Periodic Solutions for Generalized Higher-Order Neutral Differential Equations

Volume 2011, Article ID 635767,21pages doi:10.1155/2011/635767

Research Article

Existence and Lyapunov Stability of

Periodic Solutions for Generalized Higher-Order Neutral Differential Equations

Jingli Ren,

Wing-Sum Cheung,

and Zhibo Cheng

1Department of Mathematics, Zhengzhou University, Zhengzhou 450001, China

2Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong

Correspondence should be addressed to Wing-Sum Cheung,wscheung@hku.hk Received 17 May 2010; Accepted 23 June 2010

Academic Editor: Feliz Manuel Minh´os

Copyrightq2011 Jingli Ren et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Existence and Lyapunov stability of periodic solutions for a generalized higher-order neutral differential equation are established.

1. Introduction

In recent years, there is a good amount of work on periodic solutions for neutral differential equations see1–11and the references cited therein. For example, the following neutral differential equations

d

duut−kut−τ g1ut g2ut−τ1 pt, xt cxt−rf

xt

gxt−τt pt, xt−cxt−σⁿfxtxt g

₀

−rxtsdαs

pt

1.1

have been studied in1,3,8, respectively, and existence criteria of periodic solutions were established for these equations. Afterwards, along with intensive research on thep-Laplacian,

some authors4,11start to consider the followingp-Laplacian neutral functional differential equations:

φpxt−cxt−σ

gt, xt−τt pt, φ_p

xt−cxt−σ f

xt

gxt−τt et, 1.2 and by using topological degree theory and some analysis skills, existence results of periodic solutions for1.2have been presented.

In general, most of the existing results are concentrated on lower-order neutral functional differential equations, while studies on higher-order neutral functional differential equations are rather infrequent, especially on higher-order p-Laplacian neutral functional differential equations. In this paper, we consider the following generalized higher-order neutral functional differential equation:

ϕpxt−cxt−σ^l_n−l F

t, xt, xt, . . . , x^l−1t

, 1.3

whereϕp:R → Ris given byϕps |s|^p−2swithp ≥2 being a constant,Fis a continuous function defined onR^land is periodic with respect totwith periodT, that is,Ft,·, . . . ,· FtT,·, . . . ,·, Ft, a,0, . . . ,0/≡0 for alla∈R, andc,σare constants.

Since the neutral operator is divided into two cases |c|/1 and |c| 1, it is natural to study the neutral differential equation separately according to these two cases. The case

|c| 1 has been studied in 5. Now we consider1.3 for the case|c|/1. So throughout this paper, we always assume that |c|/1, and the paper is organized as follows. We first transform 1.3 into a system of first-order differential equations, and then by applying Mawhin’s continuation theory and some new inequalities, we obtain sufficient conditions for the existence of periodic solutions for1.3. The Lyapunov stability of periodic solutions for the equation will then be established. Finally, an example is given to illustrate our results.

2. Preparation

First, we recall two lemmas. LetXandYbe real Banach spaces and letL:DL⊂X → Ybe a Fredholm operator with index zero; hereDLdenotes the domain ofL. This means that ImL is closed inY and dim KerLdimY/ImL<∞. Consider supplementary subspacesX₁, Y₁ ofX,Y, respectively, such thatX KerL⊕X₁,Y ImL⊕Y₁. LetP : X → KerLand Q : Y → Y1 denote the natural projections. Clearly, KerL∩DL∩X1 {0}and so the restrictionL_P :L|_DL∩X1is invertible. LetKdenote the inverse ofL_P.

LetΩbe an open bounded subset ofXwithDL∩Ω/∅. A mapN:Ω → Y is said to beL-compact inΩifQNΩis bounded and the operatorKI−QN:Ω → Xis compact.

