Stable manifolds for non-instantaneous impulsive nonautonomous differential equations

Stable manifolds for non-instantaneous impulsive nonautonomous differential equations

Mengmeng Li

, JinRong Wang

and Donal O’Regan

1Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China

2School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China

3School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

Received 17 January 2019, appeared 7 November 2019 Communicated by Nickolai Kosmatov

Abstract. In this paper, we study stable invariant manifolds for a class of nonau- tonomous non-instantaneous impulsive equations where the homogeneous part has a nonuniform exponential dichotomy. We establish a stable invariant manifold result for sufficiently small perturbations by constructing stable and unstable invariant man- ifolds and we also show that the stable invariant manifolds are of classC¹outside the jumping times using the continuous Fiber contraction principle technique.

Keywords: non-instantaneous impulsive equations, nonuniform exponential dichoto- my, invariant manifolds.

2010 Mathematics Subject Classification: 34D09, 34D35.

1 Introduction

Instantaneous impulsive effects arise naturally in physics, biology and control theory [1–4,14].

Non-instantaneous impulsive differential equations (impulse effects start at an arbitrary point and remain active on a finite time interval) was introduced by Hernández and O’Regan [10]

and is an extension of classical instantaneous impulsive differential equations [19,21]; we refer the reader to [9,11,13,15–18,22] and the reference therein for results on qualitative and stability theory.

Invariant manifold theory plays an important role in the theory of dynamical systems. To construct stable and unstable invariant manifolds without assuming the existence of uniform exponential dichotomy for associated linear systems is of interest. As a result it is natural to discuss the notion of nonuniform exponential dichotomy as it seems to be the weakest assumption needed to find weak sufficient conditions to guarantee the existence of stable and unstable invariant manifolds. The concept of invariant manifolds was defined first for nonuniformly hyperbolic trajectories in [12] and in [5] the authors established the existence of stable invariant manifold for nonautonomous differential equations without impulses in

BCorresponding author. Email: jrwang@gzu.edu.cn

Banach spaces. The authors in [7] studied the existence of stable invariant manifolds and stable invariant manifolds ofC¹ regularity for instantaneous impulsive differential equations. The existence of stable invariant manifold for non-instantaneous impulsive differential equations has not been discussed.

In this paper, we consider the ideas in [5,7] to discuss the existence of stable invariant man- ifolds for non-instantaneous nonlinear impulsive differential equations, where the linear part has a nonuniform exponential dichotomy. Recently [20] the authors studied Lyapunov reg- ularity, the relation between the Lyapunov characteristic exponent and stability, and nonuni- form exponential behavior for the following non-instantaneous linear impulsive differential equations:















y⁰(t) =A(t)y(t), t ∈(s_i,t_i+1], i=0, 1, 2, . . . , y(t⁺_i ) =B_i(t⁺_i )y(t⁻_i ), i=1, 2, . . . ,

y(t) =B_i(t)y(t⁻_i ), t∈ (t_i,s_i], i=1, 2, . . . , y(s⁺_i ) =y(s⁻_i )_, i=1, 2, . . . ,

(1.1)

inRⁿ, where we consider n×n matrices A(t)and B_i(t) varying continuously fort ≥ 0 and i∈_Nand impulsive pointt_i and junction points_i satisfying the relations_i−1<t_i <s_i,i∈_N. The symbolsy($⁺_i )andy($⁻_i )represent the right and left limits of y(t)att = $_i, respectively and sety($⁻_i ) =y($_i).

In this paper we study the following perturbed equations:















y⁰(t) = A(t)y(t) + f(t,y(t)), t ∈(s_i,t_i+1], i=0, 1, 2, . . . , y(t⁺_i ) =B_i(t_i⁺)y(t⁻_i ) +g_i(t_i⁺,y(t_i⁻)), i=1, 2, . . . , y(t) = B_i(t)y(t⁻_i ) +g_i(t,y(t⁻_i )), t∈(t_i,s_i], i=1, 2, . . . , y(s⁺_i ) =y(s⁻_i ), i=1, 2, . . . ,

(1.2)

where f : R⁺₀ ×_Rⁿ →_Rⁿandg_i :R⁺₀ ×_Rⁿ →_Rⁿ satisfy f(t, 0) =0 and g_i(t, 0) =0 for each t≥0,i∈N. We assume f is piecewise continuous intwith at most discontinuities of the first kind att_i andg_i is of classC¹.

