# Parameter Dependence of Stable Invariant Manifolds for Delay Differential Equations under(μ,ν)-Dichotomies

Authors: Lijun Pan
Journal: Journal of Mathematics
 (2014)

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## Abstract

We obtain the existence of stable invariant manifolds for the nonlinear equationx′=L(t)xt+f(t,xt,λ) provided that the linear delay equation x′=L(t)xt admits a nonuniform (μ,ν)-dichotomy and f is a sufficiently small Lipschitz perturbation. We show that the stable invariant manifolds are dependent on parameter λ. Namely, the stable invariant manifolds are Lipschitz in the parameter λ. In addition, we also show that nonuniform (μ,ν)-contraction persists under sufficiently small nonlinear perturbations.


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## Body

## 1. Introduction

The theory of exponential dichotomy is attracting much attention in recent years. This is mostly because the concept of exponential dichotomy plays an important role in the obtention of invariant manifolds of differential equations. The notion of exponential dichotomy can be traced back to Perron in [1]. Since then, Sacker and Sell [2–5] investigated sufficient conditions for the existence of an exponential dichotomy, also in the infinite-dimensional setting. In [6], Barreira and Valls discussed the much weaker notion of nonuniform exponential dichotomy. This theory of nonuniform hyperbolicity was introduced in the seventies by Barreira and Pesin [7]. For background material on exponential dichotomy for differential equations, the books [8–10] may be consulted.We note that the study of robustness in the case of exponential behavior has a long history. For the early work we can refer to Perron [1] and Massera and Schäffer [11]. For more recent works we refer to [12–15]. In particular, in [12], Barreira and Valls studied the existence of stable invariant manifold for any sufficiently small nonlinear perturbation of the linear equation. In [13], the authors discussed the parameter dependence of stable manifolds under nonuniform exponential dichotomy.Recently, general stable and unstable behaviors with growth rates given by two increasing functions are exhibited by Bento and Silva [16]. The results in this work rely on the notion of nonuniform (μ,ν)-dichotomy, which includes the traditional exponential dichotomy and the polynomial dichotomy [17–19] and the ρ-nonuniform exponential dichotomy [20–22]. Since the nonuniform (μ,ν)-dichotomy is more general than nonuniform dichotomy before, so it is an emerging field drawing attention from both theoretical and applied disciplines. For example, in [23], the authors showed the robustness of nonuniform (μ,ν)-dichotomy provided that μ and ν were two differentiable functions. Authors of [24] showed the existence of invariant manifolds for sufficiently small Lipschitz perturbations of a linear equation with nonuniform (μ,ν)-dichotomy.In the implementation of differential equations, time delay is a common phenomenon due to instantaneous perturbations. We can refer to the book [25] for functional differential equations. Recently, Barreira et al. [26] investigated the parameter dependence of stable manifolds for delay equations with polynomial dichotomies. It is assumed that f is a sufficiently Lipschitz perturbation.Our main aim is to show that the general behavior exhibited by a linear nonuniform(μ,ν)-dichotomy persists under nonlinear perturbed equations with parameter λ, where μ, ν are two increasing functions. More precisely, we establish the existence of Lipschitz stable invariant for nonlinear perturbed equations with parameter λ provided that the linear part has a nonuniform (μ,ν)-dichotomy and the nonlinear perturbations are sufficiently small. Our method is inspired in the former work in [26]. Meanwhile, an example is given to illustrate the applicability of the results.The organization of this paper is as follows. In Section2, we introduce delay linear differential equations and the perturbed equation with parameter λ. In Section 3, we consider the case of nonuniform (μ,ν)-contraction. In Section 4, we show that the asymptotic stability of a nonuniform (μ,ν)-dichotomy persists under sufficiently small nonlinear perturbations with parameter λ. Finally, an example is provided to illustrate our theorems in Section 5.