Lemma 2.1see12. Suppose thatX andY are two Banach spaces, and suppose thatL:DL⊂ X → Yis a Fredholm operator with index zero. LetΩ⊂Xbe an open bounded set and letN:Ω → Y beL-compact onΩ. Assume that the following conditions hold:

1Lx /λNx, for allx∈∂Ω∩DL, λ∈0,1, 2Nx /∈Im L, for allx∈∂Ω∩KerL,

3deg{JQN,Ω∩KerL,0}/0, whereJ: ImQ → KerLis an isomorphism.

Then, the equationLxNxhas a solution inΩ∩DL.

Lemma 2.2see13. Ifω∈C¹R,Randω0 ωT 0, then _T

0

|ωt|^pdt≤ T

πp

_pT

0

ωt ^pdt, 2.1

wherepis a fixed real number withp >1 and

πp2

_p−1/p

0

ds 1−s^p/

p−1_1/p 2π

p−1_1/p psin

π/p . 2.2

For the sake of convenience, throughout this paper we denote byT a positive real number, and for any continuous functionu, we write

|u|₀: max

t∈0,T|ut|. 2.3

LetA:C_T → C_Tbe the operator onC_T :{x∈CR,R:xtT xtfor allt∈R}

given by

Axt:xt−cxt−σ, ∀x∈C_T, t∈R. 2.4 Lemma 2.3. The operatorAhas a continuous inverseA⁻¹onCTsatisfying the following:

1 A⁻¹f

t

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

ft ^∞

j1

c^jf t−jσ

, for|c|<1, ∀f∈C_T,

−ftσ

c −^∞

j1

1 c^j1f

t j1

σ

, for|c|>1, ∀f∈C_T,

2.5

2 A⁻¹f

t ≤ f

0

|1− |c||, ∀f∈C_T, 2.6

3 _T

0

A⁻¹f

t dt≤ 1

|1− |c||

_T

0

ft dt, ∀f ∈CT. 2.7

Remark 2.4. This lemma is basically proved in3,10. For the convenience of the readers, we present a detailed proof here as follows.

Proof. We split it into the following two cases.

Case 1|c|<1. Define an operatorB:CT → CTby

Bxt:cxt−σ, ∀x∈CT, t∈R. 2.8

Clearly,B^jxt c^jxt−jσandA I−B. Note also thatB< |c|<1. Therefore,Ahas a continuous inverseA⁻¹:C_T → C_TwithA⁻¹ I−B⁻¹_∞

j0B^j; hereB⁰xt:xt. Hence, A⁻¹f

t ^∞

j0

B^jf

t ft ^∞

j1

c^jf t−jσ

, 2.9

and so

A⁻¹ft

∞ j0

B^jf

t

∞ j0

c^jf t−jσ

≤ f

0

1− |c| , _T

0

A⁻¹f

t dt≤^∞

j0

_T

0

B^jf

t dt^∞

j0

_T

0

c^jf

t−jσ dt≤ 1 1− |c|

_T

0

ft dt.

2.10

Case 2|c|>1. Define operators

E:CT−→CT, Ext:xt−1

cxtσ, B₁:C_T−→C_T, B1xt: 1

cxtσ.

2.11

From the definition of the linear operatorB1, we have

B₁^jf t 1

c^jf tjσ

, ∞

j0

B₁^jf

t ft ^∞

j1

1 c^jf

tjσ .

2.12

SinceB1<1, the operatorEhas a bounded inverseE⁻¹:C_T → C_T with

E⁻¹ I−B1⁻¹I^∞

j1

B₁^j, 2.13

and so, for anyf∈C_T,

E⁻¹f

t ft ^∞

j1

B^j₁f

t. 2.14

On the other hand, fromAxt xt−cxt−σ, we have Axt xt−cxt−σ −c

xt−σ−1 cxt

. 2.15

That is,

Axt −cExt−σ. 2.16

Now, for anyf∈C_T, ifxtsatisfies

Axt ft, 2.17

then we have

−cExt−σ ft, 2.18

or

Ext −ftσ

c f₁t. 2.19

So, we have

xt

E⁻¹f1

t f1t ^∞

j1

B₁^jf1t −ftσ

c −^∞

j1

B₁^jftσ

c . 2.20

So,A⁻¹exists and satisfies

A⁻¹f

t −ftσ

c −^∞

j1

B₁^jftσ

c −ftσ

c −^∞

j1

1 c^j1f

t j1

σ ,

A⁻¹f t

−ftσ

c −^∞

j1

1 c^j1f

t j1

σ ≤ f

0

|c| −1.

