几类微分方程的解
摘要
非线性泛函分析是一门既悠久又现代的学科，它的丰富理论和先进方法为解决当今科技领域层出不穷的非线性问题提供了卓有成效的工具，作为自非线性泛函分析中衍生发展起来的新的分支，Banach空间微分方程和积分方程理论虽经历了不足三十年的发展过程，然而它已被广泛应用于诸如临界点理论，偏微分方程理论，特征值问题等许多领域，其重要性日益凸现出来．
郭大钧先生在专著[7]中对非线性泛函分析的几个重要课题及其应用，诸如典型的非线性算子、Hammerstein积分方程、常、偏微分方程、迁移方程、锥理论及非线性算子方程的正解、非线性算子拓扑度和不动点定理以及固有值、解的个数与分支，都做了系统的概括和总结．[1]包括的内容非常多，它包含了非线性泛函分析这一领域各个方面的成果．
本文利用锥理论、单调迭代技术、锥拉伸压缩不动点原理、上下解方法、不等式迭代技术、M(?)nch不动点理论等，研究了几类微分方程解的情况．所得结果或是新的，或是采用新方法在更弱条件下推广和改进以前的结果．
本文共分四节．
第一节我们讨论了实Banach空间(E，儿㈨)中下列非线性混合型微分-
积分方程
{u=f苟t,t，Tu，Su)，t I,k =C[Do，R+)， (1．1．1)
其中
(Tu)(t)=人k(t，s)u(s)ds，t I,K AC[Do2，R+]；
(Su)(t)=九h(t，s)u(s)ds，t／I，HC[I×I2,R+]．
这里H一二m．－丁）、o；＝｛f．｝》EHl ＜爿＜I＜叫l（；＝二11＊二川人叫：
f．XEDO｝．人0＝。。。＊x…U．人。：I．／E川．
我们需要以下条件：
（HI） 存在no E厂p．厂二是1P（1） 的下解＠ ＊。1二；S．f《f。l【j·T。I；。·从j＊》。
。。（／》S．I”；；：
。尸叫） 三任给4壬＊ 上＊厂三厂＝｛｛／ECJ厂 U＞／／；；｝回JI＜l都有
／（f。T。从）一人I。1．TI．＼U）＞一」八。一川一‘＼（T卜－。。l
其中－＼I．X三口是满足引理中条件 n或（i）的常数；
（H3） 存在L＊＊esp且＊可积函数＊川．尸川回Q卜）三＊使任意卜．含E D．
；！三。都有
J（Z．乙·．＊＊．JI·卜J（I．＊～TI．JU）＜L（川＊一U）十P（重）（T（L’一U》十Q（）尸（。·一U》·
定理 1．3．1设厂是 BQnQCh空间，P是 E中的正规锥．假设（HI）（HZ）（H3）
成立，则11。T．1．1）在D中存在唯一解。”，且对任慧。。E D，迭代序列
，l、＿一＊It＿，J｛I＿＿，J＿＼I门口．．＼I。＼／口。。V＿八、、J、．．J。＼
ti，。《t）＝6“”“110十J V（S，11”nl（）八IW。一l八S）八b川。一l八S川十 j’1出。一八SI
一NT（u＊一＊，＊一）s）e人”ds｝（1．3．l）
在I上依E中范数一致收敛于。＊t），并且有误差估计
A”
帅＊一w＊【【＊S人＊＊I卜＊一＊＊11＊＋人0一卜1一＊口11＊＊ ＊三N．（I．3．2）
”－’‘－”“‘”””’‘一””‘’“”l 人““”’“’‘
其中 01二 e一‘”仁口＋亿rt，0。（S）；（T＊。）（8），（S＊0）（S））一 N（Tuo）（。）］eM”ds）．
定理 1．3．2设 E是空间，P是中的正规锥，若 f满足下列假设：
（凤）’存在＊口Ec＊贝是＊＊＊1．1．二）的上解，即v＊’三／（土＊＊，＊＊口；仇＊）；
UO（ 三工。；
（HZ）’同（HZ）；
4
占
（H／ 同（从）．
则n一P：1回1川在D二毛E＊厂厂11仑，三互，：可｝中存在唯一解乙上寡回且对任意
；I·（，ED．迭代序列
11＿Ill72t ti，、。IlllSll．llslllll－ill．＼ll．3II’、IllSllW！Ill．IISI
一八（I（·。一乙1、可；一l）（人）］了1厂水d／｝
在 I上依厂中范数一致收敛于；l·．（1）．并且有误差估计
／l，一〔l、．＜儿、入（h 一；．．一＼————I，一l；．门＞＼
”—””””””l＜““”一
其中江·l二。一·＼’｛TO十厂／（J回。、C；（。）．〔T。·0《＞多．（J。，）仁上》一回＼（T。·0）（叫I可·土’A。／叶．
注 l．3二 文问定理4的一个关健条件为（H汁存在常数 R．r三（）使得
任给t EI间。咆灭·Eb丐u。0工雹U S。·都有
／（I。，T。，）一f儿。I回T。1）＜D卜一?