## 2. Preliminaries

Givenr>0, let χ=C([-r,0],Rn) be the Banach space of continuous functions φ:[-r,0]→Rn endowed with the norm
(1)∥φ∥=sup⁡θ∈[-r,0]|φ(θ)|,
where |·| is the norm in Rn. Let χ¯ be the set of functions φ:[-r,0]→Rn such that for each s∈[-r,0] the limits lim⁡θ→s-φ(θ) and lim⁡θ→s+φ(θ) exist, and lim⁡θ→s+φ(θ)=φ(s). By [26], we see that χ¯ is a Banach space with the norm.Givens≥0 and φ∈χ, we consider the initial value problem
(2)x′=L(t)xt,xs=φ,t≥s,
where xt(θ)=x(t+θ) for θ∈[-r,0], L(t):χ→Rn are linear operators, and there is a ϑ>0 such that
(3)∫tt+r‍∥L(τ)∥dτ≤ϑ(1+t),
for all t≥0. For each (s,φ)∈R+×χ, there is a unique solution t→xt(·,s,φ) of the initial value problem (2) with xs(·,s,φ)=φ. Define the evolution operator T(t,s):χ→χ associated with (2) by
(4)T(t,s)φ=xt(·,s,φ),t≥s.IfL(t) is written in the form
(5)L(t)φ=∫-r0‍dθ[η(t,θ)]φ(θ),
where η(t,θ) is n×n matrices and is measurable in (t,θ)∈R+×[-r,0] and continuous from the left in θ, by [26], each linear operator L(t) can be extended to χ¯ with the help of the integral in (5) in case that the Riemann-Stieltjes sums take value [η(t,b)-η(t,a)]φ(b-) for each subinterval t→[a,b]⊂[-r,0]. Thus for each (s,φ)∈R+×χ¯, there is a unique solution t→xt(·,s,φ)⊂χ¯ of the integral equation obtained from (2) with xs(·,s,φ)=φ and the corresponding evolution operator T¯(t,s):χ¯→χ¯ is defined by
(6)T¯(t,s)φ=xt(·,s,φ),t≥s.
It is easy to see that T¯(t,s)∣χ=T(t,s) and T¯(t,s)χ¯⊂χ for t≥s+r. We also consider the perturbed equation
(7)x′=L(t)xt+f(t,xt,λ),xs=φ,t≥s,
where f:R+×χ¯×Y→Rn and Y is an open subset of a Banach space (the parameter space).


## 3. Stability for Nonuniform(μ,ν)-Contraction

In this section, we show that the asymptotic stability of a nonuniform(μ,ν) contraction persists under sufficiently small nonlinear perturbations. We say that an increasing function u:[0,∞)→[1,∞) is said to be a growth rate if u(0)=1 and lim⁡t→∞u(t)=∞. Let μ(t),ν(t) be growth rates. Then we recall the definition of nonuniform (μ,ν)-contraction for (2).Definition 1.
Equation (2) is said to admit a nonuniform (μ,ν)-contraction if there exist constants a<0, ɛ≥0, and K>1 such that
(8)∥T¯(t,s)∥≤K[μ(t)μ(s)]aν(s)ɛ,
for t≥s≥0.We assume that the following conditions hold:(H1)
f(t,0,λ)=0 for all t≥0 and λ∈Y;(H2)
there exist constantsc>0,γ>0 such that for t≥0, φ,ψ∈χ¯, λ,η∈Y(9)|f(t,φ,λ)-f(t,ψ,λ)|≤cν(t)-γ∥x-y∥,|f(t,φ,λ)-f(t,φ,η)|≤cν(t)-γ|λ-η|·∥φ∥;(H3)
there existsM≥0 such that M=∫0∞‍ν(τ)ɛ-γdτ<∞;(H4)
cKM≤1/2.Theorem 2.
Assume that (2) admits a nonuniform (μ,ν)-contraction. If (H1)–(H4) hold, then for each s≥0,φ∈χ¯, the solutions xtλ of (7) with initial condition xsλ=φ satisfy
(10)∥xtλ∥≤K1-cKM[μ(t)μ(s)]aν(s)ɛ∥φ∥,∥xtλ-xtμ∥≤cK2M(1-cKM)2[μ(t)μ(s)]aν(s)ɛ|λ-μ|·∥φ∥,
for every t≥s and λ,μ∈Y.Proof.
We see that the solution of (7) satisfies the following variation-of-parameters formula:
(11)xt=T¯(t,s)φ+∫stT¯(t,τ)S0f(τ,xτ,λ)dτ,
where
(12)S0(θ)={0,-r≤θ<0,Id,θ=0.
We consider the space C of function x=xλ:[s-r,+∞)→Rn such that xs=φ, xt∈χ¯ for each t≥s and ∥x∥′≤2K, where
(13)∥x∥′=sup⁡{∥xt∥[μ(t)/μ(s)]aν(s)ɛ∥φ∥:t≥s}.
It is easy to see that C is a complete metric space. Set
(14)(Jλx)(t+θ)=[T¯(t,s)φ](θ)+∫st‍T¯(t,τ)(S0f(τ,xτ,λ))(θ)dτ,
for each λ∈Y, x∈C, and t≥s. Clearly, Jλx is continuous and (Jλx)s=φ and (Jλx)t∈χ¯ for each t≥s. By (H1), for each x,y∈C, we have
(15)∥(Jλx)t-(Jλy)t∥≤∫st‍∥T¯(t,τ)∥·|f(τ,xτ,λ)-f(τ,yτ,λ)|dτ≤cK[μ(t)μ(s)]aν(s)ɛ∥φ∥·∥x-y∥′∫st‍ν(τ)ɛ-γdτ≤cKM[μ(t)μ(s)]aν(s)ɛ∥φ∥·∥x-y∥′.
Thus we have
(16)∥(Jλx)t-(Jλy)t∥′≤cKM∥x-y∥′≤12∥x-y∥′.
Then the operator Jλ becomes a contraction. Furthermore, by (H1), (H2), and (8), we get
(17)∥(Jλx)t∥≤∥T¯(t,s)φ∥+∫st‍∥T¯(t,τ)∥·|f(τ,xτ,λ)|dτwwwwiii≤K[μ(t)ν(t)]aν(s)ɛ∥φ∥wwwwwww+cKM[μ(t)μ(s)]aν(s)ɛ∥φ∥·∥x∥′.
Thus
(18)∥Jλx∥′≤K+cKM∥x∥′≤K+12∥x∥′≤2K,
which implies that Jλ(C)⊂C. Therefore, there exists a unique function x∈C such that Jλx=x. By (18), we have
(19)∥xtλ∥≤K1-cKM[μ(t)μ(s)]aν(s)ɛ∥φ∥,
for each t≥s. By (19) and (H2), we get
(20)|f(τ,xτλ,λ)-f(τ,xτη,η)|=|f(τ,xτλ,λ)-f(τ,xτλ,η)|+|f(τ,xτλ,η)-f(τ,xτη,η)|≤cν(τ)-γ∥xτη∥·|λ-η|+cν(τ)-γ∥xτλ-xτη∥≤cK1-cKM[μ(τ)μ(s)]aν(s)ɛν(τ)-γ∥φ∥·|λ-η|+c[μ(τ)μ(s)]aν(s)ɛν(τ)-γ∥φ∥·∥xλ-xη∥′.
Therefore, it follows that
(21)∥xtλ-xtμ∥≤∫st‍∥T¯(t,τ)∥·|f(τ,xτλ,λ)-f(τ,xτμ,μ)|dτ≤cK21-cKM[μ(t)μ(s)]aν(s)ɛ∥φ∥·|λ-η|∫st‍ν(τ)ɛ-γdτ+cK[μ(t)μ(s)]aν(s)ɛ∥φ∥·∥xλ-xη∥′∫st‍ν(τ)ɛ-γdτ≤cK2M1-cKM[μ(t)μ(s)]aν(s)ɛ∥φ∥·|λ-η|+cKM[μ(t)μ(s)]aν(s)ɛ∥φ∥·∥xλ-xη∥′,
which yields that
(22)∥xλ-xμ∥′≤cK2M1-cKM|λ-η|+cKM∥xλ-xη∥′.
This completes the proof of the theorem.