2.21

This proves1and2of Lemma2.3. Finally,3is easily verified.

By Hale’s terminology14, a solution xtof 1.3is thatxt ∈ C¹R,Rsuch that Ax ∈ C¹R,Rand 1.3is satisfied onR. In general,xtdoes not belong toC¹R,R.But we can see easily fromAxt Axtthat a solutionxtof1.3must belong toC¹R,R.

Equation1.3is transformed into

ϕp

Ax^l

t_n−l F

t, xt, xt, . . . , x^l−1t

. 2.22

Lemma 2.5see4. Ifp >1, then _T

0

A⁻¹f

t ^pdt≤ 1

|1− |c||^p _T

0

ft ^pdt, ∀f∈C_T. 2.23

Now we consider 2.22. Define the conjugate index q ∈ 1,2 by 1/p1/q 1.

Introducing new variables

y₁t xt, y₂t xt, y₃t xt, . . . , ylt x^l−1t, y_l1t ϕ_p

Ax^lt

, y_l2t

ϕ_p

Ax^lt

, . . . , y_nt ϕ_p

Ax^lt_n−l−1 . 2.24

Using the fact thatϕ_q◦ϕ_p≡id and by Lemma2.3,1.3can be rewritten as

y₁t y₂t, y₂t y3t,

... y_l−1t y_lt, y_lt A⁻¹ϕq

y_l1t , y_l1 t y_l2t,

... y_n−1t ynt, y_nt F

t, y₁t, y2t, . . . , ylt .

2.25

It is clear that, ifyt y1t, y2t, . . . , ynt is aT-periodic solution to2.25, theny1t must be aT-periodic solution to1.3. Thus, the problem of finding aT-periodic solution for 1.3reduces to finding one for2.25.

Define the linear spaces

X Y

y

y₁·, y2·, . . . , yn·

∈C⁰R,Rⁿ:ytT≡yt

2.26

with normymax{y1,y2, . . . ,yn}. Obviously,XandY are Banach spaces. Define

L:DL

y∈C¹R,Rⁿ:ytT yt

⊂X−→Y 2.27

by

Lyy

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎝ y₁ y₂ ... y_l

... y_n

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎠

. 2.28

Moreover, define

N:X−→Y 2.29

by

Ny

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎝

y2t y₃t

... A⁻¹ϕ_q

y_l1t y_l2t

... F

t, y₁t, y2t, . . . , ylt

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎠

. 2.30

Then, 2.25can be rewritten as the abstract equationLy Ny. From the definition ofL, one can easily see that KerL {y ∈ C¹R,Rⁿ : yis constant} Rⁿ and ImL {y : y ∈ X,_T

0 ysds 0}. So,Lis a Fredholm operator with index zero. Let P : X → KerLand Q:Y → ImQbe defined by

P y 1 T

_T

0

ysds, Qy 1

T _T

0

ysds. 2.31

It is easy to see that KerL ImQRⁿ. Moreover, for ally∈Y, if we writey^∗ y−Qy, we have_T

0 y^∗sds0 and soy^∗∈ImL. This is to sayY ImQ⊕ImLand dimY/ImL dim ImQ dim KerL. So,L is a Fredholm operator with index zero. Let K denote the inverse ofL|Kerp∩DL, then we have

Ky t

T 0

G1t, sy1sds, _T

0

G2t, sy2sds, . . . , _T

0

Gnt, synsds

, 2.32

where

Git, s

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ s

T, 0≤s < t≤T, s−T

T , 0≤t≤s≤T,

i1,2, . . . , n. 2.33

From2.30and2.33, it is clear thatQN andKI−QN are continuous, andQNΩis bounded, and soKI−QNΩis compact for any open boundedΩ ⊂ X. Hence,N isL- compact onΩ. For the functionyt y1t, y2t, . . . , yntdefined as2.24, we have the following.

