非线性微分方程初边值问题的解
|
摘要
本文主要使用非线性泛函分析中的不动点定理与不动点指数理论以及迭代方法和锥理论研究了几类整数阶及分数阶非线性微分方程初边值问题解与多解的存在性。全文共分六章。
第一章简要地介绍了本文的研究背景和研究工作。
第二章是预备知识,主要介绍了分数阶积分与导数的基本概念和一些性质及有关锥的一些基础知识和相应的定理。
第三章研究了三阶三点边值问题(?)借助于u与u之间的关系,运用不动点指数定理和不动点定理,我们得到了多个正解的存在性。
第四章研究了Banach空间中两类脉冲微分方程边值问题。
4.1节研究了Banach空间中二阶奇异脉冲微分方程(?)通过构造一个凸闭集来克服奇异性,利用连续算子的不动点定理我们得到了解的存在性。
4.2节研究了Banach空间中带积分边值条件的二阶脉冲微分方程本节利用Green函数的性质结合锥理论与不动点指数定理得到了多个正解的存在性。
第五章考虑了分数阶微分方程m点边值问题(?)
我们用迭代的方法结合Green函数的性质和分析技巧得到最大最小正解,与上下解方法不同。
第六章在Banach空间中研究了两类分数阶微分方程的初值问题。6.1节在Banach空间中研究了分数阶微分方程(?)通过一个估计,利用Mo¨nch不动点定理我们得到了解的存在性。6.2节在Banach空间中利用锥理论研究了分数阶微分方程(?)利用比较定理,迭代技术,结合锥理论得到了最大最小解及最小正解的存在性。
This thesis mainly investigates the existence of solutions to initial and boundaryvalue problems for several nonlinear differential equations of integer order and frac-tional order by utilizing the fixed point theorems, fixed point index theory of nonlinearfunctional analysis, iterative technique and cone theory. There are six chapters in thisthesis.
In Chapter 1, the research background and results of this thesis is brie?y pre-sented.
In Chapter 2, the contents are about preliminaries, mainly including the defini-tions, some properties of fractional integral and derivative and the basic definitions andtheorems of cone.
In Chapter 3, the following third-order three-point boundary value problems(?)is studied. By utilizing the connection between u and u , the fixed point indextheory and the fixed point theorem, we get the existence of multiple positive solutions.
In Chapter 4, two classes of boundary value problems of impulsive differentialequations in Banach spaces are discussed.In Section 4.1, the existence of solutions for the following second order singulardifferential equations in Banach spaces(?)is considered. By constructing a convex closed set to overcome the singularity, andusing the fixed point theorem for continuous operators, we obtain the existence ofsolution.
In Section 4.2, the following second order differential equations with integralboundary condition in Banach spaces(?)is investigated. Combining the properties of Green function with cone theory and thefixed point index theory, we show the existence of multiple positive solutions.
In Chapter 5, we deal with m-point boundary value problems for nonlinear frac-tional differential equations(?)
Using the properties of the Green function, analysis technique and iterative technique,we get the existence of minimal and maximal positive solutions, and the method usedis different from the method of upper and lower solutions.
In Chapter 6, we consider two classes of initial value problems of fractional dif-ferential equations.
In Section 6.1, the following fractional differential equations with integral bound-ary condition (?)is studied. By an estimation and the Mo¨nch fixed point theorem, we obtain the exis-tence of solution.
In Section 6.2, the existence of solutions for fractional differential equations(?)is investigated. By using a comparability theorem, iterative technique and cone theory,we get the existence of minimal and maximal solutions and minimal positive solution.
第一章简要地介绍了本文的研究背景和研究工作。
第二章是预备知识,主要介绍了分数阶积分与导数的基本概念和一些性质及有关锥的一些基础知识和相应的定理。
第三章研究了三阶三点边值问题(?)借助于u与u之间的关系,运用不动点指数定理和不动点定理,我们得到了多个正解的存在性。
第四章研究了Banach空间中两类脉冲微分方程边值问题。
4.1节研究了Banach空间中二阶奇异脉冲微分方程(?)通过构造一个凸闭集来克服奇异性,利用连续算子的不动点定理我们得到了解的存在性。
4.2节研究了Banach空间中带积分边值条件的二阶脉冲微分方程本节利用Green函数的性质结合锥理论与不动点指数定理得到了多个正解的存在性。
第五章考虑了分数阶微分方程m点边值问题(?)
