几类分数阶微分方程边值问题解的存在性
|
摘要
分数阶微分方程是常微分方程的一个重要分支.近年来,因其自身理论体系的不断完善以及与许多实际应用(如：物理学、机械力学、化学和工程学等等)密切的联系,受到了国内外数学界和自然科学界的重视并不断深入研究,分式微分方程已成为现代数学中一个重要研究方向之一
分数阶微分方程的边值问题是近年讨论的热点,是目前这方面研究中一个十分重要的领域.本文主要利用锥理论,不动点定理等非线性泛函的方法讨论了几类非线性分式微分方程(组)积分边值问题正解的存在性,得到了一些新的结果.根据内容本文分为以下四章：
第一章绪论,主要介绍了本文的研究课题,给出了相关概念及重要引理.
第二章在本章中,主要讨论了如下形式的分数阶微分方程积分边值问题其中Dα表示Riemann-Liouville分式导数,1<α(?)2,f:(0,1)×(0,∞)→[0,∞)且连续.ξ(s)递减且为黎曼积分.易知,非线性项f(x)在t=0,1与x=0时奇异.通过锥理论知识构造锥,由混合单调算子相关理论,得到方程正解的存在.
第三章在本章中,研究了如下形式的非线性分数阶微分方程Sturm-Liouville积分边值问题其中,cDα表示Caputo分式导数,并且1<α(?)2,而非线性项f∶(0,1)×R→R连续,g0,g91∶[0,1]→[0,∞)连续且为正.a,b均为非负实参数.运用Krasnoskii's不动点定理,在不同条件下,讨论方程的解的存在性.
第四章在前两章的基础上,本章讨论了一类分数阶微分方程组的积分边值问题.其中cDα为Caputo分式导数,1<α(?)2,以及连续,连续,并且a1(t)f1(t,0,0)或a2(t)f2(t,0,0)在(0,1)任何子区间都不全等；αi(?)0,βi(?)0,γi(?)0,δi(?)0,且为Riemann-Stieltjes积分,其中i=1,2.运用Leffett-Williams不动点定理,讨论方程组多个解的存在性.
Fractional differential equation is an important branch of ordinary differential equations. In recent years, fractional differential equation has been continue to in-depth study because of its theoretical system of continuous improvement and the close contact with many practical applications (such as:physics, mechanics, chemistry and engineering, etc.), by the international mathematical community and the importance of natural science. Fractional differential equations has become an important modern mathematics research direction.
Fractional differential equations is a hot topic in recent years, research in this area is a very important area. In this paper, using the cone theory, fixed point theorem for nonlinear functional methods such as the nonlinear fractional differential equation(for Systems) integral existence of positive solutions for boundary value problems, and obtained some new results. The thesis is divided into four sections according to contents.
Chapter 1 Preference, we introduce the main contents, then give the related concepts and important lemma of this paper.
Chapter 2 We consider the following fractional differential equations with integral boundary where 0<α≤2, the nonlinear items f:(0,1)×[0,+∞)→[0,+∞), and can be singular at t=0,1. By cone theory of knowledge structure, related by a mixed monotone operator theory, the existence of positive solutions.
Chapter 3 nonlinear fractional differential Sturm - Liouville integral boundary Where cDαis the Caputo fractional derivative of order 1<α≤2,f:[0,1]×R→R is continuous, g0, g1:[0,1]→[0,∞) are continuous and positive, a and b are nonnegative real parameter. Use Krasnoskii's fixed point theorem, under different conditions to discuss the existence of solutions of equations, this chapter gives the existence and multiplicity of (3.1.1).
Chapter 4 At the basis of the former two chapters, in this section we consider the Fractional Differential Equations with Integral Boundary Where cDαis the Caputo fractional derivative of order 1
分数阶微分方程的边值问题是近年讨论的热点,是目前这方面研究中一个十分重要的领域.本文主要利用锥理论,不动点定理等非线性泛函的方法讨论了几类非线性分式微分方程(组)积分边值问题正解的存在性,得到了一些新的结果.根据内容本文分为以下四章：
第一章绪论,主要介绍了本文的研究课题,给出了相关概念及重要引理.
第二章在本章中,主要讨论了如下形式的分数阶微分方程积分边值问题其中Dα表示Riemann-Liouville分式导数,1<α(?)2,f:(0,1)×(0,∞)→[0,∞)且连续.ξ(s)递减且为黎曼积分.易知,非线性项f(x)在t=0,1与x=0时奇异.通过锥理论知识构造锥,由混合单调算子相关理论,得到方程正解的存在.