Lemma 2.6. Ifyt∈C¹R,RⁿandytT yt, then _T

0

y_it ^pdt≤ 1

|1−|c^p T

πp

_pl−i T πq

_qn−lT

0

y_nt ^qdt, 2.34

where 1/p1/q1, p≥2, i1,2, . . . , l−1.

Proof. Fromy₁0 y₁T, there is a pointt₁∈0, Tsuch thaty₁t1 0. Letω₁t y₁tt₁. Then,ω₁0 ω₁T 0. Fromy₂0 y₂T, there is a pointt₂ ∈0, Tsuch thaty₂t2 0.

Letω2t y₂tt2. Then,ω20 ω2T 0.Continuing this way, we get fromy_l−10 y_l−1Ta pointt_l−1 ∈0, Tsuch thaty_l−1t_l−1 0. Letω_l−1t y_l−1 tt_l−1. Then,ω_l−10 ω_l−1T 0.Fromy_lt y_ltT, we have_T

0Ay_ltdt _T

0 Ayltdt Aylt|^T₀ 0, so there is a pointtl ∈ 0, T such thatAy_ltl 0; hence, we haveϕpAy_ltl 0. Let ω_lt ϕ_pAy_ltt_l y_l1tt_l. Then,ω_l0 ω_lT 0. Continuing this way, we get fromy_n−10 y_n−1T that there is a pointt_n−1 ∈ 0, Tsuch that y_n−1tn−1 0. Let ω_n−1t y_n−1tt_n−1. Then,ω_n−10 ω_n−1T 0.By Lemma2.2, we have

_T

0

y₁t ^pdt _T

0

|ω1t|^pdt

≤ T

π_p p_T

0

ω₁t ^pdt

T π_p

_pT

0

y₂t ^pdt

T πp

_pT

0

|ω2t|^pdt

≤ T

π_{p 2pT}

0

ω₂t ^pdt ...

≤ T

πp

_pl−1T

0

ω_l−1t ^pdt

T π_p

_pl−1T

0

y_lt ^pdt.

2.35

By Lemma2.5and Lemma2.2, we have

_T

0

y_lt ^pdt _T

0

A⁻¹ϕq

y_l1t ^pdt

≤ 1

|1− |c||^p _T

0

ϕ_q

y_l1t ^pdt 1

|1− |c||^p _T

0

y_l1t ^pq−pdt 1

|1− |c||^p _T

0

y_l1t ^qdt 1

|1− |c||^p _T

0

|ωlt|^qdt

≤ 1

|1− |c||^p T

πq

_qT

0

ω_lt ^qdt 1

|1− |c||^p T

π_q q_T

0

y_l2t ^qdt ...

≤ 1

|1− |c||^p T

πq

_qn−lT

0

ω_n−1t ^qdt 1

|1− |c||^p T

π_q

_qn−lT

0

y_nt ^qdt.

2.36

Combining2.35and2.36, we get

_T

0

y₁t ^pdt≤ 1

|1− |c||^p T

π_p

_pl−1 T π_q

_qn−lT

0

y_nt ^qdt. 2.37

Similarly, we get _T

0

y_it ^pdt≤ 1

|1− |c||^p T

π_p

_pl−i T π_q

_qn−lT

0

y_nt ^qdt. 2.38

This completes the proof of Lemma2.6.

Remark 2.7. In particular, if we takep2, thenq2 and

π_pπ_qπ₂2 _2−1/2

0

ds

1−s²/2−1^1/2 2π2−1^1/2

2 sinπ/2 π. 2.39

In this case,2.34is transformed into _T

0

y_it ²dt≤ 1

|1−|c² T

π

2n−i_T

0

y_nt ²dt. 2.40

3. Main Results

For the sake of convenience, we list the following assumptions which will be used repeatedly in the sequel.