我们用迭代的方法结合Green函数的性质和分析技巧得到最大最小正解,与上下解方法不同。
第六章在Banach空间中研究了两类分数阶微分方程的初值问题。6.1节在Banach空间中研究了分数阶微分方程(?)通过一个估计,利用Mo¨nch不动点定理我们得到了解的存在性。6.2节在Banach空间中利用锥理论研究了分数阶微分方程(?)利用比较定理,迭代技术,结合锥理论得到了最大最小解及最小正解的存在性。
This thesis mainly investigates the existence of solutions to initial and boundaryvalue problems for several nonlinear differential equations of integer order and frac-tional order by utilizing the fixed point theorems, fixed point index theory of nonlinearfunctional analysis, iterative technique and cone theory. There are six chapters in thisthesis.
In Chapter 1, the research background and results of this thesis is brie?y pre-sented.
In Chapter 2, the contents are about preliminaries, mainly including the defini-tions, some properties of fractional integral and derivative and the basic definitions andtheorems of cone.
In Chapter 3, the following third-order three-point boundary value problems(?)is studied. By utilizing the connection between u and u , the fixed point indextheory and the fixed point theorem, we get the existence of multiple positive solutions.
In Chapter 4, two classes of boundary value problems of impulsive differentialequations in Banach spaces are discussed.In Section 4.1, the existence of solutions for the following second order singulardifferential equations in Banach spaces(?)is considered. By constructing a convex closed set to overcome the singularity, andusing the fixed point theorem for continuous operators, we obtain the existence ofsolution.
In Section 4.2, the following second order differential equations with integralboundary condition in Banach spaces(?)is investigated. Combining the properties of Green function with cone theory and thefixed point index theory, we show the existence of multiple positive solutions.
In Chapter 5, we deal with m-point boundary value problems for nonlinear frac-tional differential equations(?)
Using the properties of the Green function, analysis technique and iterative technique,we get the existence of minimal and maximal positive solutions, and the method usedis different from the method of upper and lower solutions.
In Chapter 6, we consider two classes of initial value problems of fractional dif-ferential equations.
In Section 6.1, the following fractional differential equations with integral bound-ary condition (?)is studied. By an estimation and the Mo¨nch fixed point theorem, we obtain the exis-tence of solution.
In Section 6.2, the existence of solutions for fractional differential equations(?)is investigated. By using a comparability theorem, iterative technique and cone theory,we get the existence of minimal and maximal solutions and minimal positive solution.
引文
[1] R. P. Agarwal, Difference Equations and Inequalities, Marcel Dekker, Inc. New York, 1992.
[2] R. P. Agarwal, I. Kiguradze, On multi-point boundary value problems for linear ordinarydifferential equations with singularities, J. Math. Anal. Appl. 297 (2004) 131-151.
[3] R. P. Agarwal, V. Lakshmikantham, J.J. Nieto, On the concept of solution for fractionaldifferential equations with uncertainty, Nonlinear Anal. 72 (2010) 2859-2862.
[4] D. Anderson, Multiple positive solutions for a three-point boundary value problem, Math.Compt. Modelling 27 (1998) 49-57.
[5] D. Anderson, Green,s function for a third-order generalized right focal problem, J. Math.Anal. Appl. 288 (2003)1-14.
[6] S. Abbas, M. Benchohra, Darboux problem for perturbed partial differential equations offractional order with finite delay, Nonlinear Analysis: Hybrid Systems 3 (2009) 597-604.
[7] M. Benchohra, J. Henderson, S. K. Ntouyas, A. Ouahab, Existence results for fractionalorder functional differential equations with infinite delay, J. Math. Anal. Appl. 338 (2008)1340-1350.
[8] M. Benchohra, A. Cabada, D. Seba, An existence result for nonlinear fractional differentialequations on Banach spaces, Boundary Value Problems, 2009,doi:10.1155/2009/628916.
[9] A. Boucherif, N. Al-Malki, nonlinear three-point third order boundary value problems,Appl. Math. Compt. 190 (2007) 1168-1177.
[10] A. Boucherif, Second-order boundary value problems with integral boundary conditions,Nonlinear Anal. 70 (2009) 364-371.
[11] Z. B. Bai, H. S. Lu¨, Positive solutions for boundary value problem of nonlinear fractionaldifferential equation, J. Math. Anal. Appl. 311 (2005) 495-505.
[12] P. Drabek, J. Milota, Methods of Nonlinear Analysis Applications to Differential Equations,Birkhauser, Verlag, AG, 2007.
[13] J. Dugundji, A. Granas, Fixed Point Theory, Volume 1, Warszawa, 1982.
[14] D. Barr, T. Sherman , Existence and uniqueness of solutions to three-point bounday value,J. Diff. Eqns. 13 (1973) 197-212.
[15] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, 1985.