Nonlinear Functional Analysis is a subject .old but fashionable.Its abundant theories and advanced methods are providing powerful and fruitful tools in solving ever increasing nonlinear problems in the fields of science and technology. Though the theories of integral and differential equations in Banach spaces, as new branches of Nonlinear Functional Analysis.have developed for no more than thirty years, they are finding extensive applications in such domains as the critical point theory,the theory of partial differential equa-tions,eigenvalue problems.and so on,are attracting much more attentions from both pure and applied mathematicians.
Professor Guo Dajun has summarized in his work [7] .such several important tasks and theirs application of Nonlinear Functional Analysis as typical nonlinear operators,Hammerstein integral operations,ordinarily and partially differential equations.the cone theory,the positive solutions of nonlinear operator equations,the number and the branch of solutions,and so on.Reference [1] includes all levels of results of the domain such as Nonlinear Functional Analysis.
The present thesis employs the cone theory,monotone iterative technique, the conical expansion and compression principle,the method of upper and lower solutions,the Monch theory of fixed point,and so on,to investigate the existence of solutions of sevral differential equations .The obtained results are either new
or intrinsically generalize and improve the previous relevant ones under weaker conditions.
The paper is divided into four sections.
In section one we investigate the following nonlinear integro-differential equation in Banach spaces.
where.
We list below the conditions:
(H1) There exists as the lower solution of IVP(1).i.e.
satisfy
where, M. .V > 0 are constants satisfying such conditions as (i)(ii) of Lemma 1.2.1.
(H3) There exist Lebesgue integrable functions L(t), P(t),Q(t) > 0,such that make tt, v € D, u Theorem 1.3.1 Let E be a Banach space and P be a regular cone in E . Assume that (H1)(H2)(H3) are satisfied. Then /KP(1.1.1) possesses only one
10
solution .and the iterative sequence
(1.3.1)
converges uniformly on I to w(t). Moreover. there is error estimate
Theorem 1.3.2 Let E be a Banach space and P be a regular cone in E. Assume that / satisfys the following assumptions:
(H1)t There exists TO 6 C1[I. E] as a upper solution of
Then /YP(l.l.l) possesses only one solution w, and D, the iterative sequence
converges uniformly on I to w(t). Moreover,there is error estimate
Remark 1.3.1 The key condition of Theorem 4 in [5] is (H4):There exist constants R, r > 0,such that
the theorems in the present paper changes B.r into only integarable and non-negative functions L(t).P(1). broadening this formula and extending this theorem.
Remark 1.3.4 Those1 results presented in the given refrenccs cann't induce the Theorem 1.3.1.1.3.2.These two Theorems in the present paper im-pove and extend those corresponding theorems .Moreover.its proof methods arc different from theirs.
In section two.we ('insider the nonlinear Frdholm integral equation in Ba-nach spaces
is a real parameter.
