＊ Banach空間二階積分邊值問(wèn)題的正解

丁永宏

(西北師范大學(xué)數(shù)學(xué)與信息科學(xué)學(xué)院,甘肅蘭州 730070)

＊Banach空間二階積分邊值問(wèn)題的正解

丁永宏

(西北師范大學(xué)數(shù)學(xué)與信息科學(xué)學(xué)院,甘肅蘭州 730070)

討論了Banach空間二階邊值問(wèn)題

正解的存在性與多重性.通過(guò)對(duì)非緊性測(cè)度的計(jì)算,利用嚴(yán)格集壓縮映射的不動(dòng)點(diǎn)理論,給出了該問(wèn)題正解存在與多個(gè)正解存在的充分條件.

存在性;正解;非緊性測(cè)度;不動(dòng)點(diǎn);Banach空間

0 引言

設(shè)E為Banach空間,P?E為正規(guī)錐,正規(guī)常數(shù)為1,P導(dǎo)出E中的半序≤.考慮二階積分邊值問(wèn)題

正解的存在性,其中f∈C[I×P,P],I=[0,1],g,h∈L1[0,1]為非負(fù)函數(shù),a≥0,b≥0,c≥0,d≥0,ρ=ac+ad+bc＞0.

對(duì)于該問(wèn)題,當(dāng)E=R1時(shí),已有學(xué)者做過(guò)研究,見(jiàn)文獻(xiàn)[4-5],然而在抽象空間,結(jié)論尚不多見(jiàn).由于有限維空間與無(wú)限維空間的本質(zhì)差異,在無(wú)限維空間中,非線性項(xiàng)f(t,u)的連續(xù)性保證不了解的存在性,因此,還要對(duì)f加上一定的條件.在文獻(xiàn)[1,3]中,作者假定f一致連續(xù),且滿足非緊性測(cè)度條件,分別討論了Dirichlet邊值問(wèn)題與Sturm-Liouville邊值問(wèn)題正解的存在性.而本文通過(guò)對(duì)非緊性測(cè)度的精細(xì)計(jì)算,將文獻(xiàn)[1,3]中對(duì)f一致連續(xù)這個(gè)很強(qiáng)的條件減弱為連續(xù),運(yùn)用嚴(yán)格集壓縮映射的不動(dòng)點(diǎn)理論,獲得了問(wèn)題(1)正解的存在性與多重性結(jié)果.

下面給出一些文中用到的定義和引理:

定義0.1[11]設(shè)P為實(shí)Banach空間E中的錐,P＊={ψ∈E＊:ψ(x)≥0,?x∈P},稱P＊為P的共軛錐.

引理0.2[10]設(shè)E為Banach空間,D?E有界,則存在D的可數(shù)子集D0,使得α(D)≤2α(D0).

引理0.3[2]設(shè)E為Banach空間,B={un}?C[I,E],若存在g(t)∈L1(I),使得‖un(t)‖≤g(t),a.e.t∈I,n=1,2,…,則α(B(t))∈L1(I),且有

特別當(dāng)B有界時(shí),上式成立.

引理0.4[9]設(shè)P為實(shí)Banach空間E中的錐,Pr,s={x∈P∶r≤‖x‖≤s},其中s＞r＞0,A:Pr,s→P是嚴(yán)格集壓縮映射.如果A滿足下列兩條件之一.

(i)A x≤/x,?x∈P,‖x‖=r,A x≥/x,?x∈P,‖x‖=s,

(ii)A x≥/x,?x∈P,‖x‖=r,A x≤/x,?x∈P,‖x‖=s,

則A在P上有不動(dòng)點(diǎn)x,滿足r＜‖x‖＜s.

1 預(yù)備工作

為方便起見(jiàn),記

我們假設(shè)

(H1)H,G∈[0,1),CKD F∈[0,1),

Tl,有α(f(I,D))≤Lα(D),其中Tl={x∈E∶‖x‖≤l}.

引理1.1[7]假設(shè)條件(H1)成立,則對(duì)?y∈C[I,P],邊值問(wèn)題

綜上,γ(t,s)≥z(t)γ(τ,s),?t,τ,s∈I.

引理1.3 假設(shè)條件(H1)成立,則G(t,s)≤G(t,t)≤N,γ(t,s)≤M G(t,t)≤M N.

引理1.5 假設(shè)條件(H1),(H2)成立,Kr,R={u∈K∶r≤‖u‖c≤R},其中R＞r＞0,則A∶Kr,R→K為嚴(yán)格k-集壓縮映射.

證明 設(shè)B?Kr,R,由引理0.2知,存在B的可數(shù)子集B0={un}使得

2 結(jié)論

[1] Guo D,Lakshmikantham V.Multiple Solutions of Two-point Boundary Value Problems of Ordinary Differential Equations in Banach Spaces[J].J M ath A na l A pp l,1998,129:211-222.

[2] Heinz H R.On the Behaviour of Noncompactness with Respect to Differention and Integration of Vecto r Valued Functions[J].N on linear Ana lysis,1983,7:1351-1371.

[3] Lou Ben-dong.Solutions of Superlinear Sturm-Liouville Problems in Banach Spaces[J].J Math Anal A pp l,1996,201:169-179.

[4] Yang Z.Positive Solutions of a Second Order Integral Boundary Value Problem[J].J Math Anal Appl,2006,321:751-765.

[5] Abdelkader Boucherif.Second-order Boundary Value Problems with Integral Boundary Conditions[J].N on linear Analysis,2009,70:364-371.

[6] Liu Bing.Positive Solutions of Generalized Sturm-Liouville Fou-point Boundary Value Problem in Banach Spaces[J].Nonlinear A na lysis,2007,66:1661-1674.

[7] Kang Ping,Wei Zhong-li,Xu Juan-juan.Positive Solutions to Fouth-order Boundary Value Problem sw ith Integral Boundary Conditions in Banach Spaces[J].A pp l M ath Com put,2008,206:245-256.

[8] Guo D,Lakshmikantham V.Nonlinear Problem s in Abstract Cones[M].San Diego:Academic Press,1988.

[9] Guo D.Lakshmikantham V,Liu X.Nonlinear Integral Equations in Abstract Spaces[M].Do rdrecht:Kluwer Academic Publishers,1996.

[10] 李永祥.抽象半線性發(fā)展方程初值問(wèn)題解的存在性[J].數(shù)學(xué)學(xué)報(bào),2005,48:1089-1094.

[11] 郭大鈞,孫經(jīng)先.抽象空間常微分方程[M].濟(jì)南:山東科學(xué)技術(shù)出版社,1989.

Positive Solutions of Second Order Boundary Value Problem with Integral Boundary Conditions in Banach Spaces

DING Yong-hong
(College of Mathematics and Information Science,Northwest Normal University,Lanzhou730070,China)

The existence and multiplicity of positive solutions for second order boundary value problem

are discussed.The existence and multiplicity results by using the fixed-point index theory of strict set contraction operator and doing computation of measure of noncompactness was got.

existence;positive solution;measure of noncompactness;fixed-point;Banach spaces

O175.7

A

0253-2395(2011)01-0036-06＊

2010-04-06;

2010-06-02

甘肅省自然科學(xué)基金(0710RJZA 103)

丁永宏(1985-),男,甘肅天水人,碩士研究生,主要研究方向?yàn)榉蔷€性泛函分析.E-mail:dyh198510@126.com