A New Delay Vector Integral Inequality and Its Application

Journal of Inequalities and Applications Volume 2010, Article ID 927059,9pages doi:10.1155/2010/927059

Research Article

A New Delay Vector Integral Inequality and

Its Application

Li Xiang,

_{Lingying Teng,}

_{and Hua Wu}

1_{College of Computer Science, Civil Aviation Flight University of China, Guanghan 618307, China} 2_{College of Computer Science and Technology, Southwest University for Nationalities,}

Chengdu 610041, China

3_{Civil Aviation Flight University of China, Guanghan 618307, China}

Correspondence should be addressed to Li Xiang,xiangni [email protected]

Received 22 September 2010; Accepted 14 December 2010

Academic Editor: Mohamed A. El-Gebeily

Copyrightq2010 Li Xiang 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.

A new delay integral inequality is established. Using this inequality and the properties of nonnegative matrix, the attracting sets for the nonlinear functional differential equations with distributed delays are obtained. Our results can extend and improve earlier publications.

1. Introduction

The attracting set of dynamical systems have been extensively studied over the past few decades and various results are reported. It is well known that one of the most researching tools is inequality technique. With the establishment of various differential and integral inequalities 1–12, the sufficient conditions on the attracting sets for different differential systems are obtained 12–16. However, the inequalities mentioned above are ineffective for studying the attracting sets of a class of nonlinear functional differential equations with distributed delays.

Motivated by the above discussions, in this paper, a new delay vector integral inequality is established. Applying this inequality and the properties of nonnegative matrix, some sufficient conditions ensuring the global attracting set for a class of nonlinear functional differential equations with distributed delays are obtained. The result in16is extended.

2. Preliminaries

In this section, we introduce some notations and recall some basic definitions.

For a nonnegative matrix A ∈ Rn×n_{, let ρAdenote the spectral radius of A. Then}

ρAis an eigenvalue ofAand its eigenspace is denoted by

ΩρAz∈Rn|Az ρAz, 2.1

which includes all positive eigenvectors ofAprovided that the nonnegative matrixAhas at least one positive eigenvectorsee17.

CX, Ydenotes the space of continuous mappings from the topological spaceXto the topological spaceY. Especially, letCC−∞, t0,Rn, wheret0≥0.

Let R 0,∞. For x ∈ Rn_{,A ∈ R}n×n_{,ϕ ∈ C,τt ∈ CR,R}

, we define

x |x1|, |x2|, . . . ,|xn|T, A |aij|n×n, ϕtτt ϕ1tτt,ϕ2tτt, . . ., ϕntτtT,ϕitτt sup_0≤s≤τt|ϕit − s|,i 1,2, . . . , n. For τt ∞, we define

xt_τt xt_∞.

Definition 2.1Xu4. ft, s∈UCtmeans thatf ∈ CR×R,Rand for any givenαand anyε >0 there exist positive numbersB,T, andAsatisfying

t

α

ft, sds≤B, t−T

α

ft, sds < ε, ∀t≥A. 2.2

Especially,f∈UCtifft, s ft−sand

_∞

0 fudu <∞.

Lemma 2.2Lasalle18. IfM≥0andρM<1, thenE−M−1≥0.

3. Delay Integral Inequality

be a solution of the delay integral inequality

yt≤Gt, t0ϕt0 t

α1

Bt, sys_τsds t

α2

Ψt, s s

α3

ζs, vyv_τvdv dsD, t > t0,

3.1

yt≤ϕt, ∀t∈−∞, t0, 3.2

whereGt, t0∈CR×R,Rn×n,ϕt∈C−∞, t0,Rn×1,D d1, . . . , dnT ≥0,limt→∞t− τt ∞,αi∈−∞,0 i 1,2,3. Assume that the following conditions are satisfied:

H1Gt, t0ϕt0 → 0ast → ∞,Bt, u bijt, un×n,Ψt, u ψijt, un×n,ζt, u ζijt, u_n×n,bijt, u,ψijt, u,ζijt, u ∈ UCt, and there exists a nonnegative matrix Π πij_n×nsuch that

_t

α1

Bt, sds _t

α2

Ψt, s _s

α3

ζs, vdv ds≤Π, for∀t≥t0 3.3

If the initial condition satisfies

ϕt< k1z E−Π−1D, t∈−∞, t0, 3.4

wherek1 ≥0,z >0, andz∈ΩρΠ, then there exists a constantk≥k1such that

yt< kz E−Π−1D, fort≥t0. 3.5

Proof. By the conditionΠ ≥ 0 and Lemma 2.2, together with ρΠ < 1, this implies that

E−Π−1exists andE−Π−1≥0.

