一類非自治共振二階系統(tǒng)的多重周期解

張環(huán)環(huán)

(西北民族大學數(shù)學與計算機科學學院，甘肅 蘭州 730030)

摘要：研究了非自治共振二階系統(tǒng)周期解的存在性問題.在非線性項次線性增長時，將這類系統(tǒng)的周期解轉(zhuǎn)化為定義在一個適當空間上泛函的臨界點，然后利用臨界點理論建立了此類系統(tǒng)周期解的存在性結(jié)果.

關鍵詞：非自治二階系統(tǒng)；臨界點理論；周期解；共振；臨界點

文章編號：1007-2985(2015)05-0016-05

收稿日期：2015-01-14

基金項目：數(shù)學天元基金資助項目(11326100)；中央高?；究蒲袠I(yè)務費專項資助項目(31920130010)

作者簡介：張環(huán)環(huán)(1980—)，女，甘肅靜寧人，西北民族大學數(shù)學與計算機科學學院講師，產(chǎn)要從事非線性泛函分析研究.

中圖分類號：O175.12 文獻標志碼：A

DOI:10.3969/j.cnki.jdxb.2015.05.004

DOI:10.3969/j.cnki.jdxb.2015.05.005

1　問題的提出

考慮非自治二階系統(tǒng)

(1)

其中m為非負整數(shù)，F(xiàn)∈C1(R×RN，R)，且關于t是2π周期的.

則φ連續(xù)可微，且

(F(t，u(t))，v(t))dt.

許多重要的數(shù)學模型都可以歸結(jié)為非自治二階系統(tǒng)的周期解，利用臨界點理論研究非自治二階系統(tǒng)周期解的存在性一直是人們關注的問題[1-11].文獻[1]中研究了超二次條件下非自治二階系統(tǒng)周期解的存在性;在具有部分周期位勢時，文獻[2-3，9，11]中得到了非自治二階系統(tǒng)多重周期解的存在性;在非線性項F(t，u)有界時，文獻[4]中得到了問題(1)周期解的存在性;在非線性項在無窮遠處線性增長時，文獻[5]中得到了非自治二階系統(tǒng)周期解的存在性.

次線性條件指[9]，存在f，g∈L1(0，T;R+)，0≤α<1 ，使得

|F(t，x)|≤f(t)|x|α+g(t)，

(2)

對所有x∈RN和a.e.t∈[0，2π]成立.

稱問題(1)是共振的，指

(3)

對t∈[0，2π]一致成立.在(2)式成立時，(3)式也成立，故此時稱問題(1)是共振的.

當m=0時，在具有次線性非線性項時，文獻[6]中得到了問題(1)周期解的存在性;文獻[7]將文獻[6]中結(jié)果推廣為m不恒等于0的情形；文獻[8]中討論了次線性非自治一階Hamiltonian系統(tǒng)周期解的存在性;文獻[6-8]中均假設核空間上Ahmad-Lazer-Paul型強制性條件成立，即

筆者將Ahmad-Lazer-Paul型強制性條件推廣為

(4)

在此條件下，用臨界點理論研究問題(1)的多重周期解，在非線性項滿足次線性條件時，得到這類周期解個數(shù)的下界估計.

2　預備知識

(5)

對所有u+∈H+和t∈[0，2π]成立.由文獻[12]，有

(6)

定義1 [9]設X為Banach空間，稱泛函φ∈C1(X，R)滿足(PS)條件是指對任何點列{un}?X，由{φ(un)}有界，φ′(un)→0蘊含{un}有收斂子列.

利用Z2不變?nèi)褐笜死碚摰玫饺缦屡R界點定理：

引理1 [13]設φ∈C1(X，R)滿足(PS)條件，又是偶函數(shù)，φ(0)=0.

若m>j，則泛函φ至少有2(m-j)個不同的臨界點.