We show that for a small deviation from the classical notion of uniform exponential di- chotomy for (1.1) and for any sufficiently small perturbation term f and non-instantaneous impulsive conditions g_i, there exists a stable invariant manifold for the perturbed equation (1.2). It was emphasized in [5] that this smallness is a rather common phenomenon at least from the point of view of ergodic theory (almost all linear variational equations obtained from a measure-preserving flow admit a nonuniform exponential dichotomy with arbitrarily small nonuniformity).

The notion of nonuniform hyperbolicity plays an important role in the construction of stable and unstable invariant manifolds and we establish a stable invariant manifold result for sufficiently small perturbations by constructing stable and unstable invariant manifolds and we also show that the stable invariant manifolds are of class C¹ outside the jumping times using the continuous Fiber contraction principle technique.

The rest of the paper is organized as follows. In Section 2, we recall the notion of nonuni- form exponential dichotomy and use Example 2.2 to present nonuniform exponential di- chotomies for non-instantaneous impulsive differential equations. In Section 3, we establish the existence of stable manifolds under sufficiently small perturbations of a nonuniform ex- ponential dichotomy. Existence of stable manifolds are formulated and proved. In the final

section, we establish a C¹ regularity result, Theorem 4.7, for stable manifolds by assuming that (1.1) admits a nonuniform exponential dichotomy.

2 Preliminary

Set R⁺₀ = [0,+_∞)andPC(_R⁺₀,Rⁿ) := {x :R₀⁺→ _Rⁿ : x∈ C((t_i,t_i+1],Rⁿ),i=0, 1, 2,· · · and there exist x(t⁻_i )andx(t⁺_i )with x(t⁻_i ) =x(t_i)}with the norm kxk_PC := sup_t∈R+kx(t)k, and C(_R⁺₀,Rⁿ)denotes the Banach space of vector-valued continuous functions from R⁺₀ → _Rⁿ endowed with the normkxk_C(R⁺

0) =sup_t∈R⁺

0 kx(t)kfor a normk · konRⁿ. We assume that

0=s₀=t₀< t₁ <s₁<· · · <t_i <s_i <· · · , with lim_i→_∞t_i =_{∞, limi→∞}s_i =_{∞, and}

ρ:=lim sup

t>s>0

r(t,s)

t−s <_∞, (2.1)

wherer(t,s)denotes the number of impulsive points which belong to(s,t).

In [20], the authors introduced a bounded linear operatorW(·,·)and any nontrivial solu- tion of (1.1) can be formulated by y(t) = W(t,s)y(s)for everyt,s ∈ _R⁺₀. In addition, the fact that any nontrivial solution of (1.1) has a finite Lyapunov exponent provided (2.1) holds was obtained. Note W(t,s)W(s,τ) =W(t,τ)andW(t,t) = Id for everyt ≥ s ≥ τ ≥ 0, where Id denotes the identity operator.

Definition 2.1. (see [7]) We say that (1.1) admits a nonuniform exponential dichotomy if there exist projectionsP(t)for everyt ≥0 satisfying

W(_t,s)_P(_s) =_P(_t)_W(_t,s)_{, t}≥ s, and there exist some constants D,a,b,ε>0 such that

kW(t,s)P(s)k ≤De^{−a(t−s)+εs}, t≥s, (2.2) and

kW(t,s)Q(s)k ≤De^{−b(s−t)+εs}, s≥ t, (2.3) where Q(t) =Id−P(t)is the complementary projection ofP(t).

Let E(t) = P(t)(_Rⁿ) and F(t) = Q(t)(_Rⁿ)be the stable and unstable subspaces for each t≥0 respectively.

