临界点理论在几类离散边值问题中的应用
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摘要
本博士论文运用临界点理论分别研究了离散的二阶混合边值问题,Neumann边值问题,周期边值问题,具有p-Laplace算子二阶差分边值问题,得到一系列有关解,多个解的存在性和唯一性的结果,推广并改进了已有文献的相关存在性结论.所得主要结果概括如下;
第一章简述了问题产生的历史背景和本文的主要工作.
第二章讨论了一类离散两点混合边值问题.运用Green函数和线性算子的分解,我们构造了一个变分框架,再分别利用强单调算子原理和临界点理论。建立了若干保证问题有唯一解或至少有一个非平凡解的充分条件.这些条件包涵了超线性类型.
第三章考虑了一类离散两点Neumann边值问题.我们构造了一个新的变分结构,利用临界点理论中的环绕定理和鞍点定理建立了该问题存在一个或两个非平凡解的若干充分条件.
第四章研究了一类高维非线性离散两点边值问题.此类边值问题包含了Dirichlet边值问题和混合边值问题.我们应用山路定理解决该系统至少存在两个非平凡解的问题,大大改进和推广了已知结果.
第五章讨论了一个具有p-Laplace算子依赖于参数λ的离散Dirichlet边值问题.运用Bonanno提出的具有三个临界点的定理,当特征值λ位于确定的两个开区间内时,我们证明了该问题至少有三个解.而且,当λ位于其中一个开区间时,所有的解是一致有界的.
第六章考虑了一类离散凸Hamilton系统.我们利用对偶理论和一个新的变分原理,建立了周期解存在的若干判别准则.解所对应的临界点极小化一个对偶作用,这个对偶作用是被限制在某个确定空间的子集上的.
This dissertation studies discrete mixed boundary value problem, Neumann boundary value problem, periodic boundary value problem, Dirichlet boundary value problem with the one-dimensional p-Laplacian, and higher dimensional boundary value problem. By using critical point theory, we obtain a series of results concerning with the existence of at least one nontrivial solution or multiple solutions and the uniqueness of solution. The obtained results extend and improve some known results in the existing references.
In the first Chapter, we introduce the historical background of problems which will be investigated and the main results of this paper.
In Chapter 2, we consider a discrete nonlinear mixed boundary value problem. First, by virtue of Green's functions and separation of linear operator, we obtain variational framework. Then, by employing the strongly monotone some conditions including suplinear case, to guarantee that the problem has a unique solution or at least one nontrivial solution.
In Chapter 3, by constructing a new variational frame, using the linking theorem and the saddle point theorem in the critical point theory respectively, we study the existence of one solution or two solutions for the discrete two-point Neumann boundary value problem, and some sufficient conditions are obtained.
The purpose of Chapter 4 is to study the existence of multiple nontrivial solutions for a class of higher dimensional discrete boundary value problems including the Dirichlet boundary value problems and the mixed boundary value problems. Applying the mountain pass theorem in the critical point theory, we obtain some sufficient conditions to guarantee that these problems have at least two nontrivial solutions. Our results improve and generalize the some known results.
In Chapter 5, we deal with a Dirichlet boundary value problem for p-Laplacian difference equations depending on a parameter A. By using the three critical points theorem established by Bonanno, we verify the existence of at least three solutions when A is in two exactly determined open intervals respectively. Moreover, the norms of these solutions are uniformly bounded with respect to A belonging to one of the two open intervals.
Chapter 6 mainly consider a convex discrete Hamiltonian system. Basing on the dual theory and a new variational principle, we obtain some criteria for the existence of a periodic solution. Such a solution corresponds to a critical point minimizes the dual action restricted to a subset of some determined space.
第一章简述了问题产生的历史背景和本文的主要工作.
第二章讨论了一类离散两点混合边值问题.运用Green函数和线性算子的分解,我们构造了一个变分框架,再分别利用强单调算子原理和临界点理论。建立了若干保证问题有唯一解或至少有一个非平凡解的充分条件.这些条件包涵了超线性类型.
第三章考虑了一类离散两点Neumann边值问题.我们构造了一个新的变分结构,利用临界点理论中的环绕定理和鞍点定理建立了该问题存在一个或两个非平凡解的若干充分条件.
第四章研究了一类高维非线性离散两点边值问题.此类边值问题包含了Dirichlet边值问题和混合边值问题.我们应用山路定理解决该系统至少存在两个非平凡解的问题,大大改进和推广了已知结果.
