具有p-Laplacian算子的共振微分方程組解的存在性

江衛(wèi)華，周彩蓮，李慶敏

(河北科技大學(xué)理學(xué)院，河北石家莊 050018)

具有p-Laplacian算子的共振微分方程組解的存在性

江衛(wèi)華，周彩蓮，李慶敏

(河北科技大學(xué)理學(xué)院，河北石家莊 050018)

為了研究具有非線性分?jǐn)?shù)階微分算子的微分方程共振邊值問題解的存在性，引入了推廣的Mawhin 連續(xù)定理，通過定義合適的Banach空間及范數(shù)，給出恰當(dāng)?shù)乃阕樱\(yùn)用Mawhin 連續(xù)定理的拓展，研究了具有p-Laplacian 算子的分?jǐn)?shù)階共振微分方程組邊值問題解的存在性。通過舉例驗(yàn)證了所得結(jié)論的正確性。所得結(jié)論是共振邊值問題現(xiàn)有成果的推廣和一般化，對(duì)進(jìn)一步研究具有一定參考價(jià)值。

常微分方程；邊值問題；共振；Mawhin連續(xù)定理的拓展；p-Laplacian算子

1 問題提出

微分方程邊值問題廣泛應(yīng)用于物理學(xué)、機(jī)械學(xué)、化學(xué)、能量學(xué)等領(lǐng)域中。所謂微分方程共振邊值問題是指其相應(yīng)的齊次邊值問題具有非零解。對(duì)共振微分方程解的存在性研究已有許多的成果[1-16]，分?jǐn)?shù)階微分方程邊值問題得到了許多學(xué)者的廣泛關(guān)注[17-22]，但對(duì)帶p-Laplacian算子的共振分?jǐn)?shù)階微分方程組邊值問題解的存在性的研究成果相對(duì)較少。文獻(xiàn)[20]研究了具有p-Laplacian 算子的分?jǐn)?shù)階共振微分方程邊值問題：

受上述文獻(xiàn)啟發(fā)，筆者利用Mawhin連續(xù)定理的拓展[24]，研究具有p-Laplacian算子的共振微分方程組邊值問題：

(1)

2 預(yù)備知識(shí)

為了得到想要的結(jié)論，給出如下定義和定理。

定義1[23]設(shè)X，Y是2個(gè)Banach空間，如果連續(xù)算子M:X∩domM→Y滿足下面條件：

1)ImM:=M(X∩domM)是Y的閉子集，

2)KerM:={x∈X∩domM:Mx=0}與Rn是線性同胚的，n<∞。

則稱算子M是擬線性的，其中domM表示M的定義域。

令X1=KerM，P:X→X1是投影算子，Ω?X是一個(gè)有界開集，零元θ∈Ω。

a)KerQ=ImM,

b)QNλx=θ,λ∈(0,1)?QNx=θ,

c)R(·,0)是零算子，R(·,λ)|∑λ=(I-P)|∑λ，

d)M[P+R(·,λ)]=(I-Q)Nλ，

C1)Mx≠Nλx,?x∈?Ω∩domM,λ∈(0,1)，

C2)deg{JQN,Ω∩KerM,0}≠0，

以下是分?jǐn)?shù)階微積分的定義和性質(zhì)：

定義3[1]f:[0,∞)→R是連續(xù)函數(shù)，f的α階Riemann-Liouville分?jǐn)?shù)階積分的定義為

定義4[1]f:[0,∞)→R是連續(xù)函數(shù)，則f的α階Riemann-Liouville分?jǐn)?shù)階導(dǎo)數(shù)的定義為

在對(duì)主要問題的證明中需要用到如下不等式。

引理5[24]對(duì)于任意的u,v,w≥0，有:

1)φp(u+v)≤φp(u)+φp(v)，1

2)φp(u+v)≤2p-2(φp(u)+φp(v))，p≥2；

3)φp(u+v+w)≤φp(u)+φp(v)+φp(w)，1

4)φp(u,v,w)≤22p-4φp(u)+22p-4φp(v)+2p-2φp(w)，p≥2,

其中φp(s)=|s|p-2·s=sp-1，s≥0。

3 主要結(jié)論

定義算子M:domM→Y和Nλ:X→Y如下：

其中:

引理6 M是擬線性算子。

證明 易知KerM={c1tα1-1,c2tα2-1}，對(duì)任意(u,v)∈domM，如果M(u,v)=(y1,y2)=y，可得y1,y2滿足：

(2)

反之，若(y1,y2)∈Y滿足式(2)，取

通過簡(jiǎn)單計(jì)算，可以得到(u,v)∈domM，M(u,v)=(y1,y2)=y。因此，ImM={y=(y1,y2)∈Y|y1,y2滿足式(2)}。

根據(jù)范數(shù)‖y1‖∞和‖y2‖∞的定義可得M是擬線性算子。證畢。

為了證明需要的結(jié)論，定義算子P:X→KerM，Q:Y→R如下：

Q(y1,y2)=(Q1y1,Q2y2)=(a1,a2)，

其中a1，a2滿足：

(3)

由文獻(xiàn)[20]中引理3.1和引理3.2易知，定義Q1y1=a1，Q2y2=a2有意義且Q:Y→R2是連續(xù)有界算子。易知P:X→X1是投影算子，因此X=X1⊕X2，其中X1=ImP,X2=KerP。

