常微分方程多点边值问题的正解
|
摘要
常微分方程多点边值问题起源于许多不同的应用数学和物理领域,例如:由N部分不同密度组成的均匀截面的悬链线的振动可以转化为多点边值问题;弹性稳定性理论的许多问题也可转化为多点边值问题处理.对多点边值问题的研究,始于二十世纪八十年代,由Il'in和Moiseev首次对二阶线性常微分方程进行研究.到二十世纪九十年代,Gupta开始讨论二阶非线性常微分方程三点边值问题,此后,许多作者研究了更一般的非线性多点边值问题,并取得了丰富的成果.
对常微分方程来说,正解往往是人们注重研究的符合现实意义的一类解,人们常将研究微分方程正解的存在性问题转化为研究积分算子在锥上的不动点的存在性问题.研究积分算子不动点的存在性常用的理论是非线性泛函分析的度理论和不动点指数理论,其中最常用的定理是:Schauder不动点定理,Krasnosel'skii不动点定理,Leggett-Williams不动点定理和它的一般化—五个泛函不动点定理.
尽管很多学者应用上述定理对多点边值问题正解的存在性进行研究,并取得了丰富的成果.但由于使用这些常见的不动点定理需要假设非线性项是连续的,且Green函数需满足特定的条件要求,使得这些常用的定理适用范围具有一定的局限性.因此,仍然存在许多未解决的具有挑战性的问题.
抽象空间中的积微分方程理论是近三十年发展起来的一个重要的分支,现在已有专著多部.对于抽象空间中边值问题的研究始于二十世纪七十年代,但由于在抽象空间中研究积分算子不动点的存在性具有很大困难,使得这一领域的研究进程非常缓慢.目前未涉及到的需要研究的问题有很多.针对以上这些问题,本文分以下五个方面进行研究:
1.对二阶和三阶微分方程边值问题,我们打破常规限制,采用新的方法和技巧,分别证明非线性项中含有一阶导数且非线性项可在[0,1]中任一点具有奇性以及在定义域内可以具有不连续性的二阶和三阶多点边值问题正解的存在性.我们使用的工具是Krein-Rutman定理和不动点指数理论.该结果扩展了现有的一些结论.
2.我们利用H.Amnn不动点定理,研究非线性项中含有一阶导数,且可在[0,1]中任意一点具有奇性以及在定义域内具有不连续性的带有p-Laplacian算子的多点边值问题多个正解的存在性.这个新的方法和技巧,消除了目前多数结论必须以非线性项在定义域内连续作为条件的限制.
3.对具体空间中(k,n-k)共轭边值问题的研究已经取得了很多的成果.然而,就我们所知,对抽象空间中的这一问题还未有研究.本文通过构造适当的锥,利用锥上严格集压缩算子不动点定理,研究抽象空间中积分算子A(u(t)):=integral from n=0 to 1 G(t,s)f(s,u(s))ds的一个、两个和多个不动点存在性.然后,利用这一结论研究抽象空间中(k,n-k)共轭边值问题的一个、两个和多个正解的存在性,并且这一结论可用于一类抽象空间中的边值问题研究.
4.对于高阶多点边值问题,我们做了如下研究:
(1)利用五个泛函不动点定理研究非线性项中含有各阶导数的2n阶多点边值问题至少三个正解的存在性;
(2)利用五个泛函不动点定理研究非线性项中含有各阶导数的n阶多点边值问题多个正解的存在性.
5.在抽象空间中,我们还研究了以下三种类型的二阶边值问题:
(1)利用Darbo不动点定理,通过构造适当的锥,首先给出一个积分算子的不动点的存在和不存在条件,然后利用该结论研究抽象空间中具有非齐次边界条件的二阶两点和多点边值问题正解的存在性和不存在性条件;
(2)利用锥上严格集压缩算子不动点定理,我们研究抽象空间中具有共振的二阶多点边值问题多个正解的存在性.该问题的特点是相应的齐次边值问题具有非零解,即共振性,从而不能用通常的方法来研究.难点在于找出与其等价的一个非共振边值问题的Green函数所满足的不等式;
(3)对于抽象空间中二阶两点边值问题正解的存在性,在锥是正规的条件下已有研究.我们采用新的方法和技巧,去掉这一条件限制,证明了该边值问题正解的存在性,使得我们所得结论的使用范围更广.
The multi-point boundary value problems for ordinary differential equations arise in a variety of different areas of applied mathematics and physics.For example,the vibrations of a guy wire of uniform cross-section and composed of N parts of different densities can be set up as a multi-point boundary value problem;many problems in the theory of elastic stability can be handled by the method of multi-point problems. The study of multi-point boundary value problems for linear second-order ordinary differential equations was initiated by Il'in and Moiseev in 1980s.In 1990s,Gupta began discussing three-point boundary value problems of second-order nonlinear ordinary differential equations.Since then,many authors have studied more general nonlinear multi-point boundary value problems and obtained many achievements.
For ordinary differential equations,positive solutions are always attached importance to and are a kind of practical solutions.The study of existence of positive solutions to differential equations is often transformed into investigating the existence of fixed points for integral operators on a cone.The theories most used for the research into the existence of fixed points for integral operators are that of the degree of nonlinear functional analysis and that of fixed point index.And the theorems widely used are Schauder fixed point theorem,Krasnosel'skii fixed point theorem,Leggett-Williams fixed point theorem and its generalization—the five functional fixed point theorem.
Despite the fact that many authors have studied the existence of positive solutions by these theorems and obtained many achievements,in order to use these familiar fixed point theorems,we need to assume that the nonlinear terms are continuous and the Green's functions must satisfy given conditions,which makes the range of using these theorems restricted,so there are yet many challenging questions to be solved.
The theory of differential-integral equations in abstract space has been an important branch in the past 30 years.There are several monographs on this theory.The boundary value problems in abstract space have been researched since 1970s.But the development of the study in this field is very slow because the study of existence of fixed point for integral operators in abstract space is very difficult.At present,there are many problems which have not been dealt with but need to be studied.Aiming at these problems,in the following,we will study them in five aspects.
1.By using new methods and skills,we prove the existence of positive solutions to second-order and three-order multi-point boundary value problems depending on the first-order derivative,respectively,in which the nonlinear term may be singular at any point of[0,1]and non-continuous in its domain.The main tools used are Krein-Rutman theorem and fixed point index theory.This result extends some of the existing results.
2.By using H.Amnn fixed point theorem,the existence of multiple positive solutions to multi-point boundary value problems with p-Laplacian is discussed.In the problem,the nonlinear term contains the first-order derivative,which may be singular at any point of[0,1]and may be non-continuous in its domain.At present,in most results,the nonlinear terms must be continuous in its domain.This new method and skill eliminate this restriction.
3.In concrete spaces,the study on(k,n-k)conjugate boundary value problems has obtained many achievements.But,as far as we know,the study for these problems in abstract spaces hasn't been done.By constructing suitable cone,we research the existence of one,two and multiple fixed points for the integral operator A(u(t)):= integral from n=0 to 1 G(t,s)f(s,u(s))ds in abstract space by using fixed point theorem in a cone for strict set contraction operators.Then,using the obtained result,we investigate the existence of one,two and multiple positive solutions to the(k,n-k)conjugate boundary value problems in abstract space.And this result can be used to study a kind of boundary value problems in abstract space.
4.For high-order multi-point boundary value problems,we study them as follows:
(1)Using the five functional fixed point theorem,we discuss the existence of at least three positive solutions to 2n-order multi-point boundary value problem with all derivatives.
(2)Using the five functional fixed point theorem,we discuss the existence of multiple positive solutions to n-order multi-point boundary value problem with all derivatives.
5.In abstract spaces,we also discuss the following three kinds of second-order boundary value problems.
(1)Utilizing Darbo fixed point theorem,we firstly give the conditions of existence and non-existence of fixed point for an integral operator by constructing suitable cone. Then,applying the obtained results,we study the conditions of existence and non-existence for second-order two-point and multi-point boundary value problems with non-homogeneous boundary conditions in abstract space,respectively.
(2)Using fixed point theorem in a cone for strict set contraction operators,we research the existence of multiple positive solutions for second-order multi-point boundary value problem at resonance in abstract space.The characteristic of this problem is that the corresponding homogeneous boundary value problem possesses nonzero solutions, i.e.,resonance.So,it can't be studied by using general methods.The difficulty lies in finding of the inequality satisfied by the Green's function of a non-resonance boundary value problem which is equal to the boundary value problem at resonance.