For the sake of convenience, we list the follwing conditions:
,satisfies the Caratheodory condition,i.e. for E , H(t, s, x) is measurable in s and continous in x for almost s J.
(H2) For Vr > 0, H is boundary on J x J x Tr,so is Ik(k = 1. 2. ...... m) on Tr;
(H3) ds is continous in t,and is uniformly continous in
here. M = sup {h(t.s)}.
771
Theroem 2.3.1 Assume the conditions (H1) - (H5) are satisfied.the equation (2.1.1) has solutions in PC{J.E].
Remark 2.3.2 When cone P is regular, any ordered sector is bound-
ary.Thus the BH of (III) in the Theorem 2.3.1 is satisfied automatically.when (H5) can be omitted.
Let H(t.s.x(s)) admit a decomposition of the form
H(t.s.x(s)) = Hi(t.s.x(s)) + H2(t.s.jr(s)) (2.3.24)
Definition 2.3.1 If there exist
郭大钧先生在专著[7]中对非线性泛函分析的几个重要课题及其应用，诸如典型的非线性算子、Hammerstein积分方程、常、偏微分方程、迁移方程、锥理论及非线性算子方程的正解、非线性算子拓扑度和不动点定理以及固有值、解的个数与分支，都做了系统的概括和总结．[1]包括的内容非常多，它包含了非线性泛函分析这一领域各个方面的成果．
本文利用锥理论、单调迭代技术、锥拉伸压缩不动点原理、上下解方法、不等式迭代技术、M(?)nch不动点理论等，研究了几类微分方程解的情况．所得结果或是新的，或是采用新方法在更弱条件下推广和改进以前的结果．
本文共分四节．
第一节我们讨论了实Banach空间(E，儿㈨)中下列非线性混合型微分-
积分方程
{u=f苟t,t，Tu，Su)，t I,k =C[Do，R+)， (1．1．1)
其中
(Tu)(t)=人k(t，s)u(s)ds，t I,K AC[Do2，R+]；
(Su)(t)=九h(t，s)u(s)ds，t／I，HC[I×I2,R+]．
这里H一二m．－丁）、o；＝｛f．｝》EHl ＜爿＜I＜叫l（；＝二11＊二川人叫：
f．XEDO｝．人0＝。。。＊x…U．人。：I．／E川．
我们需要以下条件：
（HI） 存在no E厂p．厂二是1P（1） 的下解＠ ＊。1二；S．f《f。l【j·T。I；。·从j＊》。
。。（／》S．I”；；：
。尸叫） 三任给4壬＊ 上＊厂三厂＝｛｛／ECJ厂 U＞／／；；｝回JI＜l都有
／（f。T。从）一人I。1．TI．＼U）＞一」八。一川一‘＼（T卜－。。l
其中－＼I．X三口是满足引理中条件 n或（i）的常数；
（H3） 存在L＊＊esp且＊可积函数＊川．尸川回Q卜）三＊使任意卜．含E D．
；！三。都有
J（Z．乙·．＊＊．JI·卜J（I．＊～TI．JU）＜L（川＊一U）十P（重）（T（L’一U》十Q（）尸（。·一U》·
定理 1．3．1设厂是 BQnQCh空间，P是 E中的正规锥．假设（HI）（HZ）（H3）
成立，则11。T．1．1）在D中存在唯一解。”，且对任慧。。E D，迭代序列
，l、＿一＊It＿，J｛I＿＿，J＿＼I门口．．＼I。＼／口。。V＿八、、J、．．J。＼
ti，。《t）＝6“”“110十J V（S，11”nl（）八IW。一l八S）八b川。一l八S川十 j’1出。一八SI
一NT（u＊一＊，＊一）s）e人”ds｝（1．3．l）
在I上依E中范数一致收敛于。＊t），并且有误差估计
A”
帅＊一w＊【【＊S人＊＊I卜＊一＊＊11＊＋人0一卜1一＊口11＊＊ ＊三N．（I．3．2）
”－’‘－”“‘”””’‘一””‘’“”l 人““”’“’‘
其中 01二 e一‘”仁口＋亿rt，0。（S）；（T＊。）（8），（S＊0）（S））一 N（Tuo）（。）］eM”ds）．
定理 1．3．2设 E是空间，P是中的正规锥，若 f满足下列假设：
（凤）’存在＊口Ec＊贝是＊＊＊1．1．二）的上解，即v＊’三／（土＊＊，＊＊口；仇＊）；
UO（ 三工。；
（HZ）’同（HZ）；
4
占
（H／ 同（从）．
则n一P：1回1川在D二毛E＊厂厂11仑，三互，：可｝中存在唯一解乙上寡回且对任意
；I·（，ED．迭代序列
11＿Ill72t ti，、。IlllSll．llslllll－ill．＼ll．3II’、IllSllW！Ill．IISI
一八（I（·。一乙1、可；一l）（人）］了1厂水d／｝
在 I上依厂中范数一致收敛于；l·．（1）．并且有误差估计
／l，一〔l、．＜儿、入（h 一；．．一＼————I，一l；．门＞＼
”—””””””l＜““”一
其中江·l二。一·＼’｛TO十厂／（J回。、C；（。）．〔T。·0《＞多．（J。，）仁上》一回＼（T。·0）（叫I可·土’A。／叶．
注 l．3二 文问定理4的一个关健条件为（H汁存在常数 R．r三（）使得
任给t EI间。咆灭·Eb丐u。0工雹U S。·都有
／（I。，T。，）一f儿。I回T。1）＜D卜一?