Now, we consider the following examples (in the particular caseP(t) =Id) of nonuniform exponential dichotomies for non-instantaneous impulsive differential equations.

Example 2.2. Letµ,ν,b>0. We consider non-instantaneous impulsive differential equations



















y⁰(t) = (−µ−νcos(t))y(t), t∈(s_i,t_i+1], i=0, 1, 2, . . . , y(t⁺_i ) = (b+1)e^−µtiy(t⁻_i ), i=1, 2, . . . ,

y(t) = (b+1)e^−µty(t⁻_i ), t ∈(t_i,s_i], i=1, 2, . . . , y(s⁺_i ) =y(s⁻_i )_, i=1, 2, . . . ,

y(s) =y_s, t₀< s<t₁,

(2.4)

with

µ>ν+ρln(b+1). (2.5)

Fors_i <t ≤t_i+1, the solutions are given byy(t) =W(t,s)y_s, where W(t,s) = (b+₁)^r(t,s)e^−µ

r(t,s) i∑=1

ti

e^{−µ(t−s)+ν}

r(t,s) i∑=1

(sinsi−sinti)+ν(sins−sint)

. From (2.1) and (2.5), there exists constantD>0 such that

W(t,s) = (b+1)^r(t,s)e^−µ

r(t,s) i∑=1

ti

e^{−µ(t−s)+ν}

r(t,s) i∑=1

(sinsi−sinti)+ν(sins−sint)

≤D(b+1)^r(t,s)e^−µ

r(t,s) i∑=1

t_i

e^{−µ(t−s)+ν(t−s)+2νs}

≤De^{(−µ+ν+ρln(b+1))(t−s)+2νs}. (2.6) Fort_i+1<t ≤s_i+1, the solutions are given byy(t) = (b+1)e^−µty(t⁻_i+1) =W(t,s)y_s, where

W(t,s) = (b+₁)^r(t,s)e^−µ

r(t,s) i∑=1

ti

e^{−µ(t−s)+ν}

r(_ti,s) i∑=1

(sinsi−sinti)+ν(sins−sint_r(t,s))

. From (2.1) and (2.5), there exists constantD>0 such that

W(t,s) = (b+1)^r(t,s)e^−µ

r(t,s) i∑=1

ti

e^{−µ(t−s)+ν}

r(_ti,s) i∑=1

(sinsi−sinti)+ν(sins−sint_r(t,s))

≤ De^{(−µ+ν+ρln(b+1))(t−s)+2νs}. (2.7) Throughout the paper, we will always denote the normk(x,y)k= kxk+kykfor(x,y)∈ Rⁿ. We assume that there exists sufficiently smallδ > 0 such that for each t ≥ 0,i ∈ _{N, we}

have (

kf(t,x)− f(t,y)k ≤δe^−2εtkx−yk,

kg_i(t,x)−g_i(t,y)k ≤δe^−(a+2ε)tkx−yk. (2.8) Note in (2.8) the constantδ > 0 is sufficiently small so that some constants in the following Lemmas can be appropriately chosen.

Now we assume that (1.1) admits a nonuniform exponential dichotomy and the unique solution(P(t)y(t),Q(t)y(t)) = (u(t),v(t))∈ E(t)×F(t)of (1.2) with initial condition(ξ,η)∈ E(s)×F(s) and fixed point s with s_j < s < t_j+1 < _∞,j = 0, 1, 2, . . . satisfies the following conditions:

Lets_j+r(t,s) <t≤t_j+r(t,s)+1andr(t,s)≥1, and we have u(t) =W(t,s)ξ+

Z _tj+1

s W(t,τ)P(τ)f(τ,u(τ),v(τ))dτ +

Z _t

s_j+r(t,s)

W(t,τ)P(τ)f(τ,u(τ),v(τ))dτ +

r(t,s)−1 k

∑

Z _tj+k+1

s_j+k

W(t,τ)P(τ)f(τ,u(τ),v(τ))dτ +

r(t,s) k

∑

W(t,s_j+k)P(s_j+k)g_j+k(s_j+k,u(t_j+k),v(t_j+k)), (2.9)

and

v(t) =W(t,s)η+

Z _tj+1

s W(t,τ)Q(τ)f(τ,u(τ),v(τ))dτ +

Z _t

s_j+r(t,s)