(3)When the cone is normal,the existence of positive solutions to second-order two-point boundary value problems in abstract space has been researched.Eliminating this restriction,we prove the existence of positive solutions to this problem by using new methods and skills,which makes the range of using our result wider.
对常微分方程来说,正解往往是人们注重研究的符合现实意义的一类解,人们常将研究微分方程正解的存在性问题转化为研究积分算子在锥上的不动点的存在性问题.研究积分算子不动点的存在性常用的理论是非线性泛函分析的度理论和不动点指数理论,其中最常用的定理是:Schauder不动点定理,Krasnosel'skii不动点定理,Leggett-Williams不动点定理和它的一般化—五个泛函不动点定理.
尽管很多学者应用上述定理对多点边值问题正解的存在性进行研究,并取得了丰富的成果.但由于使用这些常见的不动点定理需要假设非线性项是连续的,且Green函数需满足特定的条件要求,使得这些常用的定理适用范围具有一定的局限性.因此,仍然存在许多未解决的具有挑战性的问题.
抽象空间中的积微分方程理论是近三十年发展起来的一个重要的分支,现在已有专著多部.对于抽象空间中边值问题的研究始于二十世纪七十年代,但由于在抽象空间中研究积分算子不动点的存在性具有很大困难,使得这一领域的研究进程非常缓慢.目前未涉及到的需要研究的问题有很多.针对以上这些问题,本文分以下五个方面进行研究:
1.对二阶和三阶微分方程边值问题,我们打破常规限制,采用新的方法和技巧,分别证明非线性项中含有一阶导数且非线性项可在[0,1]中任一点具有奇性以及在定义域内可以具有不连续性的二阶和三阶多点边值问题正解的存在性.我们使用的工具是Krein-Rutman定理和不动点指数理论.该结果扩展了现有的一些结论.
2.我们利用H.Amnn不动点定理,研究非线性项中含有一阶导数,且可在[0,1]中任意一点具有奇性以及在定义域内具有不连续性的带有p-Laplacian算子的多点边值问题多个正解的存在性.这个新的方法和技巧,消除了目前多数结论必须以非线性项在定义域内连续作为条件的限制.
3.对具体空间中(k,n-k)共轭边值问题的研究已经取得了很多的成果.然而,就我们所知,对抽象空间中的这一问题还未有研究.本文通过构造适当的锥,利用锥上严格集压缩算子不动点定理,研究抽象空间中积分算子A(u(t)):=integral from n=0 to 1 G(t,s)f(s,u(s))ds的一个、两个和多个不动点存在性.然后,利用这一结论研究抽象空间中(k,n-k)共轭边值问题的一个、两个和多个正解的存在性,并且这一结论可用于一类抽象空间中的边值问题研究.
4.对于高阶多点边值问题,我们做了如下研究:
(1)利用五个泛函不动点定理研究非线性项中含有各阶导数的2n阶多点边值问题至少三个正解的存在性;
(2)利用五个泛函不动点定理研究非线性项中含有各阶导数的n阶多点边值问题多个正解的存在性.
5.在抽象空间中,我们还研究了以下三种类型的二阶边值问题:
(1)利用Darbo不动点定理,通过构造适当的锥,首先给出一个积分算子的不动点的存在和不存在条件,然后利用该结论研究抽象空间中具有非齐次边界条件的二阶两点和多点边值问题正解的存在性和不存在性条件;
(2)利用锥上严格集压缩算子不动点定理,我们研究抽象空间中具有共振的二阶多点边值问题多个正解的存在性.该问题的特点是相应的齐次边值问题具有非零解,即共振性,从而不能用通常的方法来研究.难点在于找出与其等价的一个非共振边值问题的Green函数所满足的不等式;
(3)对于抽象空间中二阶两点边值问题正解的存在性,在锥是正规的条件下已有研究.我们采用新的方法和技巧,去掉这一条件限制,证明了该边值问题正解的存在性,使得我们所得结论的使用范围更广.
The multi-point boundary value problems for ordinary differential equations arise in a variety of different areas of applied mathematics and physics.For example,the vibrations of a guy wire of uniform cross-section and composed of N parts of different densities can be set up as a multi-point boundary value problem;many problems in the theory of elastic stability can be handled by the method of multi-point problems. The study of multi-point boundary value problems for linear second-order ordinary differential equations was initiated by Il'in and Moiseev in 1980s.In 1990s,Gupta began discussing three-point boundary value problems of second-order nonlinear ordinary differential equations.Since then,many authors have studied more general nonlinear multi-point boundary value problems and obtained many achievements.
For ordinary differential equations,positive solutions are always attached importance to and are a kind of practical solutions.The study of existence of positive solutions to differential equations is often transformed into investigating the existence of fixed points for integral operators on a cone.The theories most used for the research into the existence of fixed points for integral operators are that of the degree of nonlinear functional analysis and that of fixed point index.And the theorems widely used are Schauder fixed point theorem,Krasnosel'skii fixed point theorem,Leggett-Williams fixed point theorem and its generalization—the five functional fixed point theorem.
Despite the fact that many authors have studied the existence of positive solutions by these theorems and obtained many achievements,in order to use these familiar fixed point theorems,we need to assume that the nonlinear terms are continuous and the Green's functions must satisfy given conditions,which makes the range of using these theorems restricted,so there are yet many challenging questions to be solved.
The theory of differential-integral equations in abstract space has been an important branch in the past 30 years.There are several monographs on this theory.The boundary value problems in abstract space have been researched since 1970s.But the development of the study in this field is very slow because the study of existence of fixed point for integral operators in abstract space is very difficult.At present,there are many problems which have not been dealt with but need to be studied.Aiming at these problems,in the following,we will study them in five aspects.
1.By using new methods and skills,we prove the existence of positive solutions to second-order and three-order multi-point boundary value problems depending on the first-order derivative,respectively,in which the nonlinear term may be singular at any point of[0,1]and non-continuous in its domain.The main tools used are Krein-Rutman theorem and fixed point index theory.This result extends some of the existing results.
2.By using H.Amnn fixed point theorem,the existence of multiple positive solutions to multi-point boundary value problems with p-Laplacian is discussed.In the problem,the nonlinear term contains the first-order derivative,which may be singular at any point of[0,1]and may be non-continuous in its domain.At present,in most results,the nonlinear terms must be continuous in its domain.This new method and skill eliminate this restriction.
3.In concrete spaces,the study on(k,n-k)conjugate boundary value problems has obtained many achievements.But,as far as we know,the study for these problems in abstract spaces hasn't been done.By constructing suitable cone,we research the existence of one,two and multiple fixed points for the integral operator A(u(t)):= integral from n=0 to 1 G(t,s)f(s,u(s))ds in abstract space by using fixed point theorem in a cone for strict set contraction operators.Then,using the obtained result,we investigate the existence of one,two and multiple positive solutions to the(k,n-k)conjugate boundary value problems in abstract space.And this result can be used to study a kind of boundary value problems in abstract space.
4.For high-order multi-point boundary value problems,we study them as follows:
(1)Using the five functional fixed point theorem,we discuss the existence of at least three positive solutions to 2n-order multi-point boundary value problem with all derivatives.
(2)Using the five functional fixed point theorem,we discuss the existence of multiple positive solutions to n-order multi-point boundary value problem with all derivatives.
5.In abstract spaces,we also discuss the following three kinds of second-order boundary value problems.
(1)Utilizing Darbo fixed point theorem,we firstly give the conditions of existence and non-existence of fixed point for an integral operator by constructing suitable cone. Then,applying the obtained results,we study the conditions of existence and non-existence for second-order two-point and multi-point boundary value problems with non-homogeneous boundary conditions in abstract space,respectively.
(2)Using fixed point theorem in a cone for strict set contraction operators,we research the existence of multiple positive solutions for second-order multi-point boundary value problem at resonance in abstract space.The characteristic of this problem is that the corresponding homogeneous boundary value problem possesses nonzero solutions, i.e.,resonance.So,it can't be studied by using general methods.The difficulty lies in finding of the inequality satisfied by the Green's function of a non-resonance boundary value problem which is equal to the boundary value problem at resonance.
(3)When the cone is normal,the existence of positive solutions to second-order two-point boundary value problems in abstract space has been researched.Eliminating this restriction,we prove the existence of positive solutions to this problem by using new methods and skills,which makes the range of using our result wider.
引文
[1]Agarwal R P,O'Regan D.Multiplicity results for singular conjugate,focal,and (N,P) problems[J].J.Differential Equations,2001,170:142-156.
[2]Agarwal R P,O'Regan D.Positive solutions for (p,n-p) conjugate boundary value problems[J].J.Differential Equations,1998,150:462-473.