[16] M. Q. Feng, H.H. Pang, A class of three-point boundary-value problems for second-orderimpulsive integro-differential equations in Banach spaces, Nonlinear Anal. 70 (2009) 64-82.
[17] M. Q. Feng, B. Du, W. Ge, Impulsive boundary value problems with integral boundaryconditions and one-dimensional p-Laplacian, Nonlinear Anal. 70 (2009) 3119-3126.
[18] D. J. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press,New York, 1988.
[19] D. J. Guo, X. Z. Liu, Multiple positive solutions of boundary-value problems for impulsivedifferential equations, Nonlinear Anal. 25 (1995) 327-337.
[20] D. J. Guo, V. Lakshmikantham, X. Z. Liu, Nonlinear Integral Equations in Abstract Spaces,Kluwer Academic Publishers, Dordrecht, 1996.
[21] D. J. Guo, positive solutions of an infinite boundary value problem for nth-order nonlin-ear impulsive singular integro-differential equations in Banach spaces, Nonlinear Anal. 70(2009) 2078-2090.
[22] L. J. Guo, J. P. Sun, Y. H. Zhao, Existence of positive solutions for nonlinear thid-orderthree-point boundary value problems, Nonlinear Anal. 68 (2008) 3151-3158.
[23] J. R. Graef, L. Kong, Positive solutions for third order semipositone boundary value prob-lems, Appl. Math. Lett. 22 (2009) 1154-1160.
[24] J. Henderson, Uniqueness implies existence for three-point boundary value problems forsecond order differential equations, Appl. Math. Lett. 18 (2005) 905-909.
[25] J. Henderson, B. Karna, C. C. Tisdell, Existence of solutions for three-point boundary valueproblems for second order equations, Proc. Amer. Math. Soc. 133 (2005) 1365-1369.
[26] J. Henderson, Solutions of multipoint boundary value problems for second order equations,Dynam. Systems Appl. 15 (2006) 111-117.
[27] J. Henderson, S. K. Ntouyas, I. K. Purnaras, Positive solutions for systems of m-point non-linear boundary value problems, Math. Model. Anal. 13 (2008) 357-370.
[28] J. Henderson, A. Ouahab, Fractional functional differential inclusions with finite delay,Nonlinear Anal. 70 (2009) 2091-2105.
[29] J. Henderson, A. Ouahab, Impulsive differential inclusions with fractional order, Comput.Math. Appl. 59 (2010) 1191-1226.
[30] H. P. Heinz, On the behavior of measures of noncompactness with respect to differentiationand integration of vector-valued function, Nonlinear Anal. 7 (1983) 1351-1371.
[31] L. G. Hu, T. J. Xiao, J. Liang, Generalized mann iterations for approximating fixedpoints of a family of hemicontractions, Fixed Point Theory Appl., Volume 2008, 9 Pages,doi:10.1155/2008/824607.
[32] L. G. Hu, T. J. Xiao, J. Liang, Positive solutions to singular and delay higher-order dif-ferential equations on time scales, Boundary Value Problems, Volume 2009, 19 Pages,doi:10.1155/2009/937064.
[33] F. L. Huang, J. Liang, T. J. Xiao, On a generalization of Horn’s fixed point theorem, J. Math.Anal. Appl. 164 (1992) 24-29.
[34] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differ-ential Equations, in: North-Holland Mathematics Studies, Vol. 204, Elsevier, Amsterdam,2006.
[35] I. Y. Karaca, On positive solutions for fourth-order boundary value problem with impulse,J. Comput. Appl. Math. 225 (2009) 356-364.
[36] M. A. Krasnosel’skii, P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, SpringerBerlin, 1984.
[37] V. Lakshmikantham, D. Bainov, P. Simeonov, Theory of Impulsive Differential Equations,World Scientific, Singapore, 1989.
[38] V. Lakshmikantham, A. S. Vatsala, Theory of fractional differential inequalities and appli-cations, communications in applied analysis 11 (2007) 395-402.
[39] V. Lakshmikantham, A. S. Vatsala, General uniqueness and monotone iterative techniquefor fractional differential equations, Appl. Math. Lett. 21 (2008) 828-834.
[40] V. Lakshmikantham, A. S. Vatsala, Basic theory of fractional differential equations, Non-linear Anal. 69 (2008) 2677-2682.
[41] V. Lakshmikantham, S. Leela, Nagumo-type uniqueness result for fractional differentialequations, Nonlinear Anal. 71 (2009) 2886-2889.
[42] V. Lakshmikantham, S. Leela, A Krasnoselskii-Krein-type uniqueness result for fractionaldifferential equations, Nonlinear Anal. 71 (2009) 3421-3424.