## 4. Stable Invariant Manifolds

In this section, we establish stable manifolds theorem under sufficiently small perturbation of nonuniform(μ,ν)-dichotomy.Definition 3.
Equation (2) is said to admit a nonuniform (μ,ν)-dichotomy if there exist projections P(t):χ¯→χ¯ and constants a<0≤b, ɛ≥0, and K>1 such that for each t≥s≥0(1)
P(t)T¯(t,s)=T¯(t,s)P(s);(2)
lettingQ(t)=Id-P(t), the map
(23)T¯Q(t,s):=T¯(t,s)∣Q(s)χ¯:Q(s)χ¯⟶Q(t)χ¯ is invertible;(3)
(24)∥T¯(t,s)P(s)∥≤K[μ(t)μ(s)]aν(s)ɛ,∥T¯Q(t,s)-1Q(t)∥≤K[μ(t)μ(s)]-bν(t)ɛ.Setting the linear subspace
(25)E(t)=P(t)(χ¯),F(t)=Q(t)(χ¯).
LetB be the space of continuous functions Φ:E→χ¯ such that, for each s≥0, Φ(s,0)=0,Φ(s,E(s))⊂F(s), and
(26)∥Φ(s,φ)-Φ(s,ψ)∥≤∥φ-ψ∥,φ,ψ∈E(s),
with the norm
(27)∥Φ∥=sup⁡{∥Φ(s,φ)∥∥φ∥:s≥0,φ∈E(s)∖{0}}.
Moreover, we denote by B the space of continuous functions Φ^:Y→E(s), which can be written by Φ^(λ)=Φλ, such that
(28)∥Φλ(s,φ)-Φη(s,φ)∥≤|λ-η|·∥φ∥,
for each s≥0, φ∈E(s), and λ,η∈Y. Given Φ^∈B and λ∈Y, we consider the set
(29)Wλ={s,φ,Φλ(s,φ):s≥0,φ∈E(s)}.
Assume that (2) admits a nonuniform (μ,ν)-dichotomy, for each λ∈Y, the solution (xt,yt)∈E(t)×F(t) of (7) with initial condition (xs,xs)∈E(s)×F(s) satisfies
(30)xt=T¯(t,s)xs+∫st‍T¯(t,τ)P(τ)S0f(τ,xτ,yτ,λ)dτ,yt=T¯(t,s)ys+∫st‍T¯(t,τ)Q(τ)S0f(τ,xτ,yτ,λ)dτ.
We consider the semiflow
(31)Ωϵλ(s,xs,ys)=(s+ϵ,xs+ϵ,ys+ϵ),
which is generated by the following equation:
(32)t′=1,x′=L(t)xt+f(t,xt,λ).To obtain our results, we also need the following assumptions:(H5)
cKM≤1/4;(H6)
there existsN>0 such that N=sup⁡s≥0{(ν(s)ɛ/μ(s)a-b)∫s∞‍μ(τ)a-bν(τ)ɛ-γdτ}<∞;(H7)
lim⁡t→∞μ(t)a-bν(t)ɛ=0;(H8)
cK2N/(1-2cKM)≤1/2;(H9)
cK3MN/(1-2cKM)2<1/4.Theorem 4.
Assume that (2) admits a nonuniform (μ,ν)-dichotomy. If (H1)–(H3) and (H5)–(H9) hold, then for each δ>0 sufficiently small, there exists a unique function Φ^=Φλ∈B such that
(33)Ωϵλ(s,φ,Φλ(s,φ))∈Wλ,foreachϵ≥0,λ∈Y.
Furthermore, for each ϵ=t-s≥0, φ,ψ∈E(s), and λ,η∈Y(34)∥Ωϵλ(s,φ,Φλ(s,φ))-Ωϵλ(s,ψ,Φλ(s,ψ))∥≤2K1-2cKM[μ(t)μ(s)]aν(s)ɛ∥φ-ψ∥,(35)∥Ωϵλ(s,φ,Φλ(s,φ))-Ωϵη(s,φ,Φη(s,φ))∥≤[6cK2M(1-2cKM)2+K1-2cKM][μ(t)μ(s)]aν(s)ɛ∥φ∥·|λ-η|.Proof.