3　主要結(jié)果

定理1設F滿足(2)，(4)式，F(xiàn)(t，0)=0且存在常數(shù)δ及整數(shù)k>m+1，使得

(7)

下文證明中用c表示常量.

|φ(un)|≤c，φ′(un)→0n→∞，

(8)

由(2)，(6)式，有

由(8)式，有

從而有

(9)

(10)

由(10)式可以推得

(11)

由(9)，(11)式，有

(12)

同理可證

(13)

(14)

由(12)，(14)式，有

(15)

反設當n→∞時，‖un‖→∞，并注意到0≤α<1，由(15)式有

(16)

由(2)，(6)，(15)式，有

2β‖vn‖α‖wn‖+2β‖wn‖α+1+c‖wn‖≤

(17)

由(17)式可得

由(4)式，對?ε>0，存在M>0，使‖vn‖≥M時有

(18)

對所有‖vn‖≥M，由(13)，(15)，(18)，(23)式，有

由ε的任意性及0≤α<1，由(16)式，當n→∞時，φ(un)→-∞，這與(8)式矛盾.

對所有的a.e.t∈[0，T]，|u|≤C-1δ成立，其中C為Sobolev不等式‖u‖∞≤C‖u‖中的正常數(shù).

從而泛函φ至少有2(k-m-1)個不同的臨界點，因此系統(tǒng)(1)至少有2(k-m-1)個周期解.

定理2設F滿足(2)，(7)式，且

定理2證明方法與定理1類似，此省略.

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Multiplicity of Periodic Solution of a Class of Non-Automous

Second Order System at Resonance

ZHANG Huanhuan

(College of Mathematics and Computer Science，Northwest University for Nationalities，Lanzhou 730030，China)

Abstract：The existence of periodic solutions for non-automous second order systems at resonance is investigated.With the sub-linear increase of the non-linear term,the periodic solutions of the system are converted into the critical points of a functional defined on a proper space，and the existence of periodic solutions is proved through the critical point theory.

Key words：non-autonomous second order systems;critical point theory；periodic solutions;resonance;critical point

(責任編輯向陽潔)

Article ID：1007-2985(2015)05-0021-06

Mittag-Leffler Stability of a Class of Fractional

Order Hopfield Neural Networks

Received date：2014-11-26

Biography：LIU Xiaolei(1983—)，male,was born in Weifang City,Shandong Province,master of science,lecture;research area are fractional order dynamic systems and neural networks.

LIU Xiaolei,MA Cuiling,HAO Shuyan

Abstract：In this paper,we investigate the Mittag-Leffler stability of a class of fractional order Hopfield neural network with Caputo derivative.By using the Mittag-Leffler function,we get some sufficient conditions to guarantee the existence and uniqueness of the equilibrium point and its Mittag-Leffler stability for the fractional order Hopfield neural networks.Finally,we use one numerical simulation example to illustrate the correctness and effectiveness of our results.

Key words：fractional order neural networks;Mittag-Leffler function;Mittag-Leffler stability

CLC number：O211.29Document code：A

1　Introduction

The subject of fractional calculus was planted over 300 years ago.In recent years,fractional calculus has played a significant role in many areas of science and engineering[1-3].The necessary and sufficient stability conditions for linear fractional differential equations (FDEs) and linear time-delayed FDEs have already been obtained in ref. [4-6].The stability of nonlinear fractional order system for Caputo’s derivative is studied in ref. [7-9].

Currently,some excellent results about fractional-order neural networks have been investigated[10-13].Particularly,Yu Juan et al[13]studiedα-stability andα-synchronization for fractional-order Hopfield neural networks as follows:

(1)

(1)

where 0<α<1,ncorresponds to the number of units in the neural networks;xi(t) corresponds to the state of theith unit at timet;gj(t) denotes the activation function of thejth neuron;aijdenotes the constant connection weight of thejth neuron on theith neuron;ci>0 represents the rate with which theith neuron will reset its potential to the resting state when disconnected from the network andIidenotes external inputs.

In this paper,by putting the systems translating into the nonlinear Volterra integral equation of the second kind,and making use of the existence and uniqueness Theorem of FDEs and a weakly singular discrete Gronwall inequality to prove the Mittag-Leffler stability of the Hopfield neural networks,which is the generation of the exponential stability.And when 0<α<1,the asymptotic rate of convergence for the system approaching the equilibrium point is faster than the exponential stability.