[43] S. Liang, J. Zhang, Positive solutions for boundary value problems of nonlinear fractionaldifferential equations, Nonlinear Anal. 71 (2009) 5545-5550.
[44] F. Li, Mild solutions for fractional differential equations with nonlocal conditions, Advancesin Difference Equations, Vol. 2010, Article ID 287861, 9 pages.
[45] Z. W. Lv, Existence of positive solutions for a class of second-order impulsive singulardifferential equations in Banach spaces, International Journal of Evolution Equations. 4 (4)(2010) 423-432.
[46] Z. W. Lv, J. Liang, T. J. Xiao, Solutions to fractional differential equations with nonlocalinitial condition in Banach Spaces, Advances in Difference Equations, Vol. 2010, ArticleID 340349, 10 pages.
[47] Z. W. Lv, J. Liang, T. J. Xiao, Multiple positive solutions for second order impulsive bound-ary value problems in Banach spaces, Electronic Journal of Qualitative Theory of Differen-tial Equations 38 (2010) 1-15.
[48] J. Liang, T. J. Xiao, Semilinear integrodifferential equations with nonlocal initial conditions,Comput. Math. Appl. 47 (2004) 863―875.
[49] J. Liang, T. J. Xiao, Z. C. Hao, Positive solutions of singular differential equations on mea-sure chains, Comput. Math. Appl. 49 (2005) 651-663.
[50] J. Liang, R. Nagel, T. J. Xiao, Approximation theorems for the propagators of higher orderabstract cauchy problems, Tran. Amer. Math. Soc. 360 (2008) 1723-1739.
[51] J. Liang, J. H. Liu, T. J. Xiao, Nonlocal impulsive problems for nonlinear differential equa-tions in Banach spaces, Math. Comput. Modelling 49 (2009) 798-804.
[52] L. S. Liu, B. M. Liu, Y. H. Wu, Nontrivial solutions of m-point boundary value problems forsingular second-order differential eauations with a sign-changing nonlinear term, J. Com-put. Appl. Math. 224 (2009) 373-382.
[53] C. F. Li, X. N. Luo, Y. Zhou, Existence of positive solutions of the boundary value problemfor nonlinear fractional differential equations, Comput. Math. Appl. 59 (2010) 1363-1375.
[54] R. W. Leggett, L. R. Williams, Multiple positive fixed points of nonlinear operators onordered Banach spaces, Indiana Univ. Math. J. 28 (1979) 673-688.
[55] F. A. McRae, Monotone iterative technique and existence results for fractional differentialequations, Nonlinear Anal. 71 (2009) 6093-6096.
[56] G. M. Mophou, G. M. N’Gue′re′kata, Existence of mild solution for some fractional differ-ential equations with nonlocal conditions, Semigroup Forum 79 (2009) 315-322.
[57] G. M. Mophou, Existence and uniqueness of mild solutions to impulsive fractional differ-ential equations, Nonlinear Anal. 72 (2010) 1604-1615.
[58] R. Y. Ma, B. Thompson, Global behavior of positive solutions of nonlinear three-pointboundary value problems, Nonlinear Anal. TMA, 60 (2005) 685-701.
[59] R. Ma, Positive silutions for a second order three-point boundary-value problems, Appl.Math. Lett. 14 (2001) 1-5.
[60] R. Y. Ma, Y. L. An, Global structure of positive solutions for nonlocal boundary valueproblems involving integral conditions, Nonlinear Anal. TMA 71 (2009) 4364-4376.
[61] G. M. N’Gue′re′kata, A Cauchy Problem for some fractional abstract differential equationwith nonlocal conditions, Nonlinear Anal. 70 (2009) 1873-1876.
[62] K. B. Oldham, J. Spanier, The Fractional Calculus, Math. Sci. Eng., Academic Press, NewYork, 1974.
[63] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,Springer-Verlag, New York, 1983.
[64] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[65] Hussein A. H. Salem, On the fractional order m-point boundary value problem in re?exiveBanach spaces and weak topologies, J. Comput. Appl. Math. 224 (2009) 565-572.
[66] Y. P. Sun, Positive solutions for third-order three-point nonhomogeneous boundary valueproblems, Appl. Math. Lett. 22 (2009) 45-51.
[67] A. M. Samoilenko, N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Sin-gapore, 1995.
[68] J. X. Sun, X. A Xu, D. O’Regan, Nodal solutions for m-point boundary value problemsusing bifurcation methods, Nonlinear Anal. 68 (2008) 3034-3046.