Nonlinear Functional Analysis is a subject .old but fashionable.Its abundant theories and advanced methods are providing powerful and fruitful tools in solving ever increasing nonlinear problems in the fields of science and technology. Though the theories of integral and differential equations in Banach spaces, as new branches of Nonlinear Functional Analysis.have developed for no more than thirty years, they are finding extensive applications in such domains as the critical point theory,the theory of partial differential equa-tions,eigenvalue problems.and so on,are attracting much more attentions from both pure and applied mathematicians.
Professor Guo Dajun has summarized in his work [7] .such several important tasks and theirs application of Nonlinear Functional Analysis as typical nonlinear operators,Hammerstein integral operations,ordinarily and partially differential equations.the cone theory,the positive solutions of nonlinear operator equations,the number and the branch of solutions,and so on.Reference [1] includes all levels of results of the domain such as Nonlinear Functional Analysis.
The present thesis employs the cone theory,monotone iterative technique, the conical expansion and compression principle,the method of upper and lower solutions,the Monch theory of fixed point,and so on,to investigate the existence of solutions of sevral differential equations .The obtained results are either new
or intrinsically generalize and improve the previous relevant ones under weaker conditions.
The paper is divided into four sections.
In section one we investigate the following nonlinear integro-differential equation in Banach spaces.
where.
We list below the conditions:
(H1) There exists as the lower solution of IVP(1).i.e.
satisfy
where, M. .V > 0 are constants satisfying such conditions as (i)(ii) of Lemma 1.2.1.
(H3) There exist Lebesgue integrable functions L(t), P(t),Q(t) > 0,such that make tt, v € D, u
10
solution .and the iterative sequence
(1.3.1)
converges uniformly on I to w(t). Moreover. there is error estimate
Theorem 1.3.2 Let E be a Banach space and P be a regular cone in E. Assume that / satisfys the following assumptions:
(H1)t There exists TO 6 C1[I. E] as a upper solution of
Then /YP(l.l.l) possesses only one solution w, and D, the iterative sequence
converges uniformly on I to w(t). Moreover,there is error estimate
Remark 1.3.1 The key condition of Theorem 4 in [5] is (H4):There exist constants R, r > 0,such that
the theorems in the present paper changes B.r into only integarable and non-negative functions L(t).P(1). broadening this formula and extending this theorem.