The remained of this paper is organized as follows:in section 2,some necessary definitions and lemmas are presented;in section 3,we give some sufficient conditions to guarantee the existence and uniqueness of the equilibrium point and its Mittag-Leffler stability for a class of fractional order Hopfield neural networks by using the Mittag-Leffler function;in section 4,one example and corresponding numerical simulation are used to illustrate the validity and feasibility of the results obtained in section 3.

2　Preliminaries

There are several definitions of a fractional derivative of orderα,which is the extended concept of integer order derivative.The commonly used definitions are Grunwald-Letnikov,Riemann-Liouville,and Caputo definitions.In this section,we will recall the definition of Caputo fractional derivative and the several important lemmas.

Definition 1[14]The Caputo fractional derivative of orderα∈R+of a functionx(t) is defined as

Consider the Cauchy problem of the following fractional differential equation:

(2)

where x=(x1,x2,…,xn)T∈Rn,0<α<1,and f:[0,+∞)×Rn→Rnis continuous int.

Definition 2[15]LetB?Rnbe a domain containing the origin.The zero solution of (2) is said to be Mittag-Leffler stable if

‖x(t)‖≤(m(x(t0))Eα(-λ(t-t0)α))b,

Definition 3[16]The constant x*is an equilibrium point of the Caputo fractional dynamic system (2) if and only iff(t,x*)=0 for anyt∈[0,+∞).

Lemma 1[17]Consider the following equation

(3)

The homotopy perturbation technique yields that the initial value problem (3) be equivalent to the nonlinear Volterra integral equation of the second kind

(4)

In particular,if 0<α<1,then eq. (4) can be written in the following form

(5)

Lemma 2[18]Letx(t) be a continuous and nonnegative function ont∈[0,T].If

where 0≤α<1 andψ(t) is a nonnegative,monotonic increasing function ont∈[0,T],andMis a constant,then x(t)≤ψ(t)E1-α(MΓ(1-α)t1-α).

Lemma 3[14]Let 0≤α<1 andf(t,x):[0,+∞)×Rn→Rnbe a function such that,for allt∈[0,+∞) and for allx1,x2∈G?Rn,

|f(t,x1)-f(t,x2)|≤A|x1-x2|,

whereA>0 does not depend ont∈[0,+∞),then there exists a unique solution x(t) to the Cauchy problem in theC[0,+∞).

3　Mittag-Leffler Stability of a Class of Fractional Order Neural Networks

In this section,we suppose that the fractional order Hopfield neural networks satisfies the following conditions:

(A1)gj(j=1,2,…,n) are Lipschtiz-continuous on (-∞,+∞) with Lipschtiz constantsLj>0,i.e.,|gj(ξ)-gj(η)|≤Lj|ξ-η|,for allξ,η∈(-∞,+∞);

Letλ=min{λ1,λ2,…,λn},l=max{l1,l2,…,ln},then we haveλ>0,l<1.

Theorem 1Under the assumptions (A1) and (A2),system (1) has a unique equilibrium point.

By the assumptions (A1) and (A2),we have

It shows that ‖Φ(u)-Φ(v)‖

So the conclusion of the theorem is correct.

Theorem 2Under the assumptions (A1) and (A2),the unique equilibrium point of system (1) is Mittag-Leffler stable.

(6)

By the lemma 3 and the assumption (A1),the solution of the system (6) exists and is unique.It is easy to see thatei(t)≡0 is a solution of the system (6);therefore we haveei(t)ei(0)>0 fort∈[0,+∞).We divide our discussion into two cases.

Case 1Ifei(0)>0,thenei(t)>0 fort∈[0,+∞).From (5) in lemma 1,we have

Case 2Ifei(0)<0,thenei(t)<0 fort∈[0,+∞).It is similar to the case 1,we have

Therefore,

From case 1 and case 2,we get the inequality

By lemma 2,we have

which shows that the system (1) is Mittag-Leffler stable.

4　Illustrative Examples

In the system (1),let

x(0)=(x1(0),x2(0),x3(0))T=(0.1,0.1,0.1)T,

thenthesystem(1)satisfiestheconditionsoftheorem1andtheorem2;thereforeithastheuniqueMittag-Lefflerstableequilibriumpoint(seefig. 1).

a x 1(t) plane

b x 2(t) plane

c x 3(t) plane

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