W(t,τ)Q(τ)f(τ,u(τ),v(τ))dτ +

r(t,s)−1 k

∑

Z _tj+k+1

s_j+k

W(t,τ)Q(τ)f(τ,u(τ),v(τ))dτ +

r(t,s) k

∑

W(t,s_j+k)Q(s_j+k)g_j+k(s_j+k,u(t_j+k),v(t_j+k)). (2.10) Lett_j+r(t,s)< t≤s_j+r(t,s) andr(t,s)≥1, and we have

u(t) =W(t,s)ξ+

Z _tj+1

s W(t,τ)P(τ)f(τ,u(τ),v(τ))dτ +P(t)g_j+r(t,s)(t,u(t_j+r(t,s)),v(t_j+r(t,s)))

+

r(t,s)−1 k

∑

Z _tj+k+1

s_j+k

W(t,τ)P(τ)f(τ,u(τ),v(τ))dτ +

r(t,s)−1 k

∑

W(t,s_j+k)P(s_j+k)g_j+k(s_j+k,u(t_j+k),v(t_j+k)), (2.11) and

v(t) =W(t,s)η+

Z _tj+1

s W(t,τ)Q(τ)f(τ,u(τ),v(τ))dτ +Q(t)g_j+r(t,s)(t,u(t_j+r(t,s)),v(t_j+r(t,s)))

+

r(t,s)−1 k

∑

Z _tj+k+1

sj+k

W(t,τ)Q(τ)f(τ,u(τ),v(τ))dτ +

r(t,s)−1 k

∑

W(t,s_j+k)Q(s_j+k)g_j+k(s_j+k,u(t_j+k),v(t_j+k)). (2.12) For each(s,ξ,η)∈_R⁺₀ ×E(s)×F(s)we consider the semiflow

Ψt(s,ξ,η) = (s+t,u(s+t),v(s+t)).

3 Stable manifold results

In this section, using ideas from [7], we consider the existence of stable manifolds under sufficiently small perturbations of a nonuniform exponential dichotomy. We first describe a certain class of functions (in fact each stable manifold is a graph of one of these functions (see [5])).

LetZ be the space of functionsψ:R₀⁺×E(·)→F(·)having at most discontinuities of the first kind in the first variable such that for eachs ≥0, andx,y∈ E(s)_{we have:}

1. ψ(s, 0) =0 andψ(s,E(s))⊂F(s); 2. There exists a constantL>0 such that

kψ(s,x)−ψ(s,y)k ≤Lkx−yk_{. (3.1)}

We equip the spaceZ with the distance

d(ψ,ϕ) =sup{kψ(s,x)−ϕ(s,x)k/kxk:s∈_R0⁺andx∈ E(s)\{0}}, and noteZ is a complete metric space. Given a ψ∈ Z we consider the set

W_ψ^s ={(s,ξ,ψ(s,ξ)):(s,ξ)∈_R⁺₀ ×E(s)}. (3.2) Definition 3.1. W_ψ^s is called the stable manifold of (1.2) if the semiflow

Ψt(s,ξ,ψ(s,ξ))∈ W_ψ^s, for everyt≥0, whereψ∈ Z andξ ∈ E(s).

Given a constantc>0 we define R^c_j =sup

t>s r(t,s)

k

∑

e^{−c(tj+k−s)} <_∞, j∈_N. (3.3) Using Definition 3.1, each solution in W_ψ^s must be of the form (t,u(t),ψ(t,u(t))) for t ≥ s.