第五章讨论了一个具有p-Laplace算子依赖于参数λ的离散Dirichlet边值问题.运用Bonanno提出的具有三个临界点的定理,当特征值λ位于确定的两个开区间内时,我们证明了该问题至少有三个解.而且,当λ位于其中一个开区间时,所有的解是一致有界的.
第六章考虑了一类离散凸Hamilton系统.我们利用对偶理论和一个新的变分原理,建立了周期解存在的若干判别准则.解所对应的临界点极小化一个对偶作用,这个对偶作用是被限制在某个确定空间的子集上的.
This dissertation studies discrete mixed boundary value problem, Neumann boundary value problem, periodic boundary value problem, Dirichlet boundary value problem with the one-dimensional p-Laplacian, and higher dimensional boundary value problem. By using critical point theory, we obtain a series of results concerning with the existence of at least one nontrivial solution or multiple solutions and the uniqueness of solution. The obtained results extend and improve some known results in the existing references.
In the first Chapter, we introduce the historical background of problems which will be investigated and the main results of this paper.
In Chapter 2, we consider a discrete nonlinear mixed boundary value problem. First, by virtue of Green's functions and separation of linear operator, we obtain variational framework. Then, by employing the strongly monotone some conditions including suplinear case, to guarantee that the problem has a unique solution or at least one nontrivial solution.
In Chapter 3, by constructing a new variational frame, using the linking theorem and the saddle point theorem in the critical point theory respectively, we study the existence of one solution or two solutions for the discrete two-point Neumann boundary value problem, and some sufficient conditions are obtained.
The purpose of Chapter 4 is to study the existence of multiple nontrivial solutions for a class of higher dimensional discrete boundary value problems including the Dirichlet boundary value problems and the mixed boundary value problems. Applying the mountain pass theorem in the critical point theory, we obtain some sufficient conditions to guarantee that these problems have at least two nontrivial solutions. Our results improve and generalize the some known results.
In Chapter 5, we deal with a Dirichlet boundary value problem for p-Laplacian difference equations depending on a parameter A. By using the three critical points theorem established by Bonanno, we verify the existence of at least three solutions when A is in two exactly determined open intervals respectively. Moreover, the norms of these solutions are uniformly bounded with respect to A belonging to one of the two open intervals.
Chapter 6 mainly consider a convex discrete Hamiltonian system. Basing on the dual theory and a new variational principle, we obtain some criteria for the existence of a periodic solution. Such a solution corresponds to a critical point minimizes the dual action restricted to a subset of some determined space.
引文
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[2]Agarwal R P.Focal Boundary Value Problems for Differential and Difference Equa-tions.Kluwer Academic Publishers,1998
[3]Agarwal R P,O'Regan D,Wong P J.Positive solutions of Differential,Difference and Integral Equations.Dordrecht;Kluwer Academic Publishers,1999
[4]O'Regan D.Theory of Singular Boundary Value Problems,Scientific World,1994
[5]O'Regan D.Existence Theory for Nonlinear Ordinary Differential Equations.Kluwer Acdemic Publishers,1997
[6]Bonanno G,Giovanneli N.An eigenvalue Dirichlet problem involving the p-Laplacian with discontinuous nonlinearities.J.Math.Anal.Appl.,2005,308;596-604
[7]Ding W,Han M.Periodic boundary value problem for the second order impulsive functional differential equations.Appl.Math.Comput.,2004,155;709-726
[8]Ding W,Han M,Yah J.Periodic boundary value problems for the second order functional differential equations.J.Math.Anal.Appl.,2004,298;341-351
[9]Halidias N.Existence theorems for noncoercive elliptic problems with discontinu-ous nonlinearities.Nonlinear Anal.,2003,54;1355-1364
[10]Hu S,Kourogenis N C,Papageorglou N S.Nonlinear elliptic eigenvalue problems with discontinuities.J.Math.Anal.Appl.,1999,233;406-424