H1There exists a constantD >0 such that

z1Ft, z1, z2, . . . , zl>0, ∀t, z1, z2, . . . , zl∈0, T×R^l, with|z1|> D. 3.1

H2There exists a constantM >0 such that

|Ft, z1, z2, . . . , zl| ≤M, ∀t, z1, z2, . . . , zl∈0, T×R^l. 3.2

H3There exist nonnegative constantsα₁, α₂, . . . , α_l, msuch that

|Ft, z1, z₂, . . . , z_l| ≤α₁|z1|α₂|z2|· · ·α_l|zl|m, ∀t, z1, z₂, . . . , z_l∈0, T×R^l. 3.3

H4There exist nonnegative constantsγ1, γ2, . . . , γnsuch that

|Ft, u1, u2, . . . , un−Ft, v1, v2, . . . , vn| ≤γ1|u1−v1|γ2|u2−v2|· · ·γn|un−vn| 3.4

for allt, u1, u2, . . . , un,t, v1, v2, . . . , vn∈0, T×Rⁿ.

Theorem 3.1. IfH1andH2hold, then1.3has at least one nonconstantT-periodic solution.

Proof. Consider the equation

LyλNy, λ∈0,1. 3.5

LetΩ1{y∈DL:LyλNy, λ∈0,1}. Ifyt y1t, y2t, . . . , ynt∈Ω1, then

y₁t λy2t, y₂t λy₃t,

... y_l−1t λylt,

y_lt λϕ_q y_l1t

, y_l1t λy_l2t,

...

y_n−1t λy_nt, y_nt λF

t, y₁t, y2t, . . . , ylt .

3.6

We first claim that there exists a constantξ∈Rsuch that

y₁ξ ≤D. 3.7

Integrating the last equation of3.6over0, T, we have _T

0

F

t, y₁t, y2t, . . . , ylt

dt0. 3.8

By the continuity ofF, there existsξ∈0, Tsuch that

F

ξ, y₁ξ, . . . , ylξ

0. 3.9

From assumptionH1, we get3.7. As a consequence, we have

y1t y1ξ

_t

ξ

y₁sds ≤D

_T

0

y₁s ds. 3.10

On the other hand, multiplying both sides of the last equation of3.6byy_ntand integrating over0, T, using assumptionH2,we have

_T

0

y_nt ²dtλ _T

0

F

t, y₁t, y2t, . . . , ylt y_ntdt

≤ _T

0

F

t, y₁t, y2t, . . . , ylt y_nt dt

≤M _T

0

y_nt dt

≤MT^1/2 _T

0

y_nt ²dt _1/2

.

3.11

It is easy to see that there exists a constantM_n>0independent ofλsuch that

_T

0

y_nt ²dt≤M_n. 3.12

From y_n−10 y_n−1T, there exists a pointt1 ∈ 0, T such thatynt1 0. By H ¨older’s inequality, we have

ynt ≤ _T

0

y_nt dt≤T^1/2 _T

0

y_nt ²dt _1/2

≤T^1/2M_n^1/2 :Mn. 3.13

Fromy_n−20 y_n−2T, there exists a pointt₂∈0, Tsuch thaty_n−1t2 0, and we have

y_n−1t ≤ _T

0

y_n−1t dt _T

0

λy_nt dt≤ _T

0

y_nt dt≤TM_n:M_n−1. 3.14

Continuing this way fory_n−2, . . . , y_l1, we get

y_l1t ≤TM_l2:M_l1. 3.15

Hence,

ylt ≤ _T

0

y_lt dt≤ _T

0

λA⁻¹ϕq

y_l1t dt≤ 1

|1− |c||

_T

0

y_l1t ^q−1dt

≤ 1

|1− |c||TM^q−1_l1 :Ml, y_l−1t ≤TMl:M_l−1,

...

y₂t ≤TM₃:M₂.