In particular, the equations in (2.9); (2.10) for s_j+r(t,s) < t ≤ t_j+r(t,s)+1 and (2.11); (2.12) for t_j+r(t,s)< t≤s_j+r(t,s) can be replaced by

u(t) =W(t,s)ξ+

Z _tj+1

s

W(t,τ)P(τ)f(_τ,u(τ)_,ψ(_τ,u(τ)))dτ +

r(t,s)−1 k

∑

Z _tj+k+1

sj+k

W(t,τ)P(τ)f(τ,u(τ),ψ(τ,u(τ)))dτ +

Z _t

s_j+r(t,s)

W(t,τ)P(τ)f(τ,u(τ),ψ(τ,u(τ)))dτ +

r(t,s) k

∑

W(t,s_j+k)P(s_j+k)g_j+k(s_j+k,u(t_j+k),ψ(t_j+k,u(t_j+k))), (3.4)

ψ(t,u(t)) =W(t,s)ψ(s,u(s)) +

Z _tj+1

s W(t,τ)Q(τ)f(τ,u(τ),ψ(τ,u(τ)))dτ +

r(t,s)−1 k

∑

Z _tj+k+1

s_j+k

W(t,τ)Q(τ)f(_τ,u(τ)_,ψ(_τ,u(τ)))dτ +

Z _t

s_j+r(t,s)

W(t,τ)Q(τ)f(τ,u(τ),ψ(τ,u(τ)))dτ +

r(t,s) k

∑

W(t,s_j+k)Q(s_j+k)g_j+k(s_j+k,u(t_j+k),ψ(t_j+k,u(t_j+k))). (3.5) and

u(t) =W(t,s)ξ+

Z _tj+1

s W(t,τ)P(τ)f(τ,u(τ),ψ(τ,u(τ)))dτ +

r(t,s)−1 k

∑

Z _tj+k+1

s_j+k

W(t,τ)P(τ)f(_τ,u(τ)_,ψ(_τ,u(τ)))dτ +

r(t,s)−1 k

∑

W(t,s_j+k)P(s_j+k)g_j+k(s_j+k,u(t_j+k),ψ(t_j+k,u(t_j+k)))

+P(t)g_j+r(t,s)(t,u(t_j+r(t,s))_,ψ(t_j+r(t,s),u(t_j+r(t,s))))_{, (3.6)}

ψ(t,u(t)) =W(t,s)ψ(s,u(s)) +

Z _tj+1

s W(t,τ)Q(τ)f(τ,u(τ),ψ(τ,u(τ)))dτ +

r(t,s)−1 k

∑

Z _tj+k+1

sj+k

W(t,τ)Q(τ)f(τ,u(τ),ψ(τ,u(τ)))dτ +

r(t,s)−1 k

∑

W(t,s_j+k)Q(s_j+k)g_j+k(s_j+k,u(t_j+k),ψ(t_j+k,u(t_j+k)))

+Q(t)g_j+r(t,s)(t,u(t_j+r(t,s)),ψ(t_j+r(t,s),u(t_j+r(t,s)))). (3.7) Define the function u = u_ψ for ψ ∈ Z. We need the following impulsive Gronwall’s inequality results.

Lemma 3.2. Let x :R⁺₀ →_R⁺₀ be a piecewise continuous function at most with discontinuities of the first kind at the points t_i. If

x(t)≤α+

Z _t

s w(τ)x(τ)dτ+

∑

s≤t_i<t

γ_ix(t_i), t ≥s

for some constantsα,γ_i ≥0, and some function w :R⁺₀ →_R⁺₀, then the following estimate holds x(t)≤α

∏

s≤ti<t

(1+γ_i)exp Z _t

s w(τ)dτ

.

Lemma 3.3. Assume that(1.1)admits a nonuniform exponential dichotomy. Givenδ>0sufficiently small and(s,ξ)∈_R⁺₀ ×E(s), for eachψ∈ Zthere exists a unique function u_ψ:[s,+_∞)→_Rⁿwith uψ(s) =ξ and uψ(t)∈ E(t)satisfying(3.4)and(3.6)with t≥s. Moreover,

ku(t)k ≤2De^{−a(t−s)+εs}kξk for t≥s. (3.8) Proof. Given (s,ξ) ∈ _R0⁺×E(s) with ξ 6= 0, and ψ ∈ Z, we consider the space Ω := {u(·) : [s,+_∞)→_Rⁿ}such thatu(s) =ξandu(t)∈E(t)for eacht >sanduis piecewise continuous with at most discontinuities of the first kind at t_i with kuk⁰ = sup_t≥sku(t)k

kξk e^{a(t−s)−εs} ≤ 2D.