We replace (30) by
(36)xt=T¯(t,s)φ+∫st‍T¯(t,τ)P(τ)S0f(τ,xτ,Φλ(τ,xτ),λ)dτ,(37)Φλ(t,xt)=T¯(t,s)Φλ(s,φ)iiiiiiiiiiiiiiiiii+∫st‍T¯(t,τ)Q(τ)S0f(τ,xτ,Φλ(τ,xτ),λ)dτ.Lemma 5.
For eachλ∈Y, Φ^=Φλ∈B, given (s,φ)∈R+×E(s), there is a unique function x(λ,Φ^,φ):[s-r,+∞)→Rn with xs(λ,Φ^,φ)=φ and xt(λ,Φ^,φ)∈E(t), satisfying (36). Moreover, for t≥s(38)∥xt(λ,Φ^,φ)∥≤K1-2cKM[μ(t)μ(s)]aν(s)ɛ∥φ∥.Proof.
We consider the operatorJ¯λ defined by
(39)(J¯λx)t=T¯(t,s)φiiiiiiiiiiiiii+∫st‍T¯(t,τ)P(τ)S0f(τ,xτ,Φλ(τ,xτ),λ)dτ,
in the space A of functions x=x(λ,Φ^,φ):[s-r,+∞)→Rn such that xs(λ,Φ^,φ)=φ, xt(λ,Φ,φ)∈E(t) for each t≥s and ∥x∥′≤2K, where ∥x∥′ is given by (13). We see that A is a complete metric space with the norm ∥x∥′. For each x,y∈A and t≥s, by (H2) and (26), we obtain
(40)|f(τ,xτ,Φλ(τ,xτ),λ)-f(τ,yτ,Φλ(τ,yτ),λ)|≤cν(τ)-γ∥(xτ-yτ,Φλ(τ,xτ)-Φλ(τ,yτ))∥≤2c[μ(τ)μ(s)]aν(s)ɛν(τ)-γ∥φ∥·∥x-y∥′.
By (39), (40), and (24), we have
(41)∥(J¯λx)t-(J¯λy)t∥≤∫st‍∥T¯(t,τ)P(τ)∥·|f(τ,xτ,Φλ(τ,xτ),λ)iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii-f(τ,yτ,Φλ(τ,yτ),λ)|dτ≤2cK[μ(t)μ(s)]aν(s)ɛ∥φ∥·∥x-y∥′∫st‍ν(τ)ɛ-γdτ≤2cKM[μ(t)μ(s)]aν(s)ɛ∥φ∥·∥x-y∥′.
Therefore
(42)∥J¯λx-J¯λy∥′≤2cKM∥x-y∥′.
By (H2) and (26), we obtain
(43)∥f(τ,xτ,Φλ(τ,xτ),λ)∥≤cν(τ)-γ∥(xτ,Φλ(τ,xτ))∥≤cν(τ)-γ(∥xτ∥+∥Φ(τ,xτ)∥)≤2cν(τ)-γ∥xτ∥.
Thus
(44)∥(J¯λx)t∥≤∥T¯(t,s)φ∥iiiiiiiiiiiiiiiiii+∫st‍∥T¯(t,τ)P(τ)∥·|f(τ,xτ,Φλ(τ,xτ),λ)|dτ∥(J¯λx)t∥≤K[μ(t)μ(s)]aν(s)ɛ∥φ∥iiiiiiiiiiiiiiiiii+2cK[μ(t)μ(s)]aν(s)ɛ∥x∥′∫st‍ν(τ)ɛ-γdτ∥(J¯λx)t∥≤K[μ(t)μ(s)]aν(s)ɛ∥φ∥+2cKM[μ(t)μ(s)]aν(s)ɛ∥x∥′.
The remainder of proof is proceeded the same as Theorem 2; here we omit it.Lemma 6.
GivenΦ^=Φλ∈B, for each λ∈Y, φ,ψ∈E(s), one has
(45)∥xt(λ,Φ^,φ)-xt(λ,Φ^,ψ)∥≤K1-2ckM[μ(t)μ(s)]aν(s)ɛ∥φ-ψ∥.Proof.
By(H2) and (26), for each τ≥s, we get
(46)q1(t):=|f(τ,xτ(λ,Φ^,φ),Φλ(τ,xτ(λ,Φ^,φ)),λ)iiiiiiiiiiii-f(τ,xτ(λ,Φ^,ψ),Φλ(τ,xτ(λ,Φ^,ψ)),λ)|q1(t):≤2cν(τ)-γ∥xτ(λ,Φ^,φ)-xτ(λ,Φ^,ψ)∥q1(t):≤2c[μ(τ)μ(s)]aν(s)ɛν(τ)-γ∥x(λ,Φ^,φ)-x(λ,Φ^,ψ)∥′.