[69] J. Vasundhara Devi, F. A. McRae, Z. Drici, Generalized quasilinearization for fractionaldifferential equations, Comput. Math. Appl. 59 (2010) 1057-1062.
[70] X. Xu, D. Jiang, C. Yuan, Multiple positive solutions for the boundary value problem of anonlinear fractional differential equation, Nonlinear Anal. 71 (2009) 4676-4688.
[71] T. J. Xiao, J. Liang, Laplace transforms and integrated, regularized semigroups in locallyconvex spaces, J. Funct. Anal. 148 (1997) 448-479.
[72] T. J. Xiao, J. Liang, The Cauchy Problem for Higher-order Abstract Differential Equations,Lecture Notes in Mathematics, Vol. 1701, Springer, Berlin, 1998.
[73] T. J. Xiao, J. Liang, Approximations of Laplace transforms and integrated semigroups, J.Funct. Anal. 172 (2000) 202-220.
[74] T. J. Xiao, J. Liang, J. van Casteren, Time dependent Desch-Schappacher type perturbationsof Volterra integral equations, Integral Equa. Opera. Theory 44 (4) (2002) 494-506.
[75] T. J. Xiao, J. Liang, Higher order abstract Cauchy problems: their existence and uniquenessfamilies, J. London Math. Soc. 67 (2003) 149-164.
[76] T. J. Xiao, J. Liang, Semilinear integrodifferential equations with nonlocal initial conditions,Comput. Math. Appl. 47 (2004) 863-875.
[77] T. J. Xiao, J. Liang, Second order differential operators with Feller-Wentzell type boundaryconditions, J. Funct. Anal. 254 (2008) 1467-1486.
[78] T. J. Xiao, J. Liang, J. Zhang, Pseudo almost automorphic functions to semilinear differen-tial equations in Banach spaces, Semigroup Forum 76 (2008) 518-524.
[79] T. J. Xiao, J. Liang, Blow-up and global existence of solutions to integral equations withinfinite delay in Banach spaces, Nonlinear Anal. TMA 71 (2009) e1442-e1447.
[80] T. J. Xiao, X. X. Zhu, J. Liang, Pseudo-almost automorphic mild solutions to nonau-tonomous differential equations and applications, Nonlinear Anal. 70 (2009) 4079-4085.
[81] B. Q. Yan, Boundary value problems on the half-line with impules and infinite delay, J.Math. Anal. Appl. 259 (2001) 94-114.
[82] B. Q. Yan, D. O’Regan, R. P. Agarwal, Multiple positive solutions via index theory forsingular boundary value problems with derivative dependence, Positivity, 11 (2007) 687-720.
[83] Q. Yao, The existence and multiplicity of positive solutions for a third-order three-pointboundary value problems, Acta Math. Appl. Sin. 19 (2003) 117-122.
[84] Y. Zhou, F. Jiao, J. Li, Existence and uniqueness for fractional neutral differential equationswith infinite delay, Nonlinear Anal. 71 (2009) 3249-3256.
[85] E. Zeidler, Nonlinear Functional Analysis and its Applications, I. Fixed-Point Theorems,Springer-Verlag, New York, 1986.
[86] A. Zhao, Z. Bai, Existence of solutions to first-order impulsive periodic boundary valueproblems, Nonlinear Anal. 71 (2009) 1970-1977.
[87] X. M. Zhang, M. Q. Feng, W. G. Ge, Existence results for nonlinear boundary-value prob-lems with integral boundary conditions in Banach spaces, Nonlinear Anal. 69 (2008) 3310-3321.
[2] R. P. Agarwal, I. Kiguradze, On multi-point boundary value problems for linear ordinarydifferential equations with singularities, J. Math. Anal. Appl. 297 (2004) 131-151.
[3] R. P. Agarwal, V. Lakshmikantham, J.J. Nieto, On the concept of solution for fractionaldifferential equations with uncertainty, Nonlinear Anal. 72 (2010) 2859-2862.
[4] D. Anderson, Multiple positive solutions for a three-point boundary value problem, Math.Compt. Modelling 27 (1998) 49-57.
[5] D. Anderson, Green,s function for a third-order generalized right focal problem, J. Math.Anal. Appl. 288 (2003)1-14.
[6] S. Abbas, M. Benchohra, Darboux problem for perturbed partial differential equations offractional order with finite delay, Nonlinear Analysis: Hybrid Systems 3 (2009) 597-604.
[7] M. Benchohra, J. Henderson, S. K. Ntouyas, A. Ouahab, Existence results for fractionalorder functional differential equations with infinite delay, J. Math. Anal. Appl. 338 (2008)1340-1350.