From limt→∞Gt, t0ϕt0 0, there is aT > t0such that

Gt, t0ϕt0≤

1−ρΠ

2 k1z, fort≥T. 3.6

By the continuity ofyt, together with3.2and3.4, there exists a constantk≥k1such that

yt< kz E−Π−1D, fort∈−∞, T. 3.7

In the following, we will prove that

yt< kz E−Π−1D, fort≥T. 3.8

If this is not true, from3.7and the continuity ofyt, then there must be a constantt1 > T

and some integerisuch that

yit1 ei

kz E−Π−1D , 3.9

yt≤kz E−Π−1D, fort≤t1, 3.10

whereei _{0,0, . . . ,0,1}

i

Using3.1,3.3,3.6,3.10, andρΠ<1, we obtain that

yit1 eiyt1

≤ei

Gt1, t0ϕt0 t1

α1

Bt1, sys_τsds

t1

α2

Ψt1, s s

α3

ζs, vyv_τvdv dsD

≤ei

1−ρΠ

2 k1z

t1

α1

Bt1, skz E−Π−1Dds

t1

α2

Ψt1, s s

α3

ζs, vkz E−Π−1Ddv dsD

≤ei

_1−ρΠ

2 kz Π

kz E−Π−1D E−ΠE−Π−1D

≤ei

_1−ρΠ

2 kzρΠkz Π E−ΠE−Π

−1_D

≤ei

1ρΠ

2 kz E−Π

−1_D

< eikz E−Π−1D .

3.11

This contradicts the equality in3.9, and so3.8holds. The proof is complete.

4. Applications

The delay integral inequality obtained in Section 3 can be widely applied to study the attracting set of the nonlinear functional differential equations. To illustrate the theory, we consider the following differential equation with distributed delays

˙

xt Atxt F

t, xt,

_t

−∞gt, s, xsds

, t≥t0,

xt ϕt, −∞< t≤t0,

4.1

wherex x1, . . . , xnT ∈Rn_{,g ∈CR×R×R}n_,Rn_{,F ∈ CR×R}n_×Rn_,Rn_{,gt, s,0 ≡0,}

assume that for anyϕ ∈C, the system4.1has at least one solution throught0, ϕdenoted byxt, t0, ϕor simplyxtif no confusion should occur.

In order to study the attracting set, we rewrite4.1as

xt Φt, t0ϕt0 _t

t0

Φt, sF

s, xs,

_s

−∞gs, v, xvdv

ds, t≥t0,

xt ϕt, −∞< t≤t0,

4.2

whereΦt, t0be a fundamental matrix of the linear equation ˙xt Atxt. Throughout this section, we suppose the following:

A1 Ft, xt,

_t

−∞gt, s, xsds ≤Btxtτt _t

−∞ζt−sxsτsdsJt, where

Bt ∈CR,Rn×n

,ζt−u ζt, u ∈CR×R,Rn×n,Jt J1t, . . . , JntT ≥0,

Jit∈CR,R,i 1,2, . . . , n,

A2 Φt, sBs wijt, sn×n,Φt, s φijt, sn×n,ζt−s ζijt−sn×n.

wijt, s, φijt, s ∈ UCt.

_∞

0 ζijsds < ∞,i, j 1,2, . . . , n. limt→∞Φt, t0 0.

There are two matricesΠ πijn×n≥0 andD d1, . . . , dnT ≥0 such that

_t

t0

Φt, sBsds _t

t0

Φt, s _s

−∞ζs, vdv ds≤Π, t

t0

Φt, sJsds≤D, for∀t≥t0,

4.3

A3ρΠ<1.

Definition 4.1. The setS ⊂ Cis called a global attracting set of4.1, if for any initial value

ϕ∈C, the solutionxtt0, ϕconverges toSast → ∞. That is,

distxt, S−→0 ast−→∞, 4.4

where distφ, S infψ∈Sdφ, ψ,dφ, ψis the distance ofφtoψinRn.

Theorem 4.2. Assume thatA1–A3hold. ThenS {ϕ ∈ C| ϕτ ≤ E−Π−1D}is a global

attracting set of 4.1.