By (39), (46), and (24), we have
(47)∥xt(λ,Φ^,φ)-xt(λ,Φ^,ψ)∥≤∥T¯(t,s)P(s)∥·∥φ-ψ∥+∫st‍∥T¯(t,τ)P(τ)∥q1(τ)dτ≤K[μ(t)μ(s)]aν(s)ɛ∥φ-ψ∥+2cK[μ(t)μ(s)]aν(s)ɛ×∥x(λ,Φ^,φ)-x(λ,Φ^,ψ)∥′∫st‍ν(τ)ɛ-γdτ≤K[μ(t)μ(s)]aν(s)ɛ∥φ-ψ∥+2cKM[μ(t)μ(s)]aν(s)ɛ∥x(λ,Φ^,φ)-x(λ,Φ^,ψ)∥′,
which implies that inequality (45) holds.Lemma 7.
Givenφ∈E(s), for each λ∈Y, Φ^=Φλ, Ψ^=Ψλ,Φ^,Ψ^∈B, one has
(48)∥xt(λ,Φ^,φ)-xt(λ,Ψ^,φ)∥≤cK2M(1-2cKM)2[μ(t)μ(s)]aν(s)ɛ∥φ∥·∥Φ^-Ψ^∥.Proof.
By(H2) and (26), for each τ≥s, we get
(49)q2(τ):=|f(τ,xτ(λ,Φ^,φ),Φλ(τ,xτ(λ,Φ^,φ)),λ)iiiiiiiiiiiii-f(τ,xτ(λ,Ψ^,φ),Ψλ(τ,xτ(λ,Ψ^,φ)),λ)|q2(τ):≤cν(τ)-γ∥(xτ(λ,Φ^,φ)-xτ(λ,Ψ^,φ),Φλ(τ,xτ(λ,Φ^,φ))-Ψλ(τ,xτ(λ,Ψ^,φ)))iiiiiiiiiiiiiiiiiiiiiiii-Ψλ(τ,xτ(λ,Ψ^,φ)))∥q2(τ):≤cν(τ)-γ[∥xτ(λ,Φ^,φ)-xτ(λ,Ψ^,φ)∥iiiiiiiiiiiiiiiiiiiiiiiiii+∥Φλ(τ,xτ(λ,Φ^,φ))-Ψλ(τ,xτ(λ,Ψ^,φ))∥],∥Φλ(τ,xτ(λ,Φ^,φ))-Ψλ(τ,xτ(λ,Ψ^,φ))∥≤∥Φλ(τ,xτ(λ,Φ^,φ))-Φλ(τ,xτ(λ,Ψ^,φ))∥+∥Φλ(τ,xτ(λ,Ψ^,φ))-Ψλ(τ,xτ(λ,Ψ^,φ))∥≤∥xτ(λ,Φ^,φ)-xτ(λ,Ψ^,φ)∥+∥xτ(λ,Φ^,φ)∥·∥Φ^-Ψ^∥≤[μ(τ)μ(s)]aν(s)ɛ∥φ∥·∥x(λ,Φ^,φ)-x(λ,Ψ^,φ)∥′+∥xτ(λ,Φ^,φ)∥·∥Φ^-Ψ^∥.
Then, by (24), (49), and Lemma 5, we obtain
(50)∥xt(λ,Φ^,φ)-xt(λ,Ψ^,φ)∥≤∫st‍∥T¯(t,τ)P(τ)∥q2(τ)dτ≤2cK[μ(t)μ(s)]aν(s)ɛ∥φ∥·∥x(λ,Φ^,φ)-x(λ,Ψ^,φ)∥′∫st‍ν(τ)ɛ-γdτ+cK21-2cKM[μ(t)μ(s)]aν(s)ɛ∥φ∥·∥Φ^-Ψ^∥∫st‍ν(τ)ɛ-γdτ≤2cKM[μ(t)μ(s)]aν(s)ɛ∥φ∥·∥x(λ,Φ^,φ)-x(λ,Ψ^,φ)∥′+cK2M1-2cKM[μ(t)μ(s)]aν(s)ɛ∥φ∥·∥Φ^-Ψ^∥.
Therefore, inequality (48) holds for t≥s≥0.Next, we transform (37) into an equivalent problem.Lemma 8.
GivenΦ^=Φλ∈B, the following properties hold.(1)
If for eachs≥0, φ∈E(s),  t≥s(51)Φλ(t,xt(λ,Φ^,φ))=T¯(t,s)Φ^(s,φ)+∫st‍T¯(t,τ)Q(τ)S0f(τ,xτ(λ,Φ^,φ),Φλ(τ,xτ(λ,Φ^,φ)),λ)dτ,then
(52)Φλ(s,φ)=-∫s∞‍T¯(τ,s)-1Q(τ)S0f(τ,xτ(λ,Φ^,φ),iiiiiiiiiiiiiiiiiiwwwwwwwwwwwiiiΦλ(τ,xτ(λ,Φ^,φ)),λ)dτ.(2)
If identity (52) holds for each s≥0, φ∈E(s), then (51) holds for each s≥0, φ∈E(s), and t≥s.Proof.
By(H1), (H2), (26), and Lemma 5, we have
(53)|f(τ,xτ(λ,Φ^,φ),Φλ(τ,xτ(λ,Φ^,φ)),λ)|≤2cν(τ)-γ∥xτ(λ,Φ^,φ)∥≤2cK1-2cKM[μ(τ)μ(s)]aν(s)ɛν(τ)-γ∥φ∥.
Hence