[8] M. Benchohra, A. Cabada, D. Seba, An existence result for nonlinear fractional differentialequations on Banach spaces, Boundary Value Problems, 2009,doi:10.1155/2009/628916.
[9] A. Boucherif, N. Al-Malki, nonlinear three-point third order boundary value problems,Appl. Math. Compt. 190 (2007) 1168-1177.
[10] A. Boucherif, Second-order boundary value problems with integral boundary conditions,Nonlinear Anal. 70 (2009) 364-371.
[11] Z. B. Bai, H. S. Lu¨, Positive solutions for boundary value problem of nonlinear fractionaldifferential equation, J. Math. Anal. Appl. 311 (2005) 495-505.
[12] P. Drabek, J. Milota, Methods of Nonlinear Analysis Applications to Differential Equations,Birkhauser, Verlag, AG, 2007.
[13] J. Dugundji, A. Granas, Fixed Point Theory, Volume 1, Warszawa, 1982.
[14] D. Barr, T. Sherman , Existence and uniqueness of solutions to three-point bounday value,J. Diff. Eqns. 13 (1973) 197-212.
[15] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, 1985.
[16] M. Q. Feng, H.H. Pang, A class of three-point boundary-value problems for second-orderimpulsive integro-differential equations in Banach spaces, Nonlinear Anal. 70 (2009) 64-82.
[17] M. Q. Feng, B. Du, W. Ge, Impulsive boundary value problems with integral boundaryconditions and one-dimensional p-Laplacian, Nonlinear Anal. 70 (2009) 3119-3126.
[18] D. J. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press,New York, 1988.
[19] D. J. Guo, X. Z. Liu, Multiple positive solutions of boundary-value problems for impulsivedifferential equations, Nonlinear Anal. 25 (1995) 327-337.
[20] D. J. Guo, V. Lakshmikantham, X. Z. Liu, Nonlinear Integral Equations in Abstract Spaces,Kluwer Academic Publishers, Dordrecht, 1996.
[21] D. J. Guo, positive solutions of an infinite boundary value problem for nth-order nonlin-ear impulsive singular integro-differential equations in Banach spaces, Nonlinear Anal. 70(2009) 2078-2090.
[22] L. J. Guo, J. P. Sun, Y. H. Zhao, Existence of positive solutions for nonlinear thid-orderthree-point boundary value problems, Nonlinear Anal. 68 (2008) 3151-3158.
[23] J. R. Graef, L. Kong, Positive solutions for third order semipositone boundary value prob-lems, Appl. Math. Lett. 22 (2009) 1154-1160.
[24] J. Henderson, Uniqueness implies existence for three-point boundary value problems forsecond order differential equations, Appl. Math. Lett. 18 (2005) 905-909.
[25] J. Henderson, B. Karna, C. C. Tisdell, Existence of solutions for three-point boundary valueproblems for second order equations, Proc. Amer. Math. Soc. 133 (2005) 1365-1369.
[26] J. Henderson, Solutions of multipoint boundary value problems for second order equations,Dynam. Systems Appl. 15 (2006) 111-117.
[27] J. Henderson, S. K. Ntouyas, I. K. Purnaras, Positive solutions for systems of m-point non-linear boundary value problems, Math. Model. Anal. 13 (2008) 357-370.
[28] J. Henderson, A. Ouahab, Fractional functional differential inclusions with finite delay,Nonlinear Anal. 70 (2009) 2091-2105.
[29] J. Henderson, A. Ouahab, Impulsive differential inclusions with fractional order, Comput.Math. Appl. 59 (2010) 1191-1226.
[30] H. P. Heinz, On the behavior of measures of noncompactness with respect to differentiationand integration of vector-valued function, Nonlinear Anal. 7 (1983) 1351-1371.
[31] L. G. Hu, T. J. Xiao, J. Liang, Generalized mann iterations for approximating fixedpoints of a family of hemicontractions, Fixed Point Theory Appl., Volume 2008, 9 Pages,doi:10.1155/2008/824607.
[32] L. G. Hu, T. J. Xiao, J. Liang, Positive solutions to singular and delay higher-order dif-ferential equations on time scales, Boundary Value Problems, Volume 2009, 19 Pages,doi:10.1155/2009/937064.
[33] F. L. Huang, J. Liang, T. J. Xiao, On a generalization of Horn’s fixed point theorem, J. Math.Anal. Appl. 164 (1992) 24-29.
[34] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differ-ential Equations, in: North-Holland Mathematics Studies, Vol. 204, Elsevier, Amsterdam,2006.
[35] I. Y. Karaca, On positive solutions for fourth-order boundary value problem with impulse,J. Comput. Appl. Math. 225 (2009) 356-364.