Remark 1.3.4 Those1 results presented in the given refrenccs cann't induce the Theorem 1.3.1.1.3.2.These two Theorems in the present paper im-pove and extend those corresponding theorems .Moreover.its proof methods arc different from theirs.
In section two.we ('insider the nonlinear Frdholm integral equation in Ba-nach spaces
is a real parameter.
For the sake of convenience, we list the follwing conditions:
,satisfies the Caratheodory condition,i.e. for E , H(t, s, x) is measurable in s and continous in x for almost s J.
(H2) For Vr > 0, H is boundary on J x J x Tr,so is Ik(k = 1. 2. ...... m) on Tr;
(H3) ds is continous in t,and is uniformly continous in
here. M = sup {h(t.s)}.
771
Theroem 2.3.1 Assume the conditions (H1) - (H5) are satisfied.the equation (2.1.1) has solutions in PC{J.E].
Remark 2.3.2 When cone P is regular, any ordered sector is bound-
ary.Thus the BH of (III) in the Theorem 2.3.1 is satisfied automatically.when (H5) can be omitted.
Let H(t.s.x(s)) admit a decomposition of the form
H(t.s.x(s)) = Hi(t.s.x(s)) + H2(t.s.jr(s)) (2.3.24)
Definition 2.3.1 If there exist
引文
[1]K.Deimling.Nonlinear Functional Analysis.Springer-Verlag.Berlin.1985.
[2]郭大钧, Lakshmikantham V.Liu X.Nonlinear Integral Equations in Abstract Space.Boston:Kluwer Academic Publishers.1996.
[3]Du.S.W.and Lakshmikanthan.V.Monotone iterative for differentiate equations in Banach space.J.Math.Appl.87(1982):454-459.
[4]孙经先．Banach空间微分方程的解．数学学报，33(1990)：374-380．
[5]刘立山．Banach空间非线性混合型微分-积分方程的解．数学学报，1995．38(16)：721-731．
[6]郭大钧. Initial value problems for integro-differential equtions of volterra type in Banach spaces.J.Apple.Math.Stochstic Anal.7(1994):13-23.
[7]郭大钧．非线性泛函分析．山东科技出版社，济南，1985．
[8]刘立山. Iterative method for solutions and coupled quasi-solutions of nonlinear integro-differential equations of mixed type in Banach spaces.Nonlinear Analysis,42(2000):583-598
[9]柴国庆，胡松林．Banach空间微分方程解的存在与唯一性．数学研究，2000，33(4)：418-425．
[10]郭大钧，孙经先．抽象空间常微分方程．山东科技出版社，济南，1985．
[11]郭大钧．非线性分析中的半序方法．山东科技出版，社济南，2000．
[12]Erbe L H,Liu,X.Qusi-solutions of nonlinear impulsive equtions in abstract spaces.Appl.Anal.,34(1989),231-250.
[13] Lakshmidantham V,Liu Xin Zhi.Nonlinear integral equtions in abstrct spaces.Kluwer Academic Publishers.Dordrecht.1996.
[14] Guo Dajun.Impulsive integral equations in Banach spaces and appliations.J Appl.Math Stpchastic Appl.,1992,5:111-122.
[15] Guo Dajun.Extremal solutions of nonlinear Fredholm integral equtions in orderded Banach spaces.Northeastern Math J..1991.7(4):416-423.
[16] Deimling K.Nonlinear function analysis.Berlin:Springer-Verlag.1985.
[17] 戚仕硕．博士研究生学位论文．2000．
[18] GuoDajun.Extremal solutions of nonlinear Fredholm integral equtions in orderded Banach spaces.Northeastern.Math J..1991.7(4):416-423.
[19] 张小美. Multiple nonnegtive solutions of fourth order ordinary differential equations.Ann of Diff Eqs,2000(4),404-413.