[3]Agarwal R P,Wong F H.Existence of solutions to (k,n — k — 2) boundary value problems[J].Appl.Math.Comput.,1999,104:33-50.
[4]Agarwal R P,Bohner M,Wong P J Y.Positive solutions and eigenvalues of conjugate boundary value problems[J].Proc.Edinburgh.Math.Soc,1999,42:349-374.
[5]Agarwal R,L ü H,O'Regan D.Existence theorems for the one-dimensional singular p-Laplacian equation with sign changing nonlinearities[J].Appl.Math.Comput.,2003,143:15-38.
[6]Agarwal R P,L ü H,O'Regan D.A necessary and sufficient condition for the existence of positive solutions to the singular p-Laplacian[J].Z.Anal.Anwend.,2003,22:689-709.
[7]Agarwal R P,O'Regan D,Positive solutions to superliner singular boundary value problems[J],J.Comput.Appl.Math.,1998,88:129-147.
[8]Agarwal R P,O'Regan D,Existence theory for single and multiple solutions to singular position boundary value problems[J],J.Differential Equations,2001,175:393-414.
[9]Agarwal R P,Grace S R,O'Regan D.Semipositone higher-order differential equations[J].Appl.Math.Lett.,2004,17:201-207.
[10]Alarc (?) n V,Amat S,Busquier S,L (?) pez D J.A Steffensen's type method in Banach spaces with applications on boundary-value problems[J].J.Comput.Appl.Math.,2008,216(1):243-250.
[11]Amann H.Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces[J].SIAM Review,1976,18:620-709.
[12]Anderson D,Anderson T,Kleber M.Green's function and existence of solutions for a functional focal differential equation[J].Electron.J.Differ.Equat,2006,12:1-14.
[13]Anderson D R.Positivity of Green's function for an n-point right focal boundary value problem on measure chains[J].Math.Comput.Modelling,2000,31:29-50.
[14]Anderson D,Avery R.Multiple positive solutions to a third-order discrete focal boundary value problem[J].Comput.Math.Appl.,2001,42:333-340.
[15]Anderson D R.Green's function for a third-order generalized right focal problem[J].J.Math.Anal.Appl.,2003,288:1-14.
[16]Anderson D R,Davis J M.Multiple solutions and eigenvalues for three-order right focal boundary value problems[J].J.Math.Anal.Appl.,2002,267:135-157.
[17]Ashyralyev A.A note on the Bitsadze-Samarskii type nonlocal boundary value problem in a Banach space[J].J.Math.Anal.Appl.,2008,344(1):557-573.
[18]Avery R I.A generalization of the Leggett-Williams fixed point theorem.Math.Sci.Res.Hot-Line,1999,3:9-14.
[19]Bai Z,Fei X.Existence of triple positive solutions for a third order generalized right focal problem[J].Math.Inequal.Appl.,2006,9(3):437-444.
[20]Cabada A,Minhos F,Santos A I.Solvability for a third order discontinuous fully equation with nonlinear functional boundary conditions[J].J.Math.Anal.Appl.,2006,322:735-748.
[21]Cabada A,Pouso R L.Existence result for the problem (φ(u'))' = qf(t,u,u') with periodic and Neumann boundary conditions[J].Nonlinear Anal.,1997,30:1733-1742.
[22]Cac N P,Gatica J A.Fixed point theorems for mappings in ordered Banach spaces[J].J.Math.Anal.Appl.,1979,71:545-557.
[23]Chang Y,Li W.Existence results for second-order dynamic inclusion with mpoint boundary value conditions on time scales[J].Appl.Math.Lett.,2007,20:885-891.
[24]Chen H,Li P.Existence of solutions of three-point boundary value problems in Banach spaces[J].Math.Comput.Model.,In Press.
[25]Chen H,Li P.Three-point boundary value problems for second-order ordinary differential equations in Banach spaces[J].Comput.Mathe.Appl.,2008,56(7):1852-1860.
[26]Cui Y,Zou Y.Positive solutions of nonlinear singular boundary value problems in abstract spaces[J].Nonlinear Anal.,2008,69(1):287-294.
[27]Davis J M,Eloe P W,Henderson J.Triple positive solutions and dependence on higher order derivatives[J].J.Math.Anal.Appl.,1999,237:710-720.
[28]Davis J M,Henderson J,P.J.Y.Wong.General Lidstone problems:multiplicity and symmetry of solutions[J].J.Math.Anal.Appl.,2000,251:527-548.
[29]Demling K.Ordinary Differential Equations in Banach Spaces[M].Springer-Verlag,Berlin,1977.
[30]Eloe P W,Henderson J.Positive solutions for (n — 1,1) conjugate boundary value problems[J].Nonlinear Anal.,1997,28:1669-1680.
[31]Eloe P W,Henderson J.Singular nonlinear (k,n-k) conjugate boundary value problems[J].J.Differential Equations,1997,133:136-151.
[32]Feng H,Ge W.Existence of three positive solutions for m-po'mt boundary-value problems with one-dimensional p-Laplacian[J].Nonlinear Anal.,to appear
[33]Feng M,Pang H.A class of three-point boundary-value problems for secondorder impulsive integro-differential equations in Banach spaces[J].Nonlinear Anal.In Press.
[34]Feng M,Ji D,Ge W.Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces[J].J.Comput.Appl.Math.,2008,222(2):351-363.
[35]Feng Y,Liu S.Solvability of a third-order two-point boundary value problem[J].Appl.Math.Lett.,2005,18:1034-1040.
[36]Gamba IM,Jungel A.Positive solutions to singular second and third order differential equations for quantum fluids[J].Arch.Ration.Mech.Anal.,2001,156(3):183-203.
[37]Gatica J A,Oliker V,Waltman P.Singular nonlinear boundary value problems for second order ordinary differential equations[J].J.Differential Equations,1989, 79:62-78.
[38]Graef J R,Yang B.Positive solutions of a nonlinear third order eigenvalue problem[J].Dynam.Syst.Appl.,2006,15(1):97-110.
[39]Graef J R,Yang B.Multiple positive solutions to a three point third order boundary value problem[J].Discrete Contin.Dyn.Syst.(Suppl.) 2005,337-344.
[40]Guo L,Sun J,Zhao Y.Existence of Positive Solutions for Nonlinear Third-Order Three-Point Boundary Value Problem[J],Nonlinear Anal.,In Press.
[41]Guo D,V.Lakshmikantham,X.Liu.Nonlinear Integral Equations in Abstract SpacesfM].Kluwer Academic,Dordrecht,1996.
[42]Guo D,Lakshmikantham V.Nonlinear Problems in Abstract Cones[M].Academic Press,San Diego,1988.
[43]Guo D,Lakshmikantham V.Multiple solutions of two-point boundary value problems of ordinary differential equations in Banach spaces[J].J.Math.Anal.Appl.,1988,129(1):211-222.
[44]Guo D.Periodic boundary value problems for second order impulsive integrodifferential equations in banach spaces[J].Nonlinear Anal.,1997,28(6):983-997.
[45]Guo D.Initial Value Problems for Nonlinear Second Order Impulsive Integrodifferential Equations in Banach Spaces[J].J.Math.Anal.Appl.,1996,200(1):1-13.
[46]Guo D.Initial value problems for second-order integro-differential equations in Banach spaces[J].Nonlinear Anal.,1999,37(3):289-300.
[47]Guo D.Boundary value problems for impulsive integro-differential equations on unbounded domains in a Banach space[J].Appl.Math.Comput.,1999,99(1):1-15.
[48]Guo D.Second order impulsive integro-differential equations on unbounded domains in Banach spaces[J].Nonlinear Anal.,1999,35(4):413-423.
[49]Guo D.Existence of solutions for nth-order integro-differential equations in Banach spaces[J].Comput.Math.Appl.,2001,41(5-6):597-606.
[50]Guo D.Minimal nonnegative solutions for nth order integro-differential equations in Banach spaces[J].Appl.Math.Comput.,2000,113(1):55-65.
[51]Guo D.Second order integro-differential equations of Volterra type on unbounded domains in Banach spaces[J].Nonlinear Anal.,2000,41(3-4):465-476.
[52]Guo D.A class of nth-order impulsive integro-differential equations in Banach spaces[J].Comput.Math.Appl.,2002,44(10-11):1339-1356.
[53]Guo D.Multiple positive solutions for first order nonlinear integro-differential equations in Banach spaces[J].Nonlinear Anal.,2003,53(2):183-195.
