非局部边值问题正解的存在性和全局结构
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摘要
本文主要使用非线性泛函分析中的拓扑度理论研究时间测度上奇异微分方程多点边值问题和特征值问题正解的存在性、非局部边值问题正解的全局结构。全文共分六章。
第一章,系统地介绍本文的研究背景和主要的研究工作。
第二章,介绍时间测度链T上的相关定义和相应的计算公式、定理；介绍一些预备知识和引理。
第三章研究奇异2n阶微分方程三点边值问题[0,+∞)是连续的且ω在t=a和/或t=b处可能有奇性,f在u=0处有奇性。通过使用截断技术和算子逼近理论处理非线性项的奇性,再使用Kransnosel'skii不动点定理研究问题(1)正解的存在性。
第四章研究了奇异2n阶微分方程特征值问题其中λ是一个正参数,η,βi,γi,αi和非线性项ω,f都与方程(1)相同。利用Krein-Rutman定理获得了正线性算子的第一特征值和相应的正特征函数,再联合不动点指数定理,证明了特征值问题(2)正解的存在性、多重性,同时也给出了参数λ的存在区间。
第五章,考虑了二阶微分方程三点边值问题其中β≥0,0<η<1,0<αη<1和1+β-αη-αβ>0；非线性项ω∈C([0,1],(0,+∞))和f∈C(R+,R+),R+=[0,∞),满足对于u>0,有f(u)>0。使用Leray-Schauder全局连续性定理和分析技巧,研究了二阶微分方程边线性情况)和f0=∞、f∞=0(次线性情况)的条件及a∈[0,1+β/η+β)时,微分方程(3)正解的存在性和全局结构,同时也给出了不存在正解的情形。
第六章,我们研究了含积分边界条件的奇异二阶微分方程特征值问题其中u(s)dA(s)是Stieltjes积分且有A是非减的,λ是一个正参数；非线性项g∈C((0,1),(0,+∞))和f∈C([0,+∞),(0,+∞)),且夕在t=0和/或t=1可能有奇性和对于u>0有f(u)>0和f∞=+∞。利用Sturm-Liouville特征值理论、Leray-Schauder全局连续性定理、不动点指数定理和上下解方法相结合,证明了特征值问题(4)正解的存在性、多重性和不存在性,同时我们也给出了特征值问题(4)正解的渐近性态和参数λ的相应区间。
This thesis mainly investigates the existence of positive solutions to multi-point boundary value problems and the eigenvalue problems on time scales for singular dif-ferential equations,and the global structure of positive solutions for nonlocal boundary value problems by utilizing the theory of topological degrees of nonlinear function analysis.There are six chapters.
Chapter 1 presents the research background and main results of this thesis.
Chapter 2 is about preliminaries of this thesis.Basic definitions,formulas and theorems on time scales are stated.
Chapter 3 is concerned with the following three-point boundary value problems for singular 2nth-order differential equations whereη∈(a,b),βi≥0,1<γi Chapter 4 studies the eigenvalue problems for singular 2nth-order differential equations
whereλis a positive parameter,andη,βi,γi,αi,w,f are as above.We obtain the first eigenvalue of the positive linear operator and the corresponding eigenfunction by utilizing Krein-Rutman theorem. Moreover, by the fixed point index theorem, we show the existence and multiplicity of positive solution to the eigenvalue problem (6) and the existence interval of the parameterλ.
In Chapter 5, we consider the three-point boundary value problems for second-order differential equations whereβ≥0,0<η<1,0<αη<1 and 1+β-αη-αβ> 0; The nonlinearity w∈
(7) by using Leray-Schauder theorem and analysis technique. Furthermore, we discuss the case of nonexistence of positive solutions.
Chapter 6 investigates the singular second-order differential equation with integral boundary condition
where∫01u(s)dA(s) is a Stieltjes integral, A is nondecreasing,λis a positive parame-ter, g∈C((0,1), (0,+∞)),f∈C([0,+∞), (0,+∞)), and g may be singular at t= 0 and/or t=1, f(u)> 0 for u> 0, and f∞=+∞. We obtain criteria of the existence, multiplicity and nonexistence of positive solutions to the equation (8) by utilizing the Sturm-Liouville eigenvalue theory, Leray-Schauder global continuation theorem, and fixed point index theory. Meanwhile, we give the asymptotic of the solutions and the corresponding interval of the parameterλ.