定義算子R:X×[0,1]→X2如下：

R(u,v,λ)(t)=(R1(u,v,λ)(t),R2(u,v,λ)(t))，

其中：

|R1(u,v,λ)(t2)-R1(u,v,λ)(t1)|=

(4)

由于：

(5)

因此有：

定理2 假設(shè)下列條件成立：

H1)存在2個(gè)常數(shù)K1,K2>0，使得下列不等式之一成立：

H2)存在非負(fù)函數(shù)ai(t),bi(t),ci(t),di(t)，ei(t),li(t),ri(t)∈C[0,1]，i=1,2，使得：

當(dāng)1

a) (1-A1)(1-B2)-A2B1>0，其中:

當(dāng)p>2時(shí)，ai(t),bi(t),ci(t),di(t),ei(t),li(t)滿足：(1-C1)(1-D2)-C2D1>0，

其中:

那么，邊界值問題(1)至少有1個(gè)解。

為了證明定理2，給出2個(gè)引理。

引理9 假設(shè)條件H1)和條件H2)成立，那么集合Ω1={(u,v)∈domM|M(u,v)=Nλ(u,v),λ∈(0,1)}在X中是有界的。

則有：

(6)

根據(jù)u(0)=v(0)=0，可得：

綜上可得：

(7)

根據(jù)條件H2)和式(6)、式(7)可得：

1) 當(dāng)1

(8)

同理可得：

(9)

由式(8)和式(9)可得：

所以有：

2)當(dāng)p>2時(shí)，有：

(10)

同理可得：

(11)

由式(10)和式(11)可得：

因此，

綜上所述可得Ω1是有界的。證畢。

引理10 若條件H1)成立，那么Ω2={(u,v)∈KerM|QN(u,v)=θ}在X中是有界的，其中N=N1。

證明 參考文獻(xiàn)[20]。

下面證明定理2成立。

當(dāng)δ=1時(shí)，H1(u,v,1)=ρk1tα1-1，H2(u,v,1)=ρk2tα2-1，所以H(u,v,δ)≠θ。當(dāng)δ=0時(shí)，由條件H1)可得：

與條件H1)矛盾。故H(u,v,δ)≠0，其中δ∈[0,1]，(u,v)∈KerM∩?Ω。

由度的同倫性不變性，可得：

deg(JQN,Ω∩KerM,0)=deg(H(u,v,0),Ω∩KerM,0=

deg(H(u,v,1),Ω∩KerM,0)=

deg(ρI,Ω∩KerM,0)≠0。

4 例 子

例1 考慮下面具有p-Laplacian算子的共振微分方程組邊值問題:

(12)

證明 顯然條件H1)成立，根據(jù)定義可得：

|f(t,A,B,C,D,E,F)|≤|a1(t)|φp(|A|)+|b1(t)|φp(|B|)+|c1(t)|φp(|C|)+

|d1(t)|φp(|D|)+|e1(t)|φp(|E|)+|l1(t)|φp(|F|)+|r1(t)|,

|g(t,A,B,C,D,E,F)|≤|a2(t)|φp(|A|)+|b2(t)|φp(|B|)+|c2(t)|φp(|C|)+

|d2(t)|φp(|D|)+|e2(t)|φp(|E|)+|l2(t)|φp(|F|)+|rr(t)|。

通過簡(jiǎn)單的計(jì)算， 可以得到：

(1-C1)(1-D2)-C2D1=0.082 5>0，條件H2)成立，通過定理2可知，問題(12)至少有1個(gè)解。證畢。

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Existence of solutions for differential equations systemswithp-Laplacian at resonance

JIANG Weihua, ZHOU Cailian, LI Qingmin

(School of Science, Hebei University of Science and Technology, Shijiazhuang, Hebei 050018, China)

In order to study the existence of solutions for boundary value problems at resonance with nonlinear fractional differential operator, a generalization of Mawhin's continuous theorem is introduced. By defining suitable Banach space and norm, constructing the proper operators and using the extension of Mawhin continuation theorem, the existence of solutions for fractional differential equations systems boundary value problem withp-Laplacian at resonance is studied. An example is given to illustrate the main results. The results are the improvement and generalization of some existing results of boundary value problems at resonance.

ordinary differential equation; boundary value problem; resonance; the extension of Mawhin’s continuation theorem；p-Laplacian operator

2016-10-01；

2017-05-10；責(zé)任編輯：張 軍

河北省自然科學(xué)基金(A2013208108)

江衛(wèi)華(1964—)，女，河北邯鄲人，教授，博士，主要從事應(yīng)用泛函分析、常微分方程邊值問題方面的研究。

E-mail:jianghua64@163.com

1008-1542(2017)04-0341-11

10.7535/hbkd.2017yx04005

O175.8MSC(2010)主題分類：34B05

A

江衛(wèi)華,周彩蓮，李慶敏.具有p-Laplacian算子的共振微分方程組解的存在性[J].河北科技大學(xué)學(xué)報(bào),2017,38(4):341-351. JIANG Weihua, ZHOU Cailian, LI Qingmin.Existence of solutions for differential equations systems withp-Laplacian at resonance[J].Journal of Hebei University of Science and Technology,2017,38(4):341-351.