第三章在本章中,研究了如下形式的非线性分数阶微分方程Sturm-Liouville积分边值问题其中,cDα表示Caputo分式导数,并且1<α(?)2,而非线性项f∶(0,1)×R→R连续,g0,g91∶[0,1]→[0,∞)连续且为正.a,b均为非负实参数.运用Krasnoskii's不动点定理,在不同条件下,讨论方程的解的存在性.
第四章在前两章的基础上,本章讨论了一类分数阶微分方程组的积分边值问题.其中cDα为Caputo分式导数,1<α(?)2,以及连续,连续,并且a1(t)f1(t,0,0)或a2(t)f2(t,0,0)在(0,1)任何子区间都不全等；αi(?)0,βi(?)0,γi(?)0,δi(?)0,且为Riemann-Stieltjes积分,其中i=1,2.运用Leffett-Williams不动点定理,讨论方程组多个解的存在性.
Fractional differential equation is an important branch of ordinary differential equations. In recent years, fractional differential equation has been continue to in-depth study because of its theoretical system of continuous improvement and the close contact with many practical applications (such as:physics, mechanics, chemistry and engineering, etc.), by the international mathematical community and the importance of natural science. Fractional differential equations has become an important modern mathematics research direction.
Fractional differential equations is a hot topic in recent years, research in this area is a very important area. In this paper, using the cone theory, fixed point theorem for nonlinear functional methods such as the nonlinear fractional differential equation(for Systems) integral existence of positive solutions for boundary value problems, and obtained some new results. The thesis is divided into four sections according to contents.
Chapter 1 Preference, we introduce the main contents, then give the related concepts and important lemma of this paper.
Chapter 2 We consider the following fractional differential equations with integral boundary where 0<α≤2, the nonlinear items f:(0,1)×[0,+∞)→[0,+∞), and can be singular at t=0,1. By cone theory of knowledge structure, related by a mixed monotone operator theory, the existence of positive solutions.
Chapter 3 nonlinear fractional differential Sturm - Liouville integral boundary Where cDαis the Caputo fractional derivative of order 1<α≤2,f:[0,1]×R→R is continuous, g0, g1:[0,1]→[0,∞) are continuous and positive, a and b are nonnegative real parameter. Use Krasnoskii's fixed point theorem, under different conditions to discuss the existence of solutions of equations, this chapter gives the existence and multiplicity of (3.1.1).
Chapter 4 At the basis of the former two chapters, in this section we consider the Fractional Differential Equations with Integral Boundary Where cDαis the Caputo fractional derivative of order 1
引文
[1]郭大钧.非线性分析中的半序方法[M].山东科技出版社,2001.
[2]郭大钧.非线性泛函分析[M].济南：山东科技出版社,2001.
[3]郭大钧,孙经先.抽象空间常微分方程[M].济南：山东科技出版社,1989.
[4]D.J.Guo, V.Lakshmikantham. Nonlinear Problems in Abstract Cone[M], Academic Press, Sandiego,1988.
[5]A.Arara, M.Benchohra. Fractional order differential equations on an unbounded domain[J]. Nonlinear Anal,2010,72 (2010)580-586.
[6]Lingju Kong. Second order singular boundary value problem with integral boundary condi-tions[J]. Nonlinear Anal,2010,72 (2010) 2628-2638.
[7]Daqing Jiang, Chengjun Yuan. The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application[J]. Nonlinear Anal,2010,72 (2010) 710-719.
[8]Ravi, P.Agarwal, V.Lakshmiknthham, Juan J.Nieto. On the concept of solution for fractional differential equations with uncertainty [J]. Nonlinear Anal,2010,72 (2010)2859-2862.
[9]Shuqin Zhang, positive solutions to singular boundary value problem for nonlinear fractional differential equation[J]. Comput.Math.Appl,2010,59 (2010)1300-1309.
[10]Xiaojie Xu, Daqing Jiang, Chengjun Yuan. Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation[J]. Nonlinear Anal,2009,71 (2009) 4676-4688.
[11]D.Guo. Nonlinear Functional Analysis, Shangdong Science and Technology press[J]. Jinan,1985 (in Chinese).
[12]V.Lakshmiknthham, Theory of fractional functional differential equations[J]. Nonlinear Anal, 2008,69(2008) 3337-3343.
[13]E.Zeider. Nonlinear Functional Analysis and its ApplicationsI:Fixed-Point Theorems[J]. Springer-Verlag, New York,1986.
[14]Wenyong Zhong, Wei Lin. Nonlocal and multiple-point boundary value problem for fractional differential equations[J]. shanghai, hunan, Comput.Math.Appl,2010,1345-1351.
[15]A.A.Kilbas, S.Marzan. Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions [J]. Differ.Equ,2005,1141 (2005) 84-89.