3.16

Meanwhile, from3.10, we get y₁t ≤D

_T

0

y₁t dt≤DTM₂:M₁. 3.17

LetM₀max{M1, M₂, . . . , M_n}. Then, obviouslyy1 ≤M₀,y2 ≤M₀, . . . ,andyn ≤M₀. LetΩ2 {y ∈ KerL : Ny ∈ ImL}. Ify ∈ Ω2, then y ∈ KerL, which means that yconstant andQNy0. We see that

y₂0, y₃0,

... y_n0, F

t, y₁,0, . . . ,0 0.

3.18

So,

y1 ≤D≤M0, y2y3· · ·yn0≤M0. 3.19

Now takeΩ {y y1, y₂, . . . , y_n ∈X :y1< M₀1, y2 < M₀1, . . . , yn <

M₀1}. By the analysis above, it is easy to see thatΩ1 ⊂Ω,Ω2⊂Ω, and conditions1and 2of Lemma2.1are satisfied.

Next we show that condition3of Lemma2.1is also satisfied. Define an isomorphism J: ImQ → KerLas follows:

J

y1, y2, . . . , yn

:

yn, y1, . . . , y_n−1

. 3.20

LetHμ, y μy 1−μJQNy,μ, y∈0,1×Ω. Then, for allμ, y∈0,1×∂Ω∩KerL,

H μ, y

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎝

μy₁1−μ T

_T

0

F

t, y₁,0, . . . ,0 dt y₂

... A⁻¹ϕ_q

y_l1 ... y_n

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎠

. 3.21

FromH1, it is obvious thatyHμ, y>0 for allμ, y∈0,1×∂Ω∩KerL. Therefore,

deg{JQN,Ω∩KerL,0}deg! H

0, y

,Ω∩KerL,0"

deg! H

1, y

,Ω∩KerL,0"

deg{I,Ω∩KerL,0}/0,

3.22

which means that condition3of Lemma2.1is also satisfied. By applying Lemma2.1, we conclude that equationLyNyhas a solutionyt^∗ y₁^∗t, y₂^∗t, . . . , y^∗_ntonΩ; that is, 1.3has aT-periodic solutiony₁^∗twithy₁^∗< M₀1.

Finally, observe thaty^∗₁tis not constant. For, ify₁^∗≡aconstant, then from1.3we haveFt, a,0, . . . ,0≡0, which contradicts the assumption thatFt, a,0, . . . ,0/≡0. The proof is complete.

Theorem 3.2. IfH1andH3hold, then1.3has at least one nonconstant T-periodic solution if one of the following conditions holds:

1p >2,

2p2 and 1/|1− |c||α1Tα₂T/π^l−1α₃T/π^l−2· · ·α_lT/πT/π^n−l<1.

Proof. LetΩ1be defined as in Theorem3.1. Ifyt y1t, y2t, . . . , ynt ∈Ω1,then from the proof of Theorem3.1we have

y_nt λF

t, y1t, y2t, . . . , ylt

, 3.23

y1

0≤D _T

0

y₁s ds. 3.24 We claim that|yn|0is bounded.

Multiplying both sides of 3.23 byϕqy_nt and integrating over 0, T, by using assumptionH3,we have

_T

0

y_nt ^qdtλ _T

0

F

t, y₁t, y2t, . . . , ylt ϕ_q

y_nt dt

≤ _T

0

F

t, y₁t, y2t, . . . , ylt ϕ_q

y_nt dt

≤α1

_T

0

y1t ϕq

y_nt dtα2

_T

0

y2t ϕq

y_nt dt · · ·α_l

_T

0

y_lt ϕ_q

y_nt dtm _T

0

ϕ_q

y_nt dt

≤α1 y1

0

_T

0

ϕq

y_nt dtα2

_T

0

y2t ϕq

y_nt dt · · ·α_l

_T

0

y_lt ϕ_q

y_nt dtm _T

0

ϕ_q

y_nt dt

≤α1

D

_T

0

y₁t dt

T 0

ϕq

y_nt dtα2

_T

0

y2t ϕq

y_nt dt · · ·α_l

_T

0

y_lt ϕ_q

y_nt dtm _T

0

ϕ_q

y_nt dt.