[20] Ma Ruyun.On the existence of positive solutions of fourth order ordinary differential equations.Applicable Analysis,1995(59),225-231.
[21] 韦忠礼．四阶奇异边值问题的正解．数学学报，1999(7)，225-231．
[22] 郭大钧．关于锥映象的几个不动点定理．科学通报，1983(28)，1217-1219．
[23] RACHUNKVA I and STANKE S.Topologyical degree method in functional boundary value problems at resonance [J].Nonlinear Analysis,J.M.A.,1996,27(3):271-285.
[24] WANG Haiyan.On the existence of positive solutions for semilinear elliptic equations in the annuls [J].J.Differential equations,1994,109:1-7.
[25] 蒋达清，刘辉昭．二阶微分方程Neumann边值问题正解的存在性．Journal of Mathematics Research and Exposition,2000,20(3):360-364.
[26]Guo Dajun.Some fixed points theorems on cone maps.Kexue Tongbao.1984.29(5):575-578.
[27]马如云．奇异二阶边值问题的正解．数学学报，1998．41(6)：1225-1230．
[28]Du.S.W.and Lakshmikantham.V..Monotone iterative for differential equations in Banach spaces.J Math.Anal.Appl.,87(1982).454-459.
[29]Guo Dajun.Solutions of nonlinear integro-differential Equations of mixed type in Banach spaces.J.Apple.Math.Sima..2:1(1989).1-11.
[30]Erbe L H,Lir X.Qusi solutions of nonlinear impulsive equations in abstract spaces.Appl.Anal.,34(1989).231-250.
[31]路慧芹，刘立山．Banach空间非线性脉冲Volterra型积分方程整体解的存在性定理及应用．数学物理学报，2000．20(1)：101-108．
[32]Vaughn,R.Existence and compararion results for nonlinear Volterra integral equations in a Banach space.Appl.Anal.,7(1978),337-348.
[33]Tallaferro S D.A nonlinear boudary value problem [J].Nonlinear Analysis(TMA),1979,3(6),623-632.
[34]赵增勤．非线性奇异微分方程边值问题的正解．数学学报，2000，43(1)，179-188．
[35]赵增勤一类奇异次线性边值问题正解存在的充分必要条件．数学学报，5(41)1998，1025-1034．
[36]毛安民．正指数超线性Emdent-Fowlet方程奇异边值问题的正解．数学学报，2000，43(4)，623-632．
[37]钱爱侠．Banach空间非线性混合型微分-积分方程解的存在唯一性．已通过．
[38]钱爱侠．Banach空间非线性脉冲Fredholm型积分方程的解．已投．
[39]钱爱侠．奇异超线性四阶边值问题的正解，待发表．
[40]钱爱侠．奇异二阶Neumann边值问题的正解．已通过．
[2]郭大钧, Lakshmikantham V.Liu X.Nonlinear Integral Equations in Abstract Space.Boston:Kluwer Academic Publishers.1996.
[3]Du.S.W.and Lakshmikanthan.V.Monotone iterative for differentiate equations in Banach space.J.Math.Appl.87(1982):454-459.
[4]孙经先．Banach空间微分方程的解．数学学报，33(1990)：374-380．
[5]刘立山．Banach空间非线性混合型微分-积分方程的解．数学学报，1995．38(16)：721-731．
[6]郭大钧. Initial value problems for integro-differential equtions of volterra type in Banach spaces.J.Apple.Math.Stochstic Anal.7(1994):13-23.
[7]郭大钧．非线性泛函分析．山东科技出版社，济南，1985．
[8]刘立山. Iterative method for solutions and coupled quasi-solutions of nonlinear integro-differential equations of mixed type in Banach spaces.Nonlinear Analysis,42(2000):583-598
[9]柴国庆，胡松林．Banach空间微分方程解的存在与唯一性．数学研究，2000，33(4)：418-425．
[10]郭大钧，孙经先．抽象空间常微分方程．山东科技出版社，济南，1985．
[11]郭大钧．非线性分析中的半序方法．山东科技出版，社济南，2000．
[12]Erbe L H,Liu,X.Qusi-solutions of nonlinear impulsive equtions in abstract spaces.Appl.Anal.,34(1989),231-250.