第一章,系统地介绍本文的研究背景和主要的研究工作。
第二章,介绍时间测度链T上的相关定义和相应的计算公式、定理；介绍一些预备知识和引理。
第三章研究奇异2n阶微分方程三点边值问题[0,+∞)是连续的且ω在t=a和/或t=b处可能有奇性,f在u=0处有奇性。通过使用截断技术和算子逼近理论处理非线性项的奇性,再使用Kransnosel'skii不动点定理研究问题(1)正解的存在性。
第四章研究了奇异2n阶微分方程特征值问题其中λ是一个正参数,η,βi,γi,αi和非线性项ω,f都与方程(1)相同。利用Krein-Rutman定理获得了正线性算子的第一特征值和相应的正特征函数,再联合不动点指数定理,证明了特征值问题(2)正解的存在性、多重性,同时也给出了参数λ的存在区间。
第五章,考虑了二阶微分方程三点边值问题其中β≥0,0<η<1,0<αη<1和1+β-αη-αβ>0；非线性项ω∈C([0,1],(0,+∞))和f∈C(R+,R+),R+=[0,∞),满足对于u>0,有f(u)>0。使用Leray-Schauder全局连续性定理和分析技巧,研究了二阶微分方程边线性情况)和f0=∞、f∞=0(次线性情况)的条件及a∈[0,1+β/η+β)时,微分方程(3)正解的存在性和全局结构,同时也给出了不存在正解的情形。
第六章,我们研究了含积分边界条件的奇异二阶微分方程特征值问题其中u(s)dA(s)是Stieltjes积分且有A是非减的,λ是一个正参数；非线性项g∈C((0,1),(0,+∞))和f∈C([0,+∞),(0,+∞)),且夕在t=0和/或t=1可能有奇性和对于u>0有f(u)>0和f∞=+∞。利用Sturm-Liouville特征值理论、Leray-Schauder全局连续性定理、不动点指数定理和上下解方法相结合,证明了特征值问题(4)正解的存在性、多重性和不存在性,同时我们也给出了特征值问题(4)正解的渐近性态和参数λ的相应区间。
This thesis mainly investigates the existence of positive solutions to multi-point boundary value problems and the eigenvalue problems on time scales for singular dif-ferential equations,and the global structure of positive solutions for nonlocal boundary value problems by utilizing the theory of topological degrees of nonlinear function analysis.There are six chapters.
Chapter 1 presents the research background and main results of this thesis.
Chapter 2 is about preliminaries of this thesis.Basic definitions,formulas and theorems on time scales are stated.
Chapter 3 is concerned with the following three-point boundary value problems for singular 2nth-order differential equations whereη∈(a,b),βi≥0,1<γi
whereλis a positive parameter,andη,βi,γi,αi,w,f are as above.We obtain the first eigenvalue of the positive linear operator and the corresponding eigenfunction by utilizing Krein-Rutman theorem. Moreover, by the fixed point index theorem, we show the existence and multiplicity of positive solution to the eigenvalue problem (6) and the existence interval of the parameterλ.
In Chapter 5, we consider the three-point boundary value problems for second-order differential equations whereβ≥0,0<η<1,0<αη<1 and 1+β-αη-αβ> 0; The nonlinearity w∈
(7) by using Leray-Schauder theorem and analysis technique. Furthermore, we discuss the case of nonexistence of positive solutions.
Chapter 6 investigates the singular second-order differential equation with integral boundary condition
where∫01u(s)dA(s) is a Stieltjes integral, A is nondecreasing,λis a positive parame-ter, g∈C((0,1), (0,+∞)),f∈C([0,+∞), (0,+∞)), and g may be singular at t= 0 and/or t=1, f(u)> 0 for u> 0, and f∞=+∞. We obtain criteria of the existence, multiplicity and nonexistence of positive solutions to the equation (8) by utilizing the Sturm-Liouville eigenvalue theory, Leray-Schauder global continuation theorem, and fixed point index theory. Meanwhile, we give the asymptotic of the solutions and the corresponding interval of the parameterλ.
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[6]D. R. Anderson, Solutions to second order three-point problems on time scales, J. Differen-tial Equ. Appl.,8 (2002) 673-688.
[7]D. R. Anderson, R. I. Avery, An even-order three-point boundary value problem on time scales, J. Math. Anal. Appl.,291 (2004) 514-525.
[8]D. R. Anderson, I. Y. Karaca, Higher-order three-point boundary value problem on time scales, Comput. Math. Appl.,56 (2008) 2429-2443.
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[11]F. M. Atici, G. Sh. Guseinov, On Green's functions and positive solutions for boundary value problems on time scales, J. Comput. Appl. Math.,141 (2002) 75-99.
[12]F. M. Atici, D. C. Biles, A. Lebedinsky, An application of time scales to economics, Math. Comput. Modelling,43 (2006) 718-726.
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[14]R. I. Avery, J. Henderson, Two positive fixed points of nonlinear operators on ordered Ba-nach spaces, Comm. Appl. Nonlinear Anal.,8 (2001) 27-36.
[15]R. I. Avery, D. R. Anderson, Fixed point theorem of cone expansion and compression of functional type, J. Differerential Equ.Appl.,8 (2002) 1073-1083.
[16]B. Aulbach, L. Neidhart, Integration on measure chains, in:Proceedings of the Sixth Inter-national Conference on Difference Equations, CRC, Boca Raton, FL,2004, pp.239-252.
[17]P. A. Binding, B. P. Rynne, The spectrum of the periodic p-Laplacian, J. Differential Equa-tions,235 (2007) 199-218.
[18]K. L. Boey, P. J. Y. Wong, Two-point right focal eigenvalue problems on time scales, Appl. Math. Comput.,167 (2005) 1281-1303.
[19]K. L. Boey, P. J. Y. Wong, Positive solutions of two point right focal boundary value prob-lems on time scales, Comput. Math. Appl.,52 (2006),555-576.
[20]M. Bohner, A. Peterson, Dynamic Equations on Time Scales, An Introduction with Appli-cations, Birkauser Boston, Cambridge, MA,2001.
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