[36] M. A. Krasnosel’skii, P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, SpringerBerlin, 1984.
[37] V. Lakshmikantham, D. Bainov, P. Simeonov, Theory of Impulsive Differential Equations,World Scientific, Singapore, 1989.
[38] V. Lakshmikantham, A. S. Vatsala, Theory of fractional differential inequalities and appli-cations, communications in applied analysis 11 (2007) 395-402.
[39] V. Lakshmikantham, A. S. Vatsala, General uniqueness and monotone iterative techniquefor fractional differential equations, Appl. Math. Lett. 21 (2008) 828-834.
[40] V. Lakshmikantham, A. S. Vatsala, Basic theory of fractional differential equations, Non-linear Anal. 69 (2008) 2677-2682.
[41] V. Lakshmikantham, S. Leela, Nagumo-type uniqueness result for fractional differentialequations, Nonlinear Anal. 71 (2009) 2886-2889.
[42] V. Lakshmikantham, S. Leela, A Krasnoselskii-Krein-type uniqueness result for fractionaldifferential equations, Nonlinear Anal. 71 (2009) 3421-3424.
[43] S. Liang, J. Zhang, Positive solutions for boundary value problems of nonlinear fractionaldifferential equations, Nonlinear Anal. 71 (2009) 5545-5550.
[44] F. Li, Mild solutions for fractional differential equations with nonlocal conditions, Advancesin Difference Equations, Vol. 2010, Article ID 287861, 9 pages.
[45] Z. W. Lv, Existence of positive solutions for a class of second-order impulsive singulardifferential equations in Banach spaces, International Journal of Evolution Equations. 4 (4)(2010) 423-432.
[46] Z. W. Lv, J. Liang, T. J. Xiao, Solutions to fractional differential equations with nonlocalinitial condition in Banach Spaces, Advances in Difference Equations, Vol. 2010, ArticleID 340349, 10 pages.
[47] Z. W. Lv, J. Liang, T. J. Xiao, Multiple positive solutions for second order impulsive bound-ary value problems in Banach spaces, Electronic Journal of Qualitative Theory of Differen-tial Equations 38 (2010) 1-15.
[48] J. Liang, T. J. Xiao, Semilinear integrodifferential equations with nonlocal initial conditions,Comput. Math. Appl. 47 (2004) 863―875.
[49] J. Liang, T. J. Xiao, Z. C. Hao, Positive solutions of singular differential equations on mea-sure chains, Comput. Math. Appl. 49 (2005) 651-663.
[50] J. Liang, R. Nagel, T. J. Xiao, Approximation theorems for the propagators of higher orderabstract cauchy problems, Tran. Amer. Math. Soc. 360 (2008) 1723-1739.
[51] J. Liang, J. H. Liu, T. J. Xiao, Nonlocal impulsive problems for nonlinear differential equa-tions in Banach spaces, Math. Comput. Modelling 49 (2009) 798-804.
[52] L. S. Liu, B. M. Liu, Y. H. Wu, Nontrivial solutions of m-point boundary value problems forsingular second-order differential eauations with a sign-changing nonlinear term, J. Com-put. Appl. Math. 224 (2009) 373-382.
[53] C. F. Li, X. N. Luo, Y. Zhou, Existence of positive solutions of the boundary value problemfor nonlinear fractional differential equations, Comput. Math. Appl. 59 (2010) 1363-1375.
[54] R. W. Leggett, L. R. Williams, Multiple positive fixed points of nonlinear operators onordered Banach spaces, Indiana Univ. Math. J. 28 (1979) 673-688.
[55] F. A. McRae, Monotone iterative technique and existence results for fractional differentialequations, Nonlinear Anal. 71 (2009) 6093-6096.
[56] G. M. Mophou, G. M. N’Gue′re′kata, Existence of mild solution for some fractional differ-ential equations with nonlocal conditions, Semigroup Forum 79 (2009) 315-322.
[57] G. M. Mophou, Existence and uniqueness of mild solutions to impulsive fractional differ-ential equations, Nonlinear Anal. 72 (2010) 1604-1615.
[58] R. Y. Ma, B. Thompson, Global behavior of positive solutions of nonlinear three-pointboundary value problems, Nonlinear Anal. TMA, 60 (2005) 685-701.
[59] R. Ma, Positive silutions for a second order three-point boundary-value problems, Appl.Math. Lett. 14 (2001) 1-5.
[60] R. Y. Ma, Y. L. An, Global structure of positive solutions for nonlocal boundary valueproblems involving integral conditions, Nonlinear Anal. TMA 71 (2009) 4364-4376.