[13] Lakshmidantham V,Liu Xin Zhi.Nonlinear integral equtions in abstrct spaces.Kluwer Academic Publishers.Dordrecht.1996.
[14] Guo Dajun.Impulsive integral equations in Banach spaces and appliations.J Appl.Math Stpchastic Appl.,1992,5:111-122.
[15] Guo Dajun.Extremal solutions of nonlinear Fredholm integral equtions in orderded Banach spaces.Northeastern Math J..1991.7(4):416-423.
[16] Deimling K.Nonlinear function analysis.Berlin:Springer-Verlag.1985.
[17] 戚仕硕．博士研究生学位论文．2000．
[18] GuoDajun.Extremal solutions of nonlinear Fredholm integral equtions in orderded Banach spaces.Northeastern.Math J..1991.7(4):416-423.
[19] 张小美. Multiple nonnegtive solutions of fourth order ordinary differential equations.Ann of Diff Eqs,2000(4),404-413.
[20] Ma Ruyun.On the existence of positive solutions of fourth order ordinary differential equations.Applicable Analysis,1995(59),225-231.
[21] 韦忠礼．四阶奇异边值问题的正解．数学学报，1999(7)，225-231．
[22] 郭大钧．关于锥映象的几个不动点定理．科学通报，1983(28)，1217-1219．
[23] RACHUNKVA I and STANKE S.Topologyical degree method in functional boundary value problems at resonance [J].Nonlinear Analysis,J.M.A.,1996,27(3):271-285.
[24] WANG Haiyan.On the existence of positive solutions for semilinear elliptic equations in the annuls [J].J.Differential equations,1994,109:1-7.
[25] 蒋达清，刘辉昭．二阶微分方程Neumann边值问题正解的存在性．Journal of Mathematics Research and Exposition,2000,20(3):360-364.
[26]Guo Dajun.Some fixed points theorems on cone maps.Kexue Tongbao.1984.29(5):575-578.
[27]马如云．奇异二阶边值问题的正解．数学学报，1998．41(6)：1225-1230．
[28]Du.S.W.and Lakshmikantham.V..Monotone iterative for differential equations in Banach spaces.J Math.Anal.Appl.,87(1982).454-459.
[29]Guo Dajun.Solutions of nonlinear integro-differential Equations of mixed type in Banach spaces.J.Apple.Math.Sima..2:1(1989).1-11.
[30]Erbe L H,Lir X.Qusi solutions of nonlinear impulsive equations in abstract spaces.Appl.Anal.,34(1989).231-250.
[31]路慧芹，刘立山．Banach空间非线性脉冲Volterra型积分方程整体解的存在性定理及应用．数学物理学报，2000．20(1)：101-108．
[32]Vaughn,R.Existence and compararion results for nonlinear Volterra integral equations in a Banach space.Appl.Anal.,7(1978),337-348.
[33]Tallaferro S D.A nonlinear boudary value problem [J].Nonlinear Analysis(TMA),1979,3(6),623-632.
[34]赵增勤．非线性奇异微分方程边值问题的正解．数学学报，2000，43(1)，179-188．
[35]赵增勤一类奇异次线性边值问题正解存在的充分必要条件．数学学报，5(41)1998，1025-1034．
[36]毛安民．正指数超线性Emdent-Fowlet方程奇异边值问题的正解．数学学报，2000，43(4)，623-632．
[37]钱爱侠．Banach空间非线性混合型微分-积分方程解的存在唯一性．已通过．
[38]钱爱侠．Banach空间非线性脉冲Fredholm型积分方程的解．已投．
[39]钱爱侠．奇异超线性四阶边值问题的正解，待发表．
[40]钱爱侠．奇异二阶Neumann边值问题的正解．已通过．