[54]Guo D.A boundary value problem for nth order integro-differential equations in a Banach space[J],Appl.Math.Comput.,2003,136(2-3):571-592.
[55]Guo D.Extremal solutions for nth-order impulsive integro-differential equations on the half-line in Banach spaces[J].Nonlinear Anal.,2006,65(3):677-696.
[56]Guo D.Multiple positive solutions of a boundary value problem for nth-order impulsive integro-differential equations in Banach spaces[J].Nonlinear Anal.,2005,63(4):618-641.
[57]Guo D.Multiple positive solutions for nth-order impulsive integro-differential equations in Banach spaces[J].Nonlinear Anal.,2005,60(5):955-976.
[58]Guo D.Multiple positive solutions for a class of nonlinear integro-differential equations in Banach spaces[J].Appl.Math.Comput.,2004,154(2):469-485.
[59]Guo D.Multiple positive solutions of a boundary value problem for nth-order impulsive integro-differential equations in a Banach space[J].Nonlinear Anal.,2004,56(7):985-1006.
[60]Guo D.Multiple positive solutions for first order nonlinear impulsive integrodifferential equations in a Banach space[J].Appl.Math.Comput.,2003,143(2-3):233-249.
[61]Guo D.Existence of positive solutions for nth-order nonlinear impulsive singular integro-differential equations in Banach spaces[J].Nonlinear Anal.,2008,68(9):2727-2740.
[62]Guo D.Positive solutions of an infinite boundary value problem for nth-order nonlinear impulsive singular integro-differential equations in Banach spaces[J].Nonlinear Anal.,In Press.
[63]Guo D.Nonlinear Functional Analysis[M].Jinan:Shandong Science and Technology Press,2004(in Chinese).
[64]Guo D,Numbers of nontrivial solutions for nonlinear two points boundary value problems[J].Math.Res.Exposition,1984,4:55-56.
[65]Guo Y,Ge W.Positive solutions for three-point boundary value problems with dependence on the first order derivative[J].J.Math.Anal.Appl.,2004,290(1):291-301.
[66]Guo Y,Shan W,Ge W.Positive solutions for second-order m-point boundary value problems[J].J.Comput.Appl.Math.,2003,151:415-424.
[67]Guo Y,Ji Y,Zhang J.Three Positive solutions for a nonlinear nth-order m-point boundary value problem[J].Nonlinear Anal.,In press.
[68]Guo Y,Liu X,Qiu J.Three positive solutions for higher orde m-point boundary value problems[J].J.Math.Anal.Appl.,2004,289:545-553.
[69]Gupta C P.Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation[J].J.Math.Anal.Appl.,1992,168:540-551.
[70]Gupta C P.A sharper condition for the Solvability of a three-point second order boundary value problem[J].J.Math.Anal.Appl.,1997,205:586-579.
[71]Gupta C P.A generalized multi-point boundary value problem for second order ordinary differential equations[J].Appl.Math.Comput.,1998,89:133-146.
[72]Hai D D,Shivaji R.An existence result for a class of superlinear p-Laplacian semipositone systems[J].Differential Integral Equations,2001,14:231-240.
[73]Han X.Positive solutions for a three-point boundary value problems at resonance[J].J.Math.Anal.Appl.,2007,336:556-568.
[74]He X,Ge W.Positive solutions for semipositone (p,n-p) right focal conjugate boundary value problems[J].Appl.Anal.,2002,81:227-240.
[75]Hopkins B,Kosmatov N.Third-order boundary value problems with signchanging solutions[J].Nonlinear Anal.,2007,67(1):126-137.
[76]Hussein A H.Salem.On the nonlinear Hammerstein integral equations in Banach spaces and application to the boundary value problem of fractional order[J].Math.Comput.Model.,2008,48(7-8):1178-1190.
[77]Il'in V A,Moiseev E I.Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator[J].Differential Equations,1987,23(8):979-987.
[78]Il' in V A,Moiseev E I.Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects[J].Differential Equations,1987,23:803-810.
[79]Ji D,Ge W.Existence of multiple positive solutions for Sturm-Liouville-like four-point boundary value problem with p-Laplacian[J].Nonlinear Anal.,to appear
[80]Ji Y,Guo Y,Zhang J.Positive solutions for second-order quasilinear multi-point boundary value problems[J].Appl.Math.Comput,2007,189:725-731.
[81]Jiang D.Positive solutions to singular (k,n — k) conjugate boundary value problems[J].Acta.Math.Sinica,2001,3:541-548.
[82]Jiang D,Xu X.Multiple positive solutions to a class of singular boundary value problems for the one-dimensional p-Laplacian[J].Comput.Math.Appl.,2004,47:667-681.
[83]Kong L,Wang J.The Green's function for (k,n — k) conjugate boundary value problems and its applications[J].J.Math.Anal.Appl.,2001,255:404-422.
[84]Kong L,Kong Q.Multi-point boundary value problems of second-order differential equations[J].Nonlinear Anal.,2004,58:909-931.
[85]Kong L,Kong Q.Second-order boundary value problems with nonhomogeneous boundary conditions[J].J.Math.Anal.Appl.,2007,330:1393-1411.
[86]Kosmatov N.Multi-point boundary value problems on an unbounded domain at resonance[J].Nonlinear Anal.,2008,68:2158-2171.
[87]Kosmatov N.Multi-point boundary value problems on time scales at resonance[J].J.Math.Anal.Appl.,2006,323:253-266.
[88]Kosmatov N.A multi-point boundary value problem with two critical conditions[J].Nonlinear Anal.,2006,65:622-633.
[89]Kosmatov N.Symmetric solutions of a multi-point boundary value problem[J],J.Math.Anal.Appl.,2005,309:25-36.
[90]Krasnosel'skii M A.Positive solutions of operator equations[M].Noordhoff,Groningen,1964.
[91]Lakshmikantham V,Leela S.Nonlinear Differential Equation in Abstract Spaces[M].Pergamon,Oxford,1981.
[92]Lakmeche A,Hammoudi A.Multiple positive solutions of the one-dimensional p-Laplacian[J].J.Math.Anal.Appl.,2006,317:43-49.
[93]Lan K Q.Multiple positive solutions of conjugate boundary value problems with singularities[J].Appl.Math.Comput.,2004,147:461-474.
[94]Leggett R W,Williams L R.Multiple positive fixed points of nonlinear operators on order Banach spaces[J].Indiana Univ.Math.J.,1979,28:673-688.
[95]Li S.Positive solutions of nonlinear singular third-order two-point boundary value problem[J].J.Math.Anal.Appl.,2006,323:413-425.
[96]Li C,Ge W.Existence of positive solutions for p-Laplacian singular boundary value problems[J].Indian J.Pure Appl.Math.,2003,34:187-203.
[97]Li W,Song G.Nonlinear boundary value problem for second order impulsive integro-differential equations of mixed type in Banach space[J].Comput.Math.Appl.,2008,56(5):1372-1381.
[98]Liu X,Qiu J,Guo Y.Three positive solutions for second-order m-point boundary value problems[J].Appl.Math.Comput.,2004,156:733-742.
[99]Liu B.Positive solutions of a nonlinear four-point boundary value problems in Banach spaces[J].J.Math.Anal.Appl.,2005,305(1):253-276.
[100]Liu B.Positive solutions of second-order three-point boundary value problems with change of sign in Banach spaces[J].Nonlinear Anal.,2006,64(6):1336-1355.
[101]Liu B.Positive solutions of generalized Sturm-Liouville four-point boundary value problems in Banach spaces[J].Nonlinear Anal.,2007,66(8):1661-1674.
[102]Liu X,Guo D.Periodic Boundary Value Problems for a Class of Second-Order Impulsive Integro-Differential Equations in Banach Spaces[J].J.Math.Anal.Appl.,1997,216(1):284-302.
[103]Liu X,Guo D.Method of upper and lower solutions for second-order impulsive integro-differential equations in a Banach space[J].Comput.Math.Appl.,1999,38:213-223.
[104]Liu X.Existence of three monotone positive solutions of nonhomogeneous multipoint BVPs for second-order differential equations[J].Boundary Value Problems,2008,1-14.
[105]Lin X,Jiang D,Li X.Existence and uniqueness of solutions for singular (k,n-k) conjugate boundary value problem[J].Comput.Math.Appl.,2006,52:375-382.
[106]Liu Y.Boundary value problems for second order differential equations on unbounded domains in a Banach space[J].Appl.Math.Comput.,2003,135(2-3):569-583.
[107]Liu Y,Qi A.Positive solutions of nonlinear singular boundary value problem in abstract space[J].Comput.Math.Appl.,2004,47(4-5):683-688.