One can easily verify that Ω is a Banach space with the norm k · k⁰. For arbitrary t ≥ s, we consider the operator Λ (see below) defined in the two intervals (s_j+r(t,s),t_j+r(t,s)+1] and (t_j+r(t,s),s_j+r(t,s)].

Case 1. Fors_j+r(t,s) <t≤ t_j+r(t,s)+1, we consider (_Λu)(t) =W(t,s)ξ+

Z _tj+1

s W(t,τ)P(τ)f(τ,u(τ),ψ(τ,u(τ)))dτ +

r(t,s)−1 k

∑

Z _tj+k+1

sj+k

W(t,τ)P(τ)f(_τ,u(τ)_,ψ(_τ,u(τ)))dτ +

Z _t

s_j+r(t,s)W(t,τ)P(τ)f(_τ,u(τ)_,ψ(_τ,u(τ)))dτ +

r(t,s) k

∑

W(t,s_j+k)P(s_j+k)g_j+k(s_j+k,u(t_j+k),ψ(t_j+k,u(t_j+k))). Givenu₁,u₂∈ _Ωandτ≥s. Note that (2.8) and (3.1), we obtain

v(τ) =kf(τ,u₁(τ),ψ(τ,u₁(τ)))− f(τ,u2(τ),ψ(τ,u2(τ)))k

≤δ(1+L)kξke^{−a(τ−s)+εs}e^−2ετku₁−u₂k⁰, (3.9)

and

ω_i = kg_i(s_i,u₁(t_i),ψ(t_i,u₁(t_i)))−g_i(s_i,u₂(t_i),ψ(t_i,u₂(t_i)))k

≤ δ(1+L)kξke^{−a(ti−s)+εs}e^−(a+2ε)siku₁−u₂k⁰, i=j+k,k=1,· · · ,r(t,s), ω_i(t) =kg_i(t,u₁(t_i),ψ(t_i,u₁(t_i)))−g_i(t,u2(t_i),ψ(t_i,u2(t_i)))k

≤ δ(1+L)kξke^{−a(ti−s)+εs}e^−(a+2ε)tku₁−u₂k⁰, i= j+r(t,s). (3.10) Therefore, we obtain

k(_Λu₁)(t)−(_Λu₂)(t)k

≤

Z _tj+1

s

kW(t,τ)P(τ)kv(τ)dτ+

r(t,s)−1 k

∑

Z _tj+k+1

s_j+k

kW(t,τ)P(τ)kv(τ)dτ +

Z _t

s_j+r(t,s)

kW(t,τ)P(τ)kv(τ)dτ+

r(t,s) k

∑

kW(t,s_j+k)P(s_j+k)kω_j+k

≤

Z _t

s

kW(t,τ)P(τ)kv(τ)dτ+

r(t,s) k

∑

kW(t,s_j+k)P(s_j+k)kω_j+k

≤ ^Dδ(1+L)

ε kξkku₁−u₂k⁰e^−a(t−s)+Dδ(1+L)kξkku₁−u₂k⁰e^{−a(t−s)+εs}

r(t,s) k

∑

e^{−a(tj+k−s)}

≤ ^Dδ(₁+L)

ε kξkku1−u2k⁰e^−a(t−s)+_Dδ(₁+_L)kξkku1−u2k⁰e^{−a(t−s)+εs}R^a_j, which implies that

k_Λu₁−_Λu₂k⁰ ≤θku₁−u₂k⁰, whereθ =Dδ(1+L)(¹

ε +R^a_j). Takeδ sufficiently small so thatθ < ¹₂. Therefore, the operator Λbecomes a contraction mapping. Moreover

k_Λuk⁰ ≤ kW(·,s)ξk⁰+θkuk⁰ ≤D+θkuk⁰ ≤2D,

and hence, Λ(_Ω) ⊂ Ω. Therefore, Λ has a unique fixed point u ∈ _Ω such that u = _Λu.