[61] G. M. N’Gue′re′kata, A Cauchy Problem for some fractional abstract differential equationwith nonlocal conditions, Nonlinear Anal. 70 (2009) 1873-1876.
[62] K. B. Oldham, J. Spanier, The Fractional Calculus, Math. Sci. Eng., Academic Press, NewYork, 1974.
[63] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,Springer-Verlag, New York, 1983.
[64] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[65] Hussein A. H. Salem, On the fractional order m-point boundary value problem in re?exiveBanach spaces and weak topologies, J. Comput. Appl. Math. 224 (2009) 565-572.
[66] Y. P. Sun, Positive solutions for third-order three-point nonhomogeneous boundary valueproblems, Appl. Math. Lett. 22 (2009) 45-51.
[67] A. M. Samoilenko, N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Sin-gapore, 1995.
[68] J. X. Sun, X. A Xu, D. O’Regan, Nodal solutions for m-point boundary value problemsusing bifurcation methods, Nonlinear Anal. 68 (2008) 3034-3046.
[69] J. Vasundhara Devi, F. A. McRae, Z. Drici, Generalized quasilinearization for fractionaldifferential equations, Comput. Math. Appl. 59 (2010) 1057-1062.
[70] X. Xu, D. Jiang, C. Yuan, Multiple positive solutions for the boundary value problem of anonlinear fractional differential equation, Nonlinear Anal. 71 (2009) 4676-4688.
[71] T. J. Xiao, J. Liang, Laplace transforms and integrated, regularized semigroups in locallyconvex spaces, J. Funct. Anal. 148 (1997) 448-479.
[72] T. J. Xiao, J. Liang, The Cauchy Problem for Higher-order Abstract Differential Equations,Lecture Notes in Mathematics, Vol. 1701, Springer, Berlin, 1998.
[73] T. J. Xiao, J. Liang, Approximations of Laplace transforms and integrated semigroups, J.Funct. Anal. 172 (2000) 202-220.
[74] T. J. Xiao, J. Liang, J. van Casteren, Time dependent Desch-Schappacher type perturbationsof Volterra integral equations, Integral Equa. Opera. Theory 44 (4) (2002) 494-506.
[75] T. J. Xiao, J. Liang, Higher order abstract Cauchy problems: their existence and uniquenessfamilies, J. London Math. Soc. 67 (2003) 149-164.
[76] T. J. Xiao, J. Liang, Semilinear integrodifferential equations with nonlocal initial conditions,Comput. Math. Appl. 47 (2004) 863-875.
[77] T. J. Xiao, J. Liang, Second order differential operators with Feller-Wentzell type boundaryconditions, J. Funct. Anal. 254 (2008) 1467-1486.
[78] T. J. Xiao, J. Liang, J. Zhang, Pseudo almost automorphic functions to semilinear differen-tial equations in Banach spaces, Semigroup Forum 76 (2008) 518-524.
[79] T. J. Xiao, J. Liang, Blow-up and global existence of solutions to integral equations withinfinite delay in Banach spaces, Nonlinear Anal. TMA 71 (2009) e1442-e1447.
[80] T. J. Xiao, X. X. Zhu, J. Liang, Pseudo-almost automorphic mild solutions to nonau-tonomous differential equations and applications, Nonlinear Anal. 70 (2009) 4079-4085.
[81] B. Q. Yan, Boundary value problems on the half-line with impules and infinite delay, J.Math. Anal. Appl. 259 (2001) 94-114.
[82] B. Q. Yan, D. O’Regan, R. P. Agarwal, Multiple positive solutions via index theory forsingular boundary value problems with derivative dependence, Positivity, 11 (2007) 687-720.
[83] Q. Yao, The existence and multiplicity of positive solutions for a third-order three-pointboundary value problems, Acta Math. Appl. Sin. 19 (2003) 117-122.
[84] Y. Zhou, F. Jiao, J. Li, Existence and uniqueness for fractional neutral differential equationswith infinite delay, Nonlinear Anal. 71 (2009) 3249-3256.
[85] E. Zeidler, Nonlinear Functional Analysis and its Applications, I. Fixed-Point Theorems,Springer-Verlag, New York, 1986.
[86] A. Zhao, Z. Bai, Existence of solutions to first-order impulsive periodic boundary valueproblems, Nonlinear Anal. 71 (2009) 1970-1977.
[87] X. M. Zhang, M. Q. Feng, W. G. Ge, Existence results for nonlinear boundary-value prob-lems with integral boundary conditions in Banach spaces, Nonlinear Anal. 69 (2008) 3310-3321.