3.25

Applying H ¨older’s inequality, we have _T

0

y_nt ^qdt≤α₁

⎡

⎣DT^1/p _T

0

y₁t ^qdt _1/q⎤

⎦T^1/q _T

0

ϕ_q

y_nt ^pdt _1/p

α_{2 T}

0

y₂t ^qdt

_1/qT

0

ϕ_q

y_nt ^pdt _1/p

· · ·α_{l T}

0

y_lt ^qdt

1/q_T

0

ϕ_q

y_nt ^pdt 1/p

mT^1/q _T

0

ϕq

y_nt ^pdt _1/p

≤α₁T·T^p−2/qp−1 _T

0

y₁t ^pdt

_1/qp−1T

0

y_nt ^qdt _1/p

α2T^p−2/qp−1 T

0

y2t ^pdt

_1/qp−1T

0

y_nt ^qdt 1/p

· · ·

αlT^p−2/qp−1 _T

0

ylt ^pdt

_1/qp−1T

0

|y_nt|^qdt _1/p

α₁DmT^1/q _T

0

|y_nt|^qdt 1/p

α1T·T^p−2/p _T

0

y₁t ^pdt

_1/pT

0

y_nt ^qdt _1/p

α₂T^p−2/p _T

0

y₂t ^pdt

_1/pT

0

y_nt ^qdt _1/p

· · ·αlT^p−2/p T

0

ylt ^pdt

1/pT 0

y_nt ^qdt 1/p

α1DmT^1/q _T

0

y_nt ^qdt _1/p

.

3.26

Applying Lemma2.6and3.26, we have

_T

0

y_nt ^qdt≤α1T ·T^p−2/p 1

|1− |c||

T π_p

_l−1 T π_q

_qn−l/pT

0

y_nt ^qdt 2/p

α2T^p−2/p 1

|1− |c||

T πp

_l−1 T πq

_qn−l/pT

0

y_nt ^qdt _2/p

· · ·α_lT^p−2/p 1

|1− |c||

T π_p

T π_q

_qn−l/pT

0

y_nt ^qdt 2/p

α1DmT^1/q _T

0

y_nt ^qdt _1/p

≤ 1

|1− |c||

⎡

⎣α1Tα₂ T

π_{p l−1}

α₃ T

π_{p l−2}

· · ·α_l T

π_p

⎤

⎦

×T^p−2/p T

πq

_qn−l/p

· _T

0

y_nt ^qdt _2/p

α1DmT^1/q T

0

y_nt ^qdt 1/p

.

3.27

Case 1. Ifp2 and 1/|1− |c||α1T α2T/π^l−1α3T/π^l−2· · ·αlT/πT/π^n−l<1, then it is easy to see that there exists a constantM_n>0independent ofλsuch that

_T

0

y_nt ^qdt≤M_n. 3.28

Case 2. Ifp >2, then it is easy to see that there exists a constantM_n > 0independent ofλ such that

_T

0

y_nt ^qdt≤M_n. 3.29

From y_n−10 y_n−1T, there exists a pointt₁ ∈ 0, T such thaty_nt1 0. By H ¨older’s inequality, we have

ynt ≤ _T

0

y_nt dt≤T^1/p _T

0

y_nt ^qdt _1/q

≤T^1/pM_n^1/q :Mn. 3.30

This proves the claim, and the rest of the proof of the theorem is identical to that of Theorem3.1.

Remark 3.3. If1.3takes the form

ϕpxt−cxt−σ^l_n−l F

t, xt, xt, . . . , x^l−1t

et, 3.31

whereet∈ CR,R, etT etand_T

0 etdt 0, then the results of Theorems3.1and 3.2still hold.