Proof. It follows fromA1-A2and4.2that

xt≤Φt, t0

ϕt0 t

t0

Φt, sBsxs_τsds

t

t0

Φt, s s

−∞ζs−vxv

τvdv dsD, t≥t0.

For the initial conditionsxt ϕt,−∞< t≤t0, whereϕ∈C, we have

xt≤k0z, −∞< t≤t0, 4.6

wherez∈ΩρΠ, z >0,k0 ϕ/min1≤i≤nzi≥0,ϕ max1≤i≤nϕit0∞, and so

xt≤k0z E−Π−1D, −∞< t≤t0. 4.7

By4.5-4.7,A1–A3andTheorem 3.1, then there exists a constantk > k0such that

xt< kz E−Π−1D, fort≥t0. 4.8

From4.8, there exists a constant vectorσ≥0, such that

lim

t→∞xt

_{σ≤kz E−Π}−1_D.

4.9

Next we will show that σ ∈ S. From limt→∞Φt, t0 0, wij, φij ∈ UCt, and

_∞

0 ζijsds <∞, for anyε >0 ande 1,1, . . . ,1

T _∈Rn×1

, there exist a positive numberA and a positive constant matrixR∈Rn×n

such that for allt≥t0A

Φt, t0ϕt0< εe

4,

t

t0

Φt, sds≤R, 4.10

t

−∞ζt−v

kz E−Π−1Ddv≤Re,

t−A

t0

Φt, sds≤ ε

4R

−1_{, 4.11}

t−A

t0

Φt, sBskz E−Π−1Dds≤ εe

4 , 4.12

_∞

A

ζukz E−Π−1Ddu≤ εR −1_e

According to the definition of superior limit and limt→∞t−τt ∞, there exists sufficient

larget2≥t02A, such that for anyt≥t2,

xt_rt< σεe, where rt 2A sup

t−2A<s<t

τs. _4.14

So, fromA1,4.5, and4.10-4.14, whent≥t2, we obtain

xt≤Φt, t0

ϕt0 t

t0

Φt, sBsxs_τsds

t

t0

Φt, s s

−∞ζs−vxv

τvdv dsD

≤ εe

4

t−A

t0

t

t−A

Φt, sBsxs_τsds

t−A

t0

Φt, s s

−∞ζs−vxv

τvdv ds

_t

t−A

Φt, s _s−A

−∞ ζs−vxv

τvdv ds

_t

t−A

Φt, s _s

s−A

ζs−vxv_τvdv dsD

≤ εe

4

εe

4

_t

t−A

Φt, sBsxs_τsdsε

4R

−1_Re

_t

t−A

Φt, s _∞

A

ζukz E−Π−1Ddu ds

_t

t−A

Φt, s _s

s−A

ζs−vdv dsxt_rtD

≤ 3εe

4

t

t−A

Φt, sBsdsxt_rt t

t0

Φt, sdsεR −1_e

4

_t

t−A

Φt, s _s

s−A

ζs−vdv dsσεe D

≤ 3εe

4

_t

t0

Φt, sBsdsσεe ε

4RR

−1_e

t

t0

Φt, s s

−∞ζs−vdv dsσεe D ≤εe Πσεe D.

Due to4.9and the definition of superior limit, there existst3≥t2, such thatxt3> σ−εe. So,

σ−εe < εe Πσεe D, t≥t3. 4.16

Lettingε → 0, we haveσ≤E−Π−1D, that isσ∈S, and the proof is completed.

Corollary 4.3. If xt 0is an equilibrium point of system 4.1, suppose that the conditions of Theorem 4.2. hold andJt 0, then the equilibriumxt 0is globally asymptotically stable.

Remark 4.4. In Corollary 4.3., if Bt Bλ, ζt − u Kt − uλ, B bij_n×n > 0,

λ diagλ1, . . . , λn > 0,Kt−u Kt, u ∈ CR ×R, Rn×n,Kt−s kijt−s_n×n,

_∞

0 kijsds < aij,At diag−α1, . . . ,−αn,αi>0i 1,2, . . . , n, then we can get Theorems

3 and 4 in16. In factΦt, t0 diage−α1t−t0, . . . , e−αnt−t0∈_UC

t, Theorems 3 and 4 in16

satisfy all conditions ofCorollary 4.3.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant no. 10971147. Thanks are due to the instruction of Professor Xu.

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