[16]S.G.Samko, A.A.Kilbas, O.I.Marichev. Fractional Integral And Derivatives[J]. Theory And Ap-plications, Gordon and Breach, Switzerland,1993.
[17]Zhang Shuqin. Existence of solution for a boundary value problem of fractional order[J]. Beijing, Acta Mathematica Scientia,2006,26 B(2):22,228.
[18]Zhanbing Bai, Tingting Qiu, Existence of Positive Solution for Singular Fractional Differential Equation[J]. shandong,2009.
[19]Z.B.Bai. Positive solutions for boundary value problem of nonlinear frac-tional differential equa-tion[J]. J.Math.Anal.Appl,2005,311 (2005) 495-505.
[20]Z.Zhao. On the existence of positive solutions for 2n-order singular boundary value problems[J]. Nonlinear Anal,2006, (64),2553-2561.
[21]X.Su. Boundary value problem for a coupled system of nonlinear fractional differential equa-tions[J]. Beijing, Appl.Math.Lett,2009,22 (2009) 64-69.
[22]Bashir Ahmad, S. Sivasundaram. On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order[J]. Saudi Arabia, Appl.Math.Comput,2010,23 (2010) 2189-2199.
[23]Moustafa El-Shahed, J.Nicto. Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order[J]. Comput.Math.Appl,2010,59 (2010) 3438-3443.
[24]Z.Zhao, Xinguang Zhang. C(I) Positive solutions of nonlinear singular differential equations for nonmonotonic function terms[J]. Nonlinear Anal,2007,66 (1):22-37.
[25]Shuqin Zhang. Positive solutions to singular boundary value problem for nonlinear fractional differential equation[J], Beijing, Comput.Math.Appl,2010,59 (2010) 1300-1309.
[26]Xinwei Su. Boundary value problem for a coupled system of nonlinear fractional differential equations[J]. Beijing, Appl.Math.Lett,2009,22 (2009) 64-69.
[27]S.Christopher, Goodrich. Existence of a posiitive solution to a class of fractional differential equations[J]. USA, Appl.Math.Comput,2010,23(2010) 1050-1055.
[28]D. O'Regan. Fixed-point theory for the sum of two operators, Appl.Math.Lett,1996,9 (1) (1996) 1-8.
[29]Xiping Liu, Mei Jia. Multiple solutions for fractional differential equations with nonlinear boundary conditions[J]. Comput.Math.Appl,2005,59 (2005) 2880-2886.
[30]Zhanbing Bai, Haishen Lu. Positive solutions for boundary value problem of nonlinear fractional differential equation[J]. J.Math.Anal.Appl,2005,311 (2005) 495-505.
[31]E.R.Kaufmann, E.Mboumi. Positive solutions of a boundary value problem for a nonlinear fractional diferential equation[J]. Electron. J.Qual.Theory Differ,2008, Equ.3 (2008)1-11.
[32]Xinwei Su. Boundary value problem for a coupled system of nonlinear fractional differential equations[J], Beijing, Appl.Math.Lett,2009,22 (2009) 64-69.
[33]Bashir Ahmada. Existence results for a coupled system of nonlinear fractional differential equa-tions with three-point boundary conditions[J]. Saudi Arabia, Comput.Math.Appl,2009,58 (2009) 1838-1843.
[34]Z.Zhao. Solutions and Green functions for some linear second-order three-point boundary value problems[J]. Comput.Math.Appl,2008,56 (2008) 104-113.
[35]E.R. Kaufmann, E.Mboumi. Positive solutions of a boundary value problem for a nonlinear fractional diferential equation[J]. Electron.J.Qual.Theory Differ. Equ,2008,3 (2008) 1-11.
[2]郭大钧.非线性泛函分析[M].济南：山东科技出版社,2001.
[3]郭大钧,孙经先.抽象空间常微分方程[M].济南：山东科技出版社,1989.
[4]D.J.Guo, V.Lakshmikantham. Nonlinear Problems in Abstract Cone[M], Academic Press, Sandiego,1988.
[5]A.Arara, M.Benchohra. Fractional order differential equations on an unbounded domain[J]. Nonlinear Anal,2010,72 (2010)580-586.
[6]Lingju Kong. Second order singular boundary value problem with integral boundary condi-tions[J]. Nonlinear Anal,2010,72 (2010) 2628-2638.
[7]Daqing Jiang, Chengjun Yuan. The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application[J]. Nonlinear Anal,2010,72 (2010) 710-719.
[8]Ravi, P.Agarwal, V.Lakshmiknthham, Juan J.Nieto. On the concept of solution for fractional differential equations with uncertainty [J]. Nonlinear Anal,2010,72 (2010)2859-2862.