Remark 3.4. Ifp2, then1.3is transformed into xt−cxt−σⁿF

t, xt, xt, . . . , xⁿ⁻¹t

, 3.32

and the results of Theorems3.1and3.2still hold.

Next, we study the Lyapunov stability of the periodic solutions of3.32.

Theorem 3.5. Assume thatH4holds. Then everyT-periodic solution of3.32is Lyapunov stable.

Proof. Let

z1t xt, z2t xt, . . . , znt xⁿ⁻¹t. 3.33

Then, system3.32is transformed into

z₁t z2t, z₂t z₃t,

...

z_nt A⁻¹Ft, z1t, z2t, . . . , znt.

3.34

Suppose now thatz^∗t z^∗₁t, z^∗₂t, . . . , z^∗_ntis aT-periodic solution of3.34. Let zt z1t, z2t, . . . , zntbe any arbitrary solution of3.34. For anyk 1, . . . , n, write w_kt z_kt−z^∗_kt. Then, it follows from3.34that

w₁t w2t, w₂t w₃t,

...

w_nt A⁻¹

Ft, z1t, z2t, . . . , znt−F

t, z^∗₁t, z^∗₂t, . . . , z^∗_nt ,

3.35

and so

w₁t |w2t|, w₂t |w3t|,

... w_nt A⁻¹

Ft, z1t, z2t, . . . , znt−F

t, z^∗₁t, z^∗₂t, . . . , z^∗_nt .

3.36

Letu^l_k t |w_k^lt|, l0,1, k1,2, . . . , n. Then, u₁t u₂t, u₂t u₃t,

...

u_nt A⁻¹

Ft, z1t, z2t, . . . , znt−F

t, z^∗₁t, z^∗₂t, . . . , z^∗_nt .

3.37

Takeβmax{γ1/|1− |c||, γ2/|1− |c||1, . . . , γ_n/|1− |c||1}1, and define a functionV·by

Vt, u1, . . . , un:e^−βt n k1

ukt. 3.38

LetUu1, . . . , un _n

k1ykt. It is obvious thatVt, u1, . . . , un > 0 and Vt, u1, . . . , un ≥ Uu1, . . . , u_n>0. FromH4and Lemma2.3, we get

V˙t, u1, . . . , un −βe^−βt _n

k1

ukt

e^−βtu2t · · ·unt e^−βt A⁻¹

Ft, z1t, z2t, . . . , znt−F

t, z^∗₁t, z^∗₂t, . . . , z^∗_nt

≤ −βe^−βt _n

k1

u_kt

e^−βtu2t · · ·u_nt

e^−βt

|1− |c|| Ft, z1t, z2t, . . . , znt−F

t, z^∗₁t, z^∗₂t, . . . , z^∗_nt

≤ −βe^−βt _n

k1

ukt

e^−βtu2t · · ·unt

e^−βt

|1− |c||

γ₁ z₁t−z^∗₁t · · ·γ_n|znt−z^∗_nt|

−βe^−βt _n

k1

u_kt

e^−βtu2t · · ·u_nt

e^−βt

|1− |c||

γ1|w1t|· · ·γn|wnt|

−βe^−βt _n

k1

u_kt

e^−βtu2t · · ·u_nt

e^−βt

|1− |c||

γ1u1t · · ·γnunt

−β γ1

|1− |c|| u1te^−βtn

k2

−β1 γk

|1− |c|| ukte^−βt

<0.

3.39

Hence,Vis a Lyapunov function for nonautonomous3.32 see15, page 50, and so theT-periodic solutionz^∗of3.32is Lyapunov stable.

Finally, we present an example to illustrate our result.

Example 3.6. Consider then-order delay differential equation

ϕpxt−3xt−σ⁴_n−4 1

3πxt 1

8sinxt 1

8cosxtsin 2t1

8sinxt.

3.40