[108]L \i H,O'Regan D,Zhong C.Multiple positive solutions for the onedimensional singular p-Laplacian[J].Appl.Math.Comput.,2002,133:407-722.
[109]L ii H,Zhong C.A note on singular nonlinear boundary value problems for the one-dimensional p-Laplacian[J].Appl.Math.Lett.,2001,14:189-194.
[110]L u H,O'Regan D,Agarwal R P.Existence theorems for the one-dimensional singular p-Laplacian equation with a nonlinear boundary condition[J].J.Comput.Appl.Math.,2005,182:188-210.
[Ill]Ma R.Positive solutions of a nonlinear three-point boundary value problem[J].Electron.J.Differ.Equat.,1999,34:1-8.
[112]Ma R.Positive solutions for second order three-point boundary value problems[J].Appl.Math.Lett,2001,14:1-5.
[113]Ma R,O'Regan D.Nodal solutions for second-order m-point boundary value problems with nonlinearities across several eigenvalues[J].Nonlinear Anal.,2006,64:1562-1577.
[114]Ma R,N.Castaneda.Existence of solutions of boundary value problems for second order functional differential equations[J].J.Math.Anal.Appl.,2004,292:49-59.
[115]Ma R.Multiplicity results for a third order boundary value problem at resonance[J].Nonlinear Anal.TMA,1998,32(4):493-499.
[116]Ma R.Positive solutions for semipositone (k.n — k) conjugate boundary value problems[J].J.Math.Anal.Appl.,2000,252:220-229.
[117]Ma R.Positive solutions for nonhomogeneous m-point boundary value problems[J].Comput.Math.Appl.,2004,47:689-698.
[118]Manasevich R,Mawhin J.Periodic solutions for boundary value problem with singular and discontinuous nonlinearity[J].Acta Mathematica Applicatae Sinica,2000,16:87-92.
[119]Manasevich R,Sweers G.A noncooperative elliptic system with p-Laplacians that preserves positivity[J].Nonlinear Anal.,1999,36:511-528.
[120]Niksirat M A,Mehri B.On the existence of positive solution for second-order multi-points boundary value problems[J].J.Comput.Appl.Math.,2006,193:269-276.
[121]Nussbaum R D.Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem[J].Fixed Point Theory (E.Fadell and G.Fournier,eds.),Lecture Notes in Math.,1981,886:309-330.
[122]O'Regan D.Some general existence principle and results for ((?)(y'))' =qf(t,y,y').0 [123]O'Regan D.Boundary value problems singular in the solution variable with nonlinear boundary data[j].Proc.Edinburgh Math.Soc,1996,39:505-523.
[124]O'Regan D.Upper and lower solutions for singular problems arising in the theory of membrane response of a spherical cap[J].Nonlinear Anal.,2001,47:1163-1174.
[125]Palamidesa Alex P,Smyrlisb G.Positive solutions to a singular third-order threepoint boundary value problem with an indefinitely signed Green ' s function[J].Nonlinear Anal.,2008,68(7):2104-2118.
[126]Potter A J B.A fixed point theorem for positive fc-set contractions[J].Proc.Edinburgh Math.Soc,1974,19:93-102.
[127]Rynne B P.Spectral properties and nodal solutions for second-order,m-point,boundary value problems[J].Nonlinear Anal.,2007,67:3318-3327.
[128]Su H,Wei Z,Xu F.The existence of countably many positive solutions for a system of nonlinear singular boundary value problems with the p-Laplacian operator[J].J.Math.Anal.Appl.,2007,325:319-332.
[129]Su H,Wei Z.Positive solutions to semipositone {k,n — k) conjugate boundary value problems[J].Nonlinear Anal.,2008,69:3190-3201.
[130]Sun B,Ge W,Zhao D.Three positive solutions for multipoint one-dimensional p-laplacian boundary value problems with dependence on the first order derivative[J].Math.Comput.Mode].,2007,45:1170-1178.
[131]Sun W,Zhang Q,Wang C.Existence of positive solutions to n-point nonhomogeneous boundary value problem[J].J.Math.Anal.Appl.,2007,330:612-621.
[132]Sun Y.Positive Solutions of singular third-order three-point boundary value problem[J].J.Math.Anal.Appl.,2005,306:589-603.
[133]Wang J,Gao W.A singular boundary value problem for the one-dimensional p-Laplacian[J].J.Math.Anal.Appl.,1996,201:851-866.
[134]Wang W,Shen J,Luo Z.Multi-point boundary value problems for second-order functional differential equations[J].Comput.Math.Appl.,2008,56:2065-2072.
[135]Wang Y,Ge W.Existence of multiple positive solutions for multipoint boundary value problems with a one-dimensional p-Laplacian[J].Nonlinear Anal.,2007,67:476-485.
[136]Wang Y,Ge W.Multiple positive solutions for multipoint boundary value problems with one-dimensional p-Laplacian[J].J.Math.Anal.Appl.,2007,327:1381-1395.
[137]Wang Z,Liu L,Wu Y.The unique solution of boundary value problems for nonlinear second-order integral-differential equations of mixed type in Banach spaces[J].Comput.Math.Appl.,2007,54(9-10):1293-1301.
[138]Webb J R L,Lan K Q.Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type[J].Topological Methods in Nonlinear Analysis Journal of the Schauder Center,2006,27:91-115.
[139]Wong F H,Wang S P,Chen T G.Existence of positive solutions for second order functional differential equations[J].Comput.Math.Appl.,2008,56(10):2580-2587,
[140]Wong P J Y,Agarwal R P.Singular differential equations,with {n,p) boundary conditions[J].Math.Comput.Modelling,1998,28:37-44.
[141]Xu Y,Zhang H.Multiple positive solutions of a m-point boundary value problem for 2nth-order singular integro-differential equations in Banach spaces[J].Nonlinear Anal.,In Press.
[142]Yang X.Green's function and positive solutions for higher-order ODE[J].Appl.Math.Comput.,2003,136:379-393.
[143]Yan X.On the Fredholm alternative for the p-Laplacian[J].Appl.Math.Comput.,2004,153:537-556.
[144]Yao Q.The existence and multiplicity of positive solutions for a third-order three-point boundary value problem[J].Acta Math.Appl.Sin.,2003,19:117-122.
[145]Yao Q,Feng Y.The existence of solution for a third-order two-point boundary value problem[J].Appl.Math.Lett.,2002,15:227-232.
[146]Yao Q,L ii H.Positive solutions of one-dimensional singular p-Laplace equations[J].Acta Math.Sinica,1998,41(6):1253-1264.
[147]Yao Z,Zhou W,Existence of positive solutions for the one-dimensional singular p-laplacian[J].Nonlinear Anal.,to appear.
[148]Zhao J,Liu Z,Liu L.The existence of solutions of infinite boundary value problems for first-order impulsive differential systems in Banach spaces[J].J.Comput.Appl.Math.,2008,222(2):524-530.
[149]Zhao Y,Chen H.Existence of multiple positive solutions for m-point boundary value problems in Banach spaces[J].J.Comput.Appl.Math.,2008,215(1):79-90.
[150]Zhang B,Liu X.Existence of multiple symmetric positive solutions of higher order Lidstone problems[J].J.Math.Anal.Appl.,2003,284:672-689.
[151]Zhang H,Liu L,Wu Y.Positive solutions for nth-order nonlinear impulsive singular integro-differential equations on infinite intervals in Banach spaces[J].Nonlinear Anal.In Press.
[152]Zhang H,Liu L,Wu Y.A unique positive solution for nth-order nonlinear impulsive singular integro-differential equations on unbounded domains in Banach spaces[J].Appl.Math.Comput.,2008,203(2):649-659.
[153]Zhang M,Yin Y,Wei Z.Existence of positive solution for singular semi-positone {k,n-k) conjugate m-point boundary value problem[J].Comput.Math.Appl.,2008,56:1146-1154.
[154]Zhang X,Feng M,Ge W.Existence and nonexistence of positive solutions for a class of nth-order three-point boundary value problems in Banach spaces[J].Nonlinear Anal.,In Press.
[155]Zhang X,Feng M,Ge W.Existence results for nonlinear boundary-value problems with integral boundary conditions in Banach spaces[J].Nonlinear Anal.,2008,69(10):3310-3321.
[156]Zhang X,Sun Y.Monotone positive solutions of second-order multi-point boundary value problems[J].Appl.Math.Comput.,In Press.
[157]Zhang X,Liu L,Wu C.Nontrivial solution of third-order nonlinear eigenvalue problems[J].Appl.Math.Comp.,2006,176:714-721.