Moreover, fort≥ swe have

ku(t)k ≤2De^{−a(t−s)+εs}kξk_. Case 2. Fort_j+r(t,s)<t ≤s_j+r(t,s), we have

(_Λu)(t) =W(t,s)ξ+

Z _tj+1

s W(t,τ)P(τ)f(τ,u(τ),ψ(τ,u(τ)))dτ +

r(t,s)−1 k

∑

Z _tj+k+1

s_j+k

W(t,τ)P(τ)f(τ,u(τ),ψ(τ,u(τ)))dτ +

r(t,s)−1 k

∑

W(t,s_j+k)P(s_j+k)g_j+k(u(t_j+k),ψ(t_j+k,u(t_j+k))) +P(t)g_j+r(t,s)(t,u(t_j+r(t,s)),ψ(t_j+r(t,s),u(t_j+r(t,s)))). From (2.2) and (2.3) witht= s≥t₀, we obtain

kP(t)k ≤De^εt and kQ(t)k ≤De^εt.

Using (3.9) and (3.10), we have k(_Λu1)(t)−(_Λu2)(t)k

≤

Z _tj+1

s

kW(t,τ)P(τ)kv(τ)dτ+

r(t,s)−1 k

∑

Z _tj+k+1

sj+k

kW(t,τ)P(τ)kv(τ)dτ +

r(t,s)−1 k

∑

kW(t,s_j+k)P(s_j+k)kω_j+k+kP(t)ω_j+r(t,s)(t)k

≤

Z _t

s

kW(t,τ)P(τ)kv(τ)dτ+

r(t,s)−1 k

∑

kW(t,s_j+k)P(s_j+k)kω_j+k+kP(t)kω_j+r(t,s)(t)

≤ ^Dδ(1+L)

ε kξkku₁−u2k⁰e^−a(t−s)+Dδ(1+L)kξkku₁−u2k⁰e^{−a(t−s)+εs}

r(t,s) k

∑

e^{−a(tj+k−s)}

≤ ^Dδ(1+L)

ε kξkku₁−u₂k⁰e^−a(t−s)+Dδ(1+L)kξkku₁−u₂k⁰e^{−a(t−s)+εs}R^a_j, which implies that

k_Λu₁−_Λu₂k⁰ ≤θku₁−u₂k⁰, where θ= Dδ(1+L)(¹

ε +R^a_j). Takeδsufficiently small so thatθ < ¹₂. Therefore, the operator Λbecomes a contraction. Moreover

k_Λuk⁰ ≤ kW(·,s)ξk⁰+θkuk⁰ ≤D+θkuk⁰ ≤2D.

and hence, Λ(_Ω) ⊂ Ω. Therefore, Λ has a unique fixed point u ∈ _Ω such that u = _Λu.

Moreover, for t≥s we have

ku(t)k ≤2De^{−a(t−s)+εs}kξk. The proof is complete.

Now, we establish some auxiliary results for the function u_ψ. Given δ > 0 sufficiently small, ψ ∈ Z,s ≥ 0, andξ, ¯ξ ∈ E(s), from Lemma 3.3, we consider the unique functionsuψ

and ¯uψ such thatuψ(s) =ξ and ¯uψ(s) =ξ.^¯

Lemma 3.4. Assume that(1.1)admits a nonuniform exponential dichotomy. Givenδ>0sufficiently small andξ, ¯ξ ∈ E(s), we have

kuψ(t)−u¯ψ(t)k ≤2De^{(−a+ρln(1+Dδ(1+L)))(t−s)+εs}kξ−ξ^¯k for eachψ∈ Z and t≥ s≥0.

