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空間二階邊值問題

正解的存在性與多重性.通過對非緊性測度的計算,利用嚴(yán)格集壓縮映射的不動點理論,給出了該問題正解存在與多個正解存在的充分條件.

存在性;正解;非緊性測度;不動點;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.

對于該問題,當(dāng)E=R1時,已有學(xué)者做過研究,見文獻(xiàn)[4-5],然而在抽象空間,結(jié)論尚不多見.由于有限維空間與無限維空間的本質(zhì)差異,在無限維空間中,非線性項f(t,u)的連續(xù)性保證不了解的存在性,因此,還要對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),且滿足非緊性測度條件,分別討論了Dirichlet邊值問題與Sturm-Liouville邊值問題正解的存在性.而本文通過對非緊性測度的精細(xì)計算,將文獻(xiàn)[1,3]中對f一致連續(xù)這個很強的條件減弱為連續(xù),運用嚴(yán)格集壓縮映射的不動點理論,獲得了問題(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,上式成立.

引理0.4[9]設(shè)P為實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上有不動點x,滿足r＜‖x‖＜s.

1 預(yù)備工作

為方便起見,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?y∈C[I,P],邊值問題

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)展方程初值問題解的存在性[J].數(shù)學(xué)學(xué)報,2005,48:1089-1094.

[11] 郭大鈞,孫經(jīng)先.抽象空間常微分方程[M].濟南: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,主要研究方向為非線性泛函分析.E-mail:dyh198510@126.com