[2]Agarwal R P,O'Regan D.Positive solutions for (p,n-p) conjugate boundary value problems[J].J.Differential Equations,1998,150:462-473.
[3]Agarwal R P,Wong F H.Existence of solutions to (k,n — k — 2) boundary value problems[J].Appl.Math.Comput.,1999,104:33-50.
[4]Agarwal R P,Bohner M,Wong P J Y.Positive solutions and eigenvalues of conjugate boundary value problems[J].Proc.Edinburgh.Math.Soc,1999,42:349-374.
[5]Agarwal R,L ü H,O'Regan D.Existence theorems for the one-dimensional singular p-Laplacian equation with sign changing nonlinearities[J].Appl.Math.Comput.,2003,143:15-38.
[6]Agarwal R P,L ü H,O'Regan D.A necessary and sufficient condition for the existence of positive solutions to the singular p-Laplacian[J].Z.Anal.Anwend.,2003,22:689-709.
[7]Agarwal R P,O'Regan D,Positive solutions to superliner singular boundary value problems[J],J.Comput.Appl.Math.,1998,88:129-147.
[8]Agarwal R P,O'Regan D,Existence theory for single and multiple solutions to singular position boundary value problems[J],J.Differential Equations,2001,175:393-414.
[9]Agarwal R P,Grace S R,O'Regan D.Semipositone higher-order differential equations[J].Appl.Math.Lett.,2004,17:201-207.
[10]Alarc (?) n V,Amat S,Busquier S,L (?) pez D J.A Steffensen's type method in Banach spaces with applications on boundary-value problems[J].J.Comput.Appl.Math.,2008,216(1):243-250.
[11]Amann H.Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces[J].SIAM Review,1976,18:620-709.
[12]Anderson D,Anderson T,Kleber M.Green's function and existence of solutions for a functional focal differential equation[J].Electron.J.Differ.Equat,2006,12:1-14.
[13]Anderson D R.Positivity of Green's function for an n-point right focal boundary value problem on measure chains[J].Math.Comput.Modelling,2000,31:29-50.
[14]Anderson D,Avery R.Multiple positive solutions to a third-order discrete focal boundary value problem[J].Comput.Math.Appl.,2001,42:333-340.
[15]Anderson D R.Green's function for a third-order generalized right focal problem[J].J.Math.Anal.Appl.,2003,288:1-14.
[16]Anderson D R,Davis J M.Multiple solutions and eigenvalues for three-order right focal boundary value problems[J].J.Math.Anal.Appl.,2002,267:135-157.
[17]Ashyralyev A.A note on the Bitsadze-Samarskii type nonlocal boundary value problem in a Banach space[J].J.Math.Anal.Appl.,2008,344(1):557-573.
[18]Avery R I.A generalization of the Leggett-Williams fixed point theorem.Math.Sci.Res.Hot-Line,1999,3:9-14.
[19]Bai Z,Fei X.Existence of triple positive solutions for a third order generalized right focal problem[J].Math.Inequal.Appl.,2006,9(3):437-444.
[20]Cabada A,Minhos F,Santos A I.Solvability for a third order discontinuous fully equation with nonlinear functional boundary conditions[J].J.Math.Anal.Appl.,2006,322:735-748.
[21]Cabada A,Pouso R L.Existence result for the problem (φ(u'))' = qf(t,u,u') with periodic and Neumann boundary conditions[J].Nonlinear Anal.,1997,30:1733-1742.
[22]Cac N P,Gatica J A.Fixed point theorems for mappings in ordered Banach spaces[J].J.Math.Anal.Appl.,1979,71:545-557.
[23]Chang Y,Li W.Existence results for second-order dynamic inclusion with mpoint boundary value conditions on time scales[J].Appl.Math.Lett.,2007,20:885-891.
[24]Chen H,Li P.Existence of solutions of three-point boundary value problems in Banach spaces[J].Math.Comput.Model.,In Press.
[25]Chen H,Li P.Three-point boundary value problems for second-order ordinary differential equations in Banach spaces[J].Comput.Mathe.Appl.,2008,56(7):1852-1860.
[26]Cui Y,Zou Y.Positive solutions of nonlinear singular boundary value problems in abstract spaces[J].Nonlinear Anal.,2008,69(1):287-294.
[27]Davis J M,Eloe P W,Henderson J.Triple positive solutions and dependence on higher order derivatives[J].J.Math.Anal.Appl.,1999,237:710-720.
[28]Davis J M,Henderson J,P.J.Y.Wong.General Lidstone problems:multiplicity and symmetry of solutions[J].J.Math.Anal.Appl.,2000,251:527-548.
[29]Demling K.Ordinary Differential Equations in Banach Spaces[M].Springer-Verlag,Berlin,1977.
[30]Eloe P W,Henderson J.Positive solutions for (n — 1,1) conjugate boundary value problems[J].Nonlinear Anal.,1997,28:1669-1680.
[31]Eloe P W,Henderson J.Singular nonlinear (k,n-k) conjugate boundary value problems[J].J.Differential Equations,1997,133:136-151.
[32]Feng H,Ge W.Existence of three positive solutions for m-po'mt boundary-value problems with one-dimensional p-Laplacian[J].Nonlinear Anal.,to appear
[33]Feng M,Pang H.A class of three-point boundary-value problems for secondorder impulsive integro-differential equations in Banach spaces[J].Nonlinear Anal.In Press.
[34]Feng M,Ji D,Ge W.Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces[J].J.Comput.Appl.Math.,2008,222(2):351-363.
[35]Feng Y,Liu S.Solvability of a third-order two-point boundary value problem[J].Appl.Math.Lett.,2005,18:1034-1040.
[36]Gamba IM,Jungel A.Positive solutions to singular second and third order differential equations for quantum fluids[J].Arch.Ration.Mech.Anal.,2001,156(3):183-203.
[37]Gatica J A,Oliker V,Waltman P.Singular nonlinear boundary value problems for second order ordinary differential equations[J].J.Differential Equations,1989, 79:62-78.
[38]Graef J R,Yang B.Positive solutions of a nonlinear third order eigenvalue problem[J].Dynam.Syst.Appl.,2006,15(1):97-110.
[39]Graef J R,Yang B.Multiple positive solutions to a three point third order boundary value problem[J].Discrete Contin.Dyn.Syst.(Suppl.) 2005,337-344.
[40]Guo L,Sun J,Zhao Y.Existence of Positive Solutions for Nonlinear Third-Order Three-Point Boundary Value Problem[J],Nonlinear Anal.,In Press.
[41]Guo D,V.Lakshmikantham,X.Liu.Nonlinear Integral Equations in Abstract SpacesfM].Kluwer Academic,Dordrecht,1996.
[42]Guo D,Lakshmikantham V.Nonlinear Problems in Abstract Cones[M].Academic Press,San Diego,1988.
[43]Guo D,Lakshmikantham V.Multiple solutions of two-point boundary value problems of ordinary differential equations in Banach spaces[J].J.Math.Anal.Appl.,1988,129(1):211-222.
[44]Guo D.Periodic boundary value problems for second order impulsive integrodifferential equations in banach spaces[J].Nonlinear Anal.,1997,28(6):983-997.
[45]Guo D.Initial Value Problems for Nonlinear Second Order Impulsive Integrodifferential Equations in Banach Spaces[J].J.Math.Anal.Appl.,1996,200(1):1-13.
[46]Guo D.Initial value problems for second-order integro-differential equations in Banach spaces[J].Nonlinear Anal.,1999,37(3):289-300.
[47]Guo D.Boundary value problems for impulsive integro-differential equations on unbounded domains in a Banach space[J].Appl.Math.Comput.,1999,99(1):1-15.
[48]Guo D.Second order impulsive integro-differential equations on unbounded domains in Banach spaces[J].Nonlinear Anal.,1999,35(4):413-423.
[49]Guo D.Existence of solutions for nth-order integro-differential equations in Banach spaces[J].Comput.Math.Appl.,2001,41(5-6):597-606.
[50]Guo D.Minimal nonnegative solutions for nth order integro-differential equations in Banach spaces[J].Appl.Math.Comput.,2000,113(1):55-65.
[51]Guo D.Second order integro-differential equations of Volterra type on unbounded domains in Banach spaces[J].Nonlinear Anal.,2000,41(3-4):465-476.
[52]Guo D.A class of nth-order impulsive integro-differential equations in Banach spaces[J].Comput.Math.Appl.,2002,44(10-11):1339-1356.
[53]Guo D.Multiple positive solutions for first order nonlinear integro-differential equations in Banach spaces[J].Nonlinear Anal.,2003,53(2):183-195.