Proof. For eachτ≥s, we have

kf(τ,u_ψ(τ),ψ(τ,u_ψ(τ)))− f(τ, ¯u_ψ(τ),ψ(τ, ¯u_ψ(τ)))k ≤δ(1+L)e^−2ετku_ψ(τ)−u¯_ψ(τ)k, and

kg_i(s_i,uψ(t_i),ψ(t_i,uψ(t_i)))−g_i(s_i, ¯uψ(t_i),ψ(t_i, ¯uψ(t_i)))k

≤δ(1+L)e^−(a+2ε)siku_ψ(t_i)−u¯_ψ(t_i)k, i= j+k,k=1, 2, . . . ,r(t,s), kg_i(t,uψ(t_i),ψ(t_i,uψ(t_i)))−g_i(t, ¯uψ(t_i),ψ(t_i, ¯uψ(t_i)))k

≤δ(1+L)ke^−(a+2ε)tku_ψ(t_i)−u¯_ψ(t_i)k, i=j+r(t,s).

Set

φ(t) =kuψ(t)−u¯ψ(t)k. Using (3.4) and (3.6), we have two cases to consider:

Case 1. Fors_j+r(t,s)<t ≤t_j+r(t,s)+1, we have φ(t)≤ kW(t,s)P(s)kkξ−ξ^¯k+δ(1+L)

Z _t

s

kW(t,τ)P(τ)ke^−2ετφ(τ)dτ +δ(1+L)

r(t,s) k

∑

kW(t,s_j+k)P(s_j+k)ke^{−(a+2ε)sj+k}φ(t_j+k)

≤ De^{−a(t−s)+εs}kξ−ξ^¯k+Dδ(1+L) _{Z t}

s e^{−a(t−τ)−ετ}φ(τ)dτ+

r(t,s) k

∑

e^−atφ(t_j+k)

≤ De^{−a(t−s)+εs}kξ−ξ^¯k+Dδ(1+L) Z _t

s e^{−a(t−τ)−ετ}φ(τ)dτ+

r(t,s) k

∑

e^{−a(t−tj+k)}φ(t_j+k)

.

Case 2. Fort_j+r(t,s)<t ≤s_j+r(t,s), we have φ(t)≤ kW(t,s)P(s)kkξ−ξ^¯k+δ(1+L)

Z _tj+r(t,s)

s

kW(t,τ)P(τ)ke^−2ετφ(τ)dτ +δ(1+L)

r(t,s)−1 k

∑

kW(t,s_j+k)P(s_j+k)ke^{−(a+2ε)sj+k}φ(t_j+k) +Dδ(₁+L)e^εte^−(a+2ε)tφ(t_j+r(t,s))

≤ De^{−a(t−s)+εs}kξ−ξ^¯k +Dδ(1+L)

_{Z t}

s e^{−a(t−τ)−ετ}φ(τ)dτ+

r(t,s)−1 k

∑

e^−atφ(t_j+k) +e^−atφ(t_j+r(t,s))

≤ De^{−a(t−s)+εs}kξ−ξ^¯k+Dδ(1+L) _{Z t}

s e^{−a(t−τ)−ετ}φ(τ)dτ+

r(t,s) k

∑

e^−atφ(t_j+k)

≤ De^{−a(t−s)+εs}kξ−ξ^¯k+Dδ(1+L) Z _t

s e^{−a(t−τ)−ετ}φ(τ)dτ+

r(t,s) k

∑

e^{−a(t−tj+k)}φ(t_j+k)

, where we usekP(t)k ≤De^εt.

Settingv₁(t) =e^a(t−s)φ(t), we obtain v₁(t)≤ De^εskξ−ξ^¯k+Dδ(1+L)

_{Z t}

s e^a(τ−s)e^−ετφ(τ)dτ+

r(t,s) k

∑

e^a(tj+k−s)φ(t_j+k)

≤ De^εskξ−ξ^¯k+Dδ(1+L) Z _t

s e^−ετv₁(τ)dτ+

r(t,s) k

∑

v₁(t_j+k)

. Therefore, using Lemma3.2, we have

v₁(t)≤De^εskξ−ξ^¯k

r(t,s) k

∏

(1+Dδ(1+L))exp _{Z t}

s Dδ(1+L)e^−ετdτ

≤De^Dδ(1ε+L)e^εskξ−ξ^¯k(1+Dδ(1+L))^r(t,s).