(54)∫s∞‍∥T(τ,s)-1Q(τ)∥·|f(τ,xτ(λ,Φ^,φ),Φλ(τ,xτ(λ,Φ^,φ)),λ)|dτ≤2cK∥φ∥∫s∞‍(μ(τ)μ(s))a-bν(τ)ɛ-γν(s)ɛdτ≤2cKN∥φ∥<∞.
Therefore, the right-hand of (52) is well defined. We assume that (51) holds for each s≥0,φ∈E(s), and t≥s; then it follows that
(55)Φλ(s,φ)=T¯Q(t,s)-1Q(t)Φλ(t,xt(λ,Φ^,φ))IIIIIIIIIIIII-∫st‍T¯Q(τ,s)-1Q(τ)S0f(τ,xτ(λ,Φ^,φ),wwwwwwwwwwwwwwwwwwwwiΦλ(τ,xτ(λ,Φ^,φ)),λ)dτ.
By (24), (26), and Lemma 5, we have
(56)∥T¯Q(t,s)-1Q(t)Φλ(t,xt(λ,Φ^,φ))∥≤K21-2cKM[μ(t)μ(s)]a-bν(t)ɛν(s)ɛ∥φ∥.
Letting t→∞ in (55), it follows from (H7) that for each s≥0,φ∈E(s), and t≥s, (52) holds.
If (52) holds for every s≥0,φ∈E(s), and t≥s, we have
(57)T¯(t,s)Φλ(s,φ)+∫st‍T¯Q(t,τ)Q(τ)S0f(τ,xτ(λ,Φ^,φ),wwwwwwwwwwwwwwwwiΦλ(τ,xτ(λ,Φ^,φ)),λ)dτ=-∫t∞‍T¯Q(τ,t)-1Q(τ)S0f(τ,xτ(λ,Φ^,φ),wwwwwwwwwwwwwwwwwiΦλ(τ,xτ(λ,Φ^,φ)),λ)dτ.
Replacing(s,φ) by (t,xt(λ,Φ^,φ)) in (52), we have
(58)Φλ(t,xt(λ,Φ^,φ))=-∫t∞‍T¯Q(τ,t)-1Q(τ)S0f(τ,xτ(λ,Φ^,φ),iiiiiiiiiiiiwwwwwwwwwwwwwwΦλ(τ,xτ(λ,Φ^,φ)),λ)dτ.
From (57) and (58), we see that (51) holds for (s,φ)∈R+×E(s).Lemma 9.
For eachλ∈Y, there exists a unique function Φ^=Φλ∈B such that (52) holds for every s≥0, φ∈E(s).Proof.
We consider the operatorJ^λ defined for each Φ∈B by
(59)(J^λΦλ)(s,φ)=-∫s∞‍TQ(τ,s)-1Q(τ)S0f(τ,xτ(λ,Φ^,φ),iiiiiiiiiiiiiiiiiiiiiiiiiiiwwwwwiiiiiΦλ(τ,xτ(λ,Φ^,φ)),λ)dτ,
for every s≥0,φ∈E(s). When φ=0, we have xτ(λ,Φ^,φ)=0; then (J^λΦλ)(s,0)=0. By (24), (46), and Lemma 6, we have
(60)∥(J^λΦλ)(s,φ)-(J^λΦλ)(s,ψ)∥≤∫s∞‍∥T¯(τ,s)-1Q(τ)∥iiiiiiiiiiii·|f(τ,xτ(λ,Φ^,φ),Φλ(τ,xτ(λ,Φ^,φ)),λ)iiiiiiiiiiiiiiii-f(τ,xτ(λ,Φ^,ψ),Φλ(τ,xτ(λ,Φ^,ψ)),λ)|dτ≤2cK21-2cKM∥φ-ψ∥×∫s∞‍[μ(τ)μ(s)]a-bν(s)ɛν(τ)ɛ-γdτ≤2cK2N1-2cKM∥φ-ψ∥.
By (H8), for each s≥0,φ,ψ∈E(s), we can conclude that
(61)∥(JλΦλ)(s,φ)-(JλΦλ)(s,ψ)∥≤∥φ-ψ∥,
which implies that J^λ(B)⊂B.
The next step is done thatJ^λis a contraction. Given φ∈E(s), for each Φ^=Φλ,Ψ^=Ψλ, and Φλ,Ψλ∈B, by (49), Lemmas 5, and 7, we have
(62)∥(J^λΦλ)(s,φ)-(J^λΨλ)(s,φ)∥≤∫s∞‍∥T¯Q(τ,s)-1Q(τ)∥iiiiiiiiiiiii·|f(τ,xτ(λ,Φ^,φ),Φλ(τ,xτ(λ,Φ^,φ)),λ)iiiiiiiiiiiiiiii-f(τ,xτ(λ,Ψ^,φ)(τ,ξ),Ψλ(τ,xτ(λ,Ψ^,φ)),λ)|dτ≤[2cK3M(1-2cKM)2+cK21-2cKM]∥φ∥·∥Φ^-Ψ^∥∫s∞‍[μ(τ)μ(s)]a-bν(s)ɛν(τ)ɛ-γdτ≤[2cK3M(1-2cKM)2+cK21-2cKM]N∥φ∥·∥Φ^-Ψ^∥.
