneous impulsive differential equations, and some important lemmas are given.In Section 3, we establish the existence of center manifolds under sufficiently small perturbations of the linear homogeneous part which has a nonuniform exponential trichotomy.Existence of center manifolds are formulated and proved.

Preliminaries
We assume that the impulsive points t ±i and the junction points s ±i satisfy the following relation where r (t , s) denotes the number of impulsive points which belong to (s, t ).In [28], the authors introduced a bounded linear operator W ( • , • ) and any nontrivial solution of (1) can be formulated as y(t ) = W (t , s)y(s) for every t , s ∈ R. In addition, we obtained the fact that any nontrivial solution of (1) has a finite Lyapunov exponent provided (3) holds.Note W (t , s)W (s, τ) = W (t , τ) and W (t , t ) = Id for every t , s, τ ∈ R, where Id denotes the identity operator.
Definition 1 (see [5]).We say that (1) admits a nonuniform exponential trichotomy if there exist projections P for every t ≥ s, and there exist constants b Let ) be the center, stable and unstable subspaces for each t ∈ R, respectively.We now present an example of nonuniform exponential trichotomy.
Therefore, using (3), The remaining cases (when t < s) can be treated in a similar manner.Similar inequalities hold for the component v 2 .This shows that (7) admits a nonuniform exponential trichotomy.
To obtain the smoothness of the center manifolds, we consider the following result.Let X and Y be Banach spaces, and let U ⊂ X be an open set.Given constants α ∈ (0, 1] and b > 0, we consider the set The following result shows that D α b (U , Y ) is closed with respect to the supremum norm.
Lemma 3 (see [11,Lemma 2.2]).Let X and Y be Banach spaces, and let U ⊂ X be an open set.
Then the following properties hold: ) There exists sufficiently small δ > 0 such that for each t ∈ R, i ∈ Z and x, y ∈ R n we have and where ε is defined in Definition 1.
Note in (8) and ( 9) the δ > 0 is sufficiently small so that some constants in the following Lemma's can be appropriately chosen.
Without loss of generality, we consider only the case when t ≥ s, the case when t ≤ s can be proved in a similar fashion, we assume that (1) admits a nonuniform exponential trichotomy.Concerning the position relationship between the impulsive point t i and the junction point s i , the unique solution ( s) and fixed point s with 0 < s j < s < t j +1 < +∞, j ∈ N satisfies the following conditions: Let s j +r (t ,s) < t ≤ t j +r (t ,s)+1 and r (t , s) ≥ 1, and we have and Let t j +r (t ,s) < t ≤ s j +r (t ,s) and r (t , s) ≥ 1, and we have and From the properties of the projection operator, the function u( • ) defined in ( 10) is a solution of the center subspace of the phase space of (1) and the function v i ( • ) defined in (11) are the solutions of the stable subspace and the unstable subspace of the phase space of (1).Similar comments apply to the formula ( 12) and (13).Note, (10), (11), ( 12) and ( 13) are useful in studying the center manifold of the perturbed equation (2).
For each (s,

Smooth center manifold results
In this section, using ideas from [5], we consider the existence of smooth center manifolds under sufficiently small perturbations of nonuniform exponential trichotomies.We first describe a certain class of functions (in fact each center manifold is a graph of one of these functions (see [5]).
Let S be the space of functions φ : We equip the space S with the distance Lemma 3 can be used to show that S is a complete metric space with the distance in (15) (see [4,Proposition 3]).Given a φ ∈ S we consider the set Moreover, for arbitrarily fixed constant σ > 0 we let where φ ∈ S and ξ ∈ E (s).
Next, we present some auxiliary results for the function u φ ∈ B. Let Lemma 7. Assume that (1) admits a nonuniform exponential trichotomy.Given δ > 0 sufficiently small and φ, ϕ ∈ S and (s, ξ) ∈ (s j , t j +1 ) × E (s), j ∈ N, there exists a K > 0 such that for every t ≥ s.
A similar argument works if we assume (36).
A similar argument shows (36).The proof is complete.
Put (51) and ( 52) into (50) and provided that δ is sufficiently small, then the operator J is a contraction in the complete metric space S .Hence, there exists a unique function φ ∈ S such that J φ = φ.The proof is complete.
The following center manifold theorem is in the sense that we have the unique graph of the form V c φ (for some function φ ∈ S ) which is invariant under the semiflow.