[54]Guo D.A boundary value problem for nth order integro-differential equations in a Banach space[J],Appl.Math.Comput.,2003,136(2-3):571-592.
[55]Guo D.Extremal solutions for nth-order impulsive integro-differential equations on the half-line in Banach spaces[J].Nonlinear Anal.,2006,65(3):677-696.
[56]Guo D.Multiple positive solutions of a boundary value problem for nth-order impulsive integro-differential equations in Banach spaces[J].Nonlinear Anal.,2005,63(4):618-641.
[57]Guo D.Multiple positive solutions for nth-order impulsive integro-differential equations in Banach spaces[J].Nonlinear Anal.,2005,60(5):955-976.
[58]Guo D.Multiple positive solutions for a class of nonlinear integro-differential equations in Banach spaces[J].Appl.Math.Comput.,2004,154(2):469-485.
[59]Guo D.Multiple positive solutions of a boundary value problem for nth-order impulsive integro-differential equations in a Banach space[J].Nonlinear Anal.,2004,56(7):985-1006.
[60]Guo D.Multiple positive solutions for first order nonlinear impulsive integrodifferential equations in a Banach space[J].Appl.Math.Comput.,2003,143(2-3):233-249.
[61]Guo D.Existence of positive solutions for nth-order nonlinear impulsive singular integro-differential equations in Banach spaces[J].Nonlinear Anal.,2008,68(9):2727-2740.
[62]Guo D.Positive solutions of an infinite boundary value problem for nth-order nonlinear impulsive singular integro-differential equations in Banach spaces[J].Nonlinear Anal.,In Press.
[63]Guo D.Nonlinear Functional Analysis[M].Jinan:Shandong Science and Technology Press,2004(in Chinese).
[64]Guo D,Numbers of nontrivial solutions for nonlinear two points boundary value problems[J].Math.Res.Exposition,1984,4:55-56.
[65]Guo Y,Ge W.Positive solutions for three-point boundary value problems with dependence on the first order derivative[J].J.Math.Anal.Appl.,2004,290(1):291-301.
[66]Guo Y,Shan W,Ge W.Positive solutions for second-order m-point boundary value problems[J].J.Comput.Appl.Math.,2003,151:415-424.
[67]Guo Y,Ji Y,Zhang J.Three Positive solutions for a nonlinear nth-order m-point boundary value problem[J].Nonlinear Anal.,In press.
[68]Guo Y,Liu X,Qiu J.Three positive solutions for higher orde m-point boundary value problems[J].J.Math.Anal.Appl.,2004,289:545-553.
[69]Gupta C P.Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation[J].J.Math.Anal.Appl.,1992,168:540-551.
[70]Gupta C P.A sharper condition for the Solvability of a three-point second order boundary value problem[J].J.Math.Anal.Appl.,1997,205:586-579.
[71]Gupta C P.A generalized multi-point boundary value problem for second order ordinary differential equations[J].Appl.Math.Comput.,1998,89:133-146.
[72]Hai D D,Shivaji R.An existence result for a class of superlinear p-Laplacian semipositone systems[J].Differential Integral Equations,2001,14:231-240.
[73]Han X.Positive solutions for a three-point boundary value problems at resonance[J].J.Math.Anal.Appl.,2007,336:556-568.
[74]He X,Ge W.Positive solutions for semipositone (p,n-p) right focal conjugate boundary value problems[J].Appl.Anal.,2002,81:227-240.
[75]Hopkins B,Kosmatov N.Third-order boundary value problems with signchanging solutions[J].Nonlinear Anal.,2007,67(1):126-137.
[76]Hussein A H.Salem.On the nonlinear Hammerstein integral equations in Banach spaces and application to the boundary value problem of fractional order[J].Math.Comput.Model.,2008,48(7-8):1178-1190.
[77]Il'in V A,Moiseev E I.Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator[J].Differential Equations,1987,23(8):979-987.
[78]Il' in V A,Moiseev E I.Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects[J].Differential Equations,1987,23:803-810.
[79]Ji D,Ge W.Existence of multiple positive solutions for Sturm-Liouville-like four-point boundary value problem with p-Laplacian[J].Nonlinear Anal.,to appear
[80]Ji Y,Guo Y,Zhang J.Positive solutions for second-order quasilinear multi-point boundary value problems[J].Appl.Math.Comput,2007,189:725-731.
[81]Jiang D.Positive solutions to singular (k,n — k) conjugate boundary value problems[J].Acta.Math.Sinica,2001,3:541-548.
[82]Jiang D,Xu X.Multiple positive solutions to a class of singular boundary value problems for the one-dimensional p-Laplacian[J].Comput.Math.Appl.,2004,47:667-681.
[83]Kong L,Wang J.The Green's function for (k,n — k) conjugate boundary value problems and its applications[J].J.Math.Anal.Appl.,2001,255:404-422.
[84]Kong L,Kong Q.Multi-point boundary value problems of second-order differential equations[J].Nonlinear Anal.,2004,58:909-931.
[85]Kong L,Kong Q.Second-order boundary value problems with nonhomogeneous boundary conditions[J].J.Math.Anal.Appl.,2007,330:1393-1411.
[86]Kosmatov N.Multi-point boundary value problems on an unbounded domain at resonance[J].Nonlinear Anal.,2008,68:2158-2171.
[87]Kosmatov N.Multi-point boundary value problems on time scales at resonance[J].J.Math.Anal.Appl.,2006,323:253-266.
[88]Kosmatov N.A multi-point boundary value problem with two critical conditions[J].Nonlinear Anal.,2006,65:622-633.
[89]Kosmatov N.Symmetric solutions of a multi-point boundary value problem[J],J.Math.Anal.Appl.,2005,309:25-36.
[90]Krasnosel'skii M A.Positive solutions of operator equations[M].Noordhoff,Groningen,1964.
[91]Lakshmikantham V,Leela S.Nonlinear Differential Equation in Abstract Spaces[M].Pergamon,Oxford,1981.
[92]Lakmeche A,Hammoudi A.Multiple positive solutions of the one-dimensional p-Laplacian[J].J.Math.Anal.Appl.,2006,317:43-49.
[93]Lan K Q.Multiple positive solutions of conjugate boundary value problems with singularities[J].Appl.Math.Comput.,2004,147:461-474.
[94]Leggett R W,Williams L R.Multiple positive fixed points of nonlinear operators on order Banach spaces[J].Indiana Univ.Math.J.,1979,28:673-688.
[95]Li S.Positive solutions of nonlinear singular third-order two-point boundary value problem[J].J.Math.Anal.Appl.,2006,323:413-425.
[96]Li C,Ge W.Existence of positive solutions for p-Laplacian singular boundary value problems[J].Indian J.Pure Appl.Math.,2003,34:187-203.
[97]Li W,Song G.Nonlinear boundary value problem for second order impulsive integro-differential equations of mixed type in Banach space[J].Comput.Math.Appl.,2008,56(5):1372-1381.
[98]Liu X,Qiu J,Guo Y.Three positive solutions for second-order m-point boundary value problems[J].Appl.Math.Comput.,2004,156:733-742.
[99]Liu B.Positive solutions of a nonlinear four-point boundary value problems in Banach spaces[J].J.Math.Anal.Appl.,2005,305(1):253-276.
[100]Liu B.Positive solutions of second-order three-point boundary value problems with change of sign in Banach spaces[J].Nonlinear Anal.,2006,64(6):1336-1355.
[101]Liu B.Positive solutions of generalized Sturm-Liouville four-point boundary value problems in Banach spaces[J].Nonlinear Anal.,2007,66(8):1661-1674.
[102]Liu X,Guo D.Periodic Boundary Value Problems for a Class of Second-Order Impulsive Integro-Differential Equations in Banach Spaces[J].J.Math.Anal.Appl.,1997,216(1):284-302.
[103]Liu X,Guo D.Method of upper and lower solutions for second-order impulsive integro-differential equations in a Banach space[J].Comput.Math.Appl.,1999,38:213-223.
[104]Liu X.Existence of three monotone positive solutions of nonhomogeneous multipoint BVPs for second-order differential equations[J].Boundary Value Problems,2008,1-14.
[105]Lin X,Jiang D,Li X.Existence and uniqueness of solutions for singular (k,n-k) conjugate boundary value problem[J].Comput.Math.Appl.,2006,52:375-382.
[106]Liu Y.Boundary value problems for second order differential equations on unbounded domains in a Banach space[J].Appl.Math.Comput.,2003,135(2-3):569-583.