By (H8) and (H9), we see that the operator J^λ:B→B becomes a contraction. Therefore, given λ∈Y, for each φ∈E(s), there exists a unique function Φ^=Φλ satisfying (51). By (26) and Lemma 6, we obtain
(63)∥Ωϵλ(s,φ,Φλ(s,φ))-Ωϵ(s,ψ,Φλ(s,ψ))∥=∥(t,xt(λ,Φ^,φ),Φλ(t,xt(λ,Φ^,φ)))iiiiiiii-(t,xt(λ,Φ^,ψ),Φλ(t,xt(λ,Φ^,ψ)))∥≤2∥xt(λ,Φ^,φ)-xt(λ,Φ^,ψ)∥≤2K1-2ckM[μ(t)μ(s)]aν(s)ɛ∥φ-ψ∥.
Finally, we turn to the proof of inequality (35) of the theorem. The following lemma is necessary for our establishing (35).Lemma 10.
Givenφ∈E(s), for each λ,η∈Y, one has
(64)∥xt(λ,Φλ,φ)-xt(η,Φη,φ)∥≤3cK2M(1-2cKM)2[μ(t)μ(s)]aν(s)ɛ∥φ∥·|λ-η|,
for every s≥0,φ∈E(s).Proof.
By (26), (28), (H2), and Lemma 5, we have
(65)q3(τ):=|f(τ,xτ(λ,Φλ,φ),Φλ(τ,xτ(λ,Φλ,φ)),λ)iiiiiiiiii-f(τ,xτ(η,Φη,φ),Φη(τ,xτ(η,Φη,φ)),η)|≤|f(τ,xτ(λ,Φλ,φ),Φλ(τ,xτ(λ,Φλ,φ)),λ)iiiiiiiiii-f(τ,xτ(λ,Φλ,φ),Φλ(τ,xτ(λ,Φλ,φ)),η)|+|f(τ,xτ(λ,Φλ,φ),Φλ(τ,xτ(λ,Φλ,φ)),η)iiiiiiiiiii-f(τ,xτ(η,Φη,φ),Φη(τ,xτ(η,Φη,φ)),η)|≤cν(τ)-γ[2∥xτ(λ,Φλ,φ)∥·|λ-η|iiiiiiiiiiiiiiiiii+∥xτ(λ,Φλ,φ)-xτ(η,Φη,φ)∥iiiiiiiiiiiiiiiiii+∥Φλ(τ,xτ(λ,Φλ,φ))-Φη(τ,xτ(η,Φη,φ))∥]≤cν(τ)-γ[3∥xτ(λ,Φλ,φ)∥·∥φ∥·|λ-η|iiiiiiiiiiiiiiiiii+2∥xτ(λ,Φλ,φ)-xτ(η,Φη,φ)∥]≤cν(τ)-γ{3K1-2cKM[μ(τ)μ(s)]aν(s)ɛ∥φ∥iiiiiiiiiiiiiiiiiii[μ(τ)μ(s)]a·|λ-η|+2∥xτ(λ,Φλ,φ)-xτ(η,Φη,φ)∥}.
Thus, it follows from (24) and (65) that
(66)∥xτ(λ,Φλ,φ)-xτ(η,Φη,φ)∥≤∫st‍∥T¯(t,τ)P(τ)∥q3(τ)dτ≤3cK21-2cKM[μ(t)μ(s)]aν(s)ɛ∥φ∥·|λ-η|∫st‍ν(τ)ɛ-γdτ+2cK[μ(t)μ(s)]aν(s)ɛ∥φ∥·∥x(λ,Φλ,φ)-x(η,Φη,φ)∥′∫st‍ν(τ)ɛ-γdτ≤3cK2M1-2cKM[μ(t)μ(s)]aν(s)ɛ∥φ∥·|λ-η|+2cKM[μ(t)μ(s)]aν(s)ɛ∥φ∥·∥x(λ,Φλ,φ)-x(η,Φη,φ)∥′.
This proves that inequality (64) holds.
Therefore, by (28) and Lemma 10, we get
(67)∥Ωϵλ(s,φ,Φλ(s,φ))-Ωϵη(s,φ,Φη(s,φ))∥=∥(t,xt(λ,Φλ,φ),Φλ(t,xt(λ,Φλ,φ)))iiiiiiiiii-(t,xt(η,Φη,φ),Φλ(t,xt(λ,Φη,φ)))∥≤2∥xt(λ,Φλ,φ)-xt(η,Φη,φ)∥+∥xt(λ,Φλ,φ)∥·|λ-η|≤[6cK2M(1-2cKM)2+K1-2cKM][μ(t)μ(s)]aν(s)ɛ∥φ∥·|λ-η|.
This completes the proof of Theorem 4.