[11]Lee E K,Lee Y H.Multiple positive solutions of singular two point boundary value problems for second order impulsive differential equations.Appl.Math.Comput.,2004,158;745-759
[12]Liu B.Positive solutions of singular three-point boundary value problems for the one-dimensional p-Laplacian.Comput.Math.Appl.,2004,48;913-925
[13]Mavridis K G,Tsamatos P C.Positive solutions for a Floquet functional boundary value problem.J.Math.Anal.Appl.,2004,296;165-182
[14]Yang X.The Fredholm alternative for the one-dimensional p-Laplacian.J.Math.Anal.Appl.,2004,299;494-507
[15]Zhang G,Liu S.Three symmetric solutions for a class of elliptic equations involving the p-Laplacian with discontinuous nonlinearities in R~N.Nonlinear Anal.,2007,67(7);2232-2239
[16]Zhang G,Zhang W,Wang Y.Existence result for a class of singular quasilinear elliptic equations with critical growth.Nonlinear Anal.,in press
[17]Gasinski L,Papageorgiou N S.Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems.Chapman and Hall/CRC,2005
[18]郭大钧.非线性泛函分析.济南;山东科学技术出版社,2001
[19]陆文端.微分方程中的变分方法.北京;科学出版社,2003
[20]Mawhin J,Willem M.Critical Points Theory and Hamiltonian Systems.Boston;Springer-Verlag,1989
[21]Struwe M.Variational Methods.Heidelberg;Springer-Verlag,1996
[22]张恭庆.临界点理论及其应用.上海;上海科学技术出版社,1986
[23]Anderson D R,Rachunkova I,Tisdell C C.Solvability of discrete Neumann bound-ary value problems.J.Math.Anal.Appl.,2007,331;736-741
[24]Aykut N.Existence of positive solutions for boundary value problems of second-order functional difference equations.Comput.Math.Appl.,2004,48;517-527
[25]Cabada A,Otero-Espinav V.Fixed sign solutions of second-order difference equa-tions with Neumann boundary conditions.Comput.Math.Appl.,2003,45;1125-1136
[26]Cheung W S,Ren J,Wong P J Y,etc.Multiple positive solutions for discrete nonlocal boundary value problems.J.Math.Anal.Appl.,2007,330;990-915
[27]Rachunkova I,Christopher C T.Existence of non-spurious solutions to discrete Dirichlet problems with lower and upper solutions.Nonlinear Anal.,2007,67;1236-1245
[28]Thompson H B,Tisdell C.Boundary value problems for systems of difference equations associated with systems of second-order ordinary differential equations.Appl.Math.Lett.,2002,15;761-766
[29]Tisdell C C.The uniqueness of solutions to discrete,vector,two-point boundary value problems.Appl.Math.Lett.,2003,16;1321-1328
[30]Wong P J Y,Xie L.Three symmetric solutions of Lidstone boundary value prob-lems for difference and partial difference equations.Comput.Math.Appl.,2003,45;1445-1460
[31]Guo Z M,Yu J S.Existence of periodic solutions and subharmonic solutions on second order superlinear difference equations.Sci.China Ser.A.,2003,33(3);226-235
[32]Agarwal R P,Perera K,O'Regan D.Multiple positive solutions of singular and nonsingular discrete problems via variational methods.Nonlinear Anal.,2004,58;69-73
[33]Zhang G,Yang Z L.Existence of 2~n nontrivial solutions for discrete two-point boundary value problems.Nonlinear Anal.,2004,59;1181-1187
[34]Cabada A,Pouso R L.Existence result for the problem(φ(u')′ = qf(t,u,u')with periodic and Neumann boundary conditions.Nonlinear Anal.,1997,30(3);1733-1742
[35]Dang H,Oppenheimer S F.Existence and uniqueness results for some nonlinear boundary value problems.J.Math.Anal.Appl.,1996,198;35-48
[36]Manasevich R,Mawhin J.Periodic solutions for boundary value problem with sin-gular and discontinuous nonlinearity.Acta Math.Appl.Sin.,2000,16(1);87-92
[37]Manasevich R,Sweers G.A noncooperative elliptic system with p-Laplacians that preserves positivity.Nonlinear Anal.,1999,36;511-528
[38]O'Regan D.some general existence principles and results for(φ(y'))' =qf(t,y,y'),0 < t < 1.SIAM J.Math.Anal.,1993,24(3);648-668
[39]Wang J.The existence of positive solutions for the one-dimensional p-Laplacian.Proc.Amer.Math.Soc.,1997,125(8);2275-2283
[40]Wong F.Uniqueness of positive solutions for Sturm-Liouville boundary value prob-lems.Amer.Math.Soc.,1998,126(2);365-374
[41]Zhang M.Nonuniform nonresonance at the first eigenvalue of the p-Laplacian.Nonlinear Anal.,1997,29(1);41-51
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