[107]Liu Y,Qi A.Positive solutions of nonlinear singular boundary value problem in abstract space[J].Comput.Math.Appl.,2004,47(4-5):683-688.
[108]L \i H,O'Regan D,Zhong C.Multiple positive solutions for the onedimensional singular p-Laplacian[J].Appl.Math.Comput.,2002,133:407-722.
[109]L ii H,Zhong C.A note on singular nonlinear boundary value problems for the one-dimensional p-Laplacian[J].Appl.Math.Lett.,2001,14:189-194.
[110]L u H,O'Regan D,Agarwal R P.Existence theorems for the one-dimensional singular p-Laplacian equation with a nonlinear boundary condition[J].J.Comput.Appl.Math.,2005,182:188-210.
[Ill]Ma R.Positive solutions of a nonlinear three-point boundary value problem[J].Electron.J.Differ.Equat.,1999,34:1-8.
[112]Ma R.Positive solutions for second order three-point boundary value problems[J].Appl.Math.Lett,2001,14:1-5.
[113]Ma R,O'Regan D.Nodal solutions for second-order m-point boundary value problems with nonlinearities across several eigenvalues[J].Nonlinear Anal.,2006,64:1562-1577.
[114]Ma R,N.Castaneda.Existence of solutions of boundary value problems for second order functional differential equations[J].J.Math.Anal.Appl.,2004,292:49-59.
[115]Ma R.Multiplicity results for a third order boundary value problem at resonance[J].Nonlinear Anal.TMA,1998,32(4):493-499.
[116]Ma R.Positive solutions for semipositone (k.n — k) conjugate boundary value problems[J].J.Math.Anal.Appl.,2000,252:220-229.
[117]Ma R.Positive solutions for nonhomogeneous m-point boundary value problems[J].Comput.Math.Appl.,2004,47:689-698.
[118]Manasevich R,Mawhin J.Periodic solutions for boundary value problem with singular and discontinuous nonlinearity[J].Acta Mathematica Applicatae Sinica,2000,16:87-92.
[119]Manasevich R,Sweers G.A noncooperative elliptic system with p-Laplacians that preserves positivity[J].Nonlinear Anal.,1999,36:511-528.
[120]Niksirat M A,Mehri B.On the existence of positive solution for second-order multi-points boundary value problems[J].J.Comput.Appl.Math.,2006,193:269-276.
[121]Nussbaum R D.Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem[J].Fixed Point Theory (E.Fadell and G.Fournier,eds.),Lecture Notes in Math.,1981,886:309-330.
[122]O'Regan D.Some general existence principle and results for ((?)(y'))' =qf(t,y,y').0
[124]O'Regan D.Upper and lower solutions for singular problems arising in the theory of membrane response of a spherical cap[J].Nonlinear Anal.,2001,47:1163-1174.
[125]Palamidesa Alex P,Smyrlisb G.Positive solutions to a singular third-order threepoint boundary value problem with an indefinitely signed Green ' s function[J].Nonlinear Anal.,2008,68(7):2104-2118.
[126]Potter A J B.A fixed point theorem for positive fc-set contractions[J].Proc.Edinburgh Math.Soc,1974,19:93-102.
[127]Rynne B P.Spectral properties and nodal solutions for second-order,m-point,boundary value problems[J].Nonlinear Anal.,2007,67:3318-3327.
[128]Su H,Wei Z,Xu F.The existence of countably many positive solutions for a system of nonlinear singular boundary value problems with the p-Laplacian operator[J].J.Math.Anal.Appl.,2007,325:319-332.
[129]Su H,Wei Z.Positive solutions to semipositone {k,n — k) conjugate boundary value problems[J].Nonlinear Anal.,2008,69:3190-3201.
[130]Sun B,Ge W,Zhao D.Three positive solutions for multipoint one-dimensional p-laplacian boundary value problems with dependence on the first order derivative[J].Math.Comput.Mode].,2007,45:1170-1178.
[131]Sun W,Zhang Q,Wang C.Existence of positive solutions to n-point nonhomogeneous boundary value problem[J].J.Math.Anal.Appl.,2007,330:612-621.
[132]Sun Y.Positive Solutions of singular third-order three-point boundary value problem[J].J.Math.Anal.Appl.,2005,306:589-603.
[133]Wang J,Gao W.A singular boundary value problem for the one-dimensional p-Laplacian[J].J.Math.Anal.Appl.,1996,201:851-866.
[134]Wang W,Shen J,Luo Z.Multi-point boundary value problems for second-order functional differential equations[J].Comput.Math.Appl.,2008,56:2065-2072.
[135]Wang Y,Ge W.Existence of multiple positive solutions for multipoint boundary value problems with a one-dimensional p-Laplacian[J].Nonlinear Anal.,2007,67:476-485.
[136]Wang Y,Ge W.Multiple positive solutions for multipoint boundary value problems with one-dimensional p-Laplacian[J].J.Math.Anal.Appl.,2007,327:1381-1395.
[137]Wang Z,Liu L,Wu Y.The unique solution of boundary value problems for nonlinear second-order integral-differential equations of mixed type in Banach spaces[J].Comput.Math.Appl.,2007,54(9-10):1293-1301.
[138]Webb J R L,Lan K Q.Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type[J].Topological Methods in Nonlinear Analysis Journal of the Schauder Center,2006,27:91-115.
[139]Wong F H,Wang S P,Chen T G.Existence of positive solutions for second order functional differential equations[J].Comput.Math.Appl.,2008,56(10):2580-2587,
[140]Wong P J Y,Agarwal R P.Singular differential equations,with {n,p) boundary conditions[J].Math.Comput.Modelling,1998,28:37-44.
[141]Xu Y,Zhang H.Multiple positive solutions of a m-point boundary value problem for 2nth-order singular integro-differential equations in Banach spaces[J].Nonlinear Anal.,In Press.
[142]Yang X.Green's function and positive solutions for higher-order ODE[J].Appl.Math.Comput.,2003,136:379-393.
[143]Yan X.On the Fredholm alternative for the p-Laplacian[J].Appl.Math.Comput.,2004,153:537-556.
[144]Yao Q.The existence and multiplicity of positive solutions for a third-order three-point boundary value problem[J].Acta Math.Appl.Sin.,2003,19:117-122.
[145]Yao Q,Feng Y.The existence of solution for a third-order two-point boundary value problem[J].Appl.Math.Lett.,2002,15:227-232.
[146]Yao Q,L ii H.Positive solutions of one-dimensional singular p-Laplace equations[J].Acta Math.Sinica,1998,41(6):1253-1264.
[147]Yao Z,Zhou W,Existence of positive solutions for the one-dimensional singular p-laplacian[J].Nonlinear Anal.,to appear.
[148]Zhao J,Liu Z,Liu L.The existence of solutions of infinite boundary value problems for first-order impulsive differential systems in Banach spaces[J].J.Comput.Appl.Math.,2008,222(2):524-530.
[149]Zhao Y,Chen H.Existence of multiple positive solutions for m-point boundary value problems in Banach spaces[J].J.Comput.Appl.Math.,2008,215(1):79-90.
[150]Zhang B,Liu X.Existence of multiple symmetric positive solutions of higher order Lidstone problems[J].J.Math.Anal.Appl.,2003,284:672-689.
[151]Zhang H,Liu L,Wu Y.Positive solutions for nth-order nonlinear impulsive singular integro-differential equations on infinite intervals in Banach spaces[J].Nonlinear Anal.In Press.
[152]Zhang H,Liu L,Wu Y.A unique positive solution for nth-order nonlinear impulsive singular integro-differential equations on unbounded domains in Banach spaces[J].Appl.Math.Comput.,2008,203(2):649-659.
[153]Zhang M,Yin Y,Wei Z.Existence of positive solution for singular semi-positone {k,n-k) conjugate m-point boundary value problem[J].Comput.Math.Appl.,2008,56:1146-1154.
[154]Zhang X,Feng M,Ge W.Existence and nonexistence of positive solutions for a class of nth-order three-point boundary value problems in Banach spaces[J].Nonlinear Anal.,In Press.
[155]Zhang X,Feng M,Ge W.Existence results for nonlinear boundary-value problems with integral boundary conditions in Banach spaces[J].Nonlinear Anal.,2008,69(10):3310-3321.
[156]Zhang X,Sun Y.Monotone positive solutions of second-order multi-point boundary value problems[J].Appl.Math.Comput.,In Press.
[157]Zhang X,Liu L,Wu C.Nontrivial solution of third-order nonlinear eigenvalue problems[J].Appl.Math.Comp.,2006,176:714-721.