## 5. Example

In this section, we provide an example to demonstrate the derived results. Consider the delay system(68)x′=[-43+2t3(1+t2)iiiiiii×(log⁡⁡(1+t2)cos⁡⁡(log⁡⁡(1+t2))-1)2t3(1+t2)]xiiiiiii+c^(1+t2)-γ(λ1+sint)x(t-1),y′=[53+2t3(1+t2)iiiiiii×(log⁡(1+t2)cos⁡(log⁡(1+t2))-1)2t3(1+t2)]yiiiiiii+c^(1+t2)-γ(λ2+cos⁡t)x(t-1),
where B=C([-1,0],R2) and λ=(λ1,λ2)=[-1,1]×[-1,1]. Then, we have
(69)f(t,φ,λ)=(c^(1+t2)-γ(λ1+sint)00c^(1+t2)-γ(λ2+cos⁡t))×φ(-1).
Setting P(t)(x,y)=x and Q(t)(x,y)=y, we have (70)T(t,s)P(s)=(etes)-4/3e(1/3)[log⁡⁡(1+t2)sin(log⁡(1+t2)-1)+cos⁡(log⁡(1+t2))-cos⁡⁡(log⁡⁡(1+s2))-log⁡⁡(1+s2)(sin⁡(log⁡(1+s2))-1)],T(t,s)Q(s)=(etes)5/3e(1/3)[log⁡⁡(1+t2)sin(log⁡(1+t2)-1)+cos⁡(log⁡(1+t2))-cos⁡⁡(log⁡⁡(1+s2))-log⁡⁡(1+s2)(sin⁡(log⁡⁡(1+s2))-1)],which implies that
(71)∥T(t,s)P(s)∥≤e2/3(etes)-4/3(1+s2)2/3,∥T(t,s)-1Q(s)∥≤e2/3(etes)-5/3(1+t2)2/3.
This shows that the linear part of (68) admits a nonuniform (μ,ν)-dichotomy with
(72)μ(t)=et,ν(t)=1+t2,K=e2/3,a=-43,b=53,ɛ=23.
Furthermore, we see that f(t,0,λ)=0 and
(73)|f(t,φ,λ)-f(t,ψ,λ)|≤2c^(1+t2)-γ∥φ-ψ∥,|f(t,φ,λ)-f(t,φ,η)|≤c^(1+t2)-γ∥φ∥·|λ-η|.
Taking suitable γ>0 and sufficiently small c=2c^, we conclude that (68) has a stable invariant manifold provided that (H1)–(H3) and (H5)–(H9) hold.



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*Source: 989526-2014-10-27.xml*