[9]Shuqin Zhang, positive solutions to singular boundary value problem for nonlinear fractional differential equation[J]. Comput.Math.Appl,2010,59 (2010)1300-1309.
[10]Xiaojie Xu, Daqing Jiang, Chengjun Yuan. Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation[J]. Nonlinear Anal,2009,71 (2009) 4676-4688.
[11]D.Guo. Nonlinear Functional Analysis, Shangdong Science and Technology press[J]. Jinan,1985 (in Chinese).
[12]V.Lakshmiknthham, Theory of fractional functional differential equations[J]. Nonlinear Anal, 2008,69(2008) 3337-3343.
[13]E.Zeider. Nonlinear Functional Analysis and its ApplicationsI:Fixed-Point Theorems[J]. Springer-Verlag, New York,1986.
[14]Wenyong Zhong, Wei Lin. Nonlocal and multiple-point boundary value problem for fractional differential equations[J]. shanghai, hunan, Comput.Math.Appl,2010,1345-1351.
[15]A.A.Kilbas, S.Marzan. Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions [J]. Differ.Equ,2005,1141 (2005) 84-89.
[16]S.G.Samko, A.A.Kilbas, O.I.Marichev. Fractional Integral And Derivatives[J]. Theory And Ap-plications, Gordon and Breach, Switzerland,1993.
[17]Zhang Shuqin. Existence of solution for a boundary value problem of fractional order[J]. Beijing, Acta Mathematica Scientia,2006,26 B(2):22,228.
[18]Zhanbing Bai, Tingting Qiu, Existence of Positive Solution for Singular Fractional Differential Equation[J]. shandong,2009.
[19]Z.B.Bai. Positive solutions for boundary value problem of nonlinear frac-tional differential equa-tion[J]. J.Math.Anal.Appl,2005,311 (2005) 495-505.
[20]Z.Zhao. On the existence of positive solutions for 2n-order singular boundary value problems[J]. Nonlinear Anal,2006, (64),2553-2561.
[21]X.Su. Boundary value problem for a coupled system of nonlinear fractional differential equa-tions[J]. Beijing, Appl.Math.Lett,2009,22 (2009) 64-69.
[22]Bashir Ahmad, S. Sivasundaram. On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order[J]. Saudi Arabia, Appl.Math.Comput,2010,23 (2010) 2189-2199.
[23]Moustafa El-Shahed, J.Nicto. Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order[J]. Comput.Math.Appl,2010,59 (2010) 3438-3443.
[24]Z.Zhao, Xinguang Zhang. C(I) Positive solutions of nonlinear singular differential equations for nonmonotonic function terms[J]. Nonlinear Anal,2007,66 (1):22-37.
[25]Shuqin Zhang. Positive solutions to singular boundary value problem for nonlinear fractional differential equation[J], Beijing, Comput.Math.Appl,2010,59 (2010) 1300-1309.
[26]Xinwei Su. Boundary value problem for a coupled system of nonlinear fractional differential equations[J]. Beijing, Appl.Math.Lett,2009,22 (2009) 64-69.
[27]S.Christopher, Goodrich. Existence of a posiitive solution to a class of fractional differential equations[J]. USA, Appl.Math.Comput,2010,23(2010) 1050-1055.
[28]D. O'Regan. Fixed-point theory for the sum of two operators, Appl.Math.Lett,1996,9 (1) (1996) 1-8.
[29]Xiping Liu, Mei Jia. Multiple solutions for fractional differential equations with nonlinear boundary conditions[J]. Comput.Math.Appl,2005,59 (2005) 2880-2886.
[30]Zhanbing Bai, Haishen Lu. Positive solutions for boundary value problem of nonlinear fractional differential equation[J]. J.Math.Anal.Appl,2005,311 (2005) 495-505.
[31]E.R.Kaufmann, E.Mboumi. Positive solutions of a boundary value problem for a nonlinear fractional diferential equation[J]. Electron. J.Qual.Theory Differ,2008, Equ.3 (2008)1-11.
[32]Xinwei Su. Boundary value problem for a coupled system of nonlinear fractional differential equations[J], Beijing, Appl.Math.Lett,2009,22 (2009) 64-69.
[33]Bashir Ahmada. Existence results for a coupled system of nonlinear fractional differential equa-tions with three-point boundary conditions[J]. Saudi Arabia, Comput.Math.Appl,2009,58 (2009) 1838-1843.
[34]Z.Zhao. Solutions and Green functions for some linear second-order three-point boundary value problems[J]. Comput.Math.Appl,2008,56 (2008) 104-113.
[35]E.R. Kaufmann, E.Mboumi. Positive solutions of a boundary value problem for a nonlinear fractional diferential equation[J]. Electron.J.Qual.Theory Differ. Equ,2008,3 (2008) 1-11.