具共振条件下多点边值问题解的存在性
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摘要
常微分方程起源于应用学科,诸如核物理学,气体动力学,流体力学,非线性光学等.常微分方程边值问题是常微分方程理论研究中最重要的课题之一.常微分方程边值问题在共振条件下解的存在性近十年来受到学者的关注.但是对于高阶复杂边值问题与含p-Laplacian算子型方程边值问题在共振情况的可解性的研究还不多见.针对这种情况本文作了如下研究:第一章我们着重介绍了问题的起源和相关背景,以及问题的发展现状和趋势.第二章,我们用Mawhin重合度定理研究了共振条件下一类四阶四点边值问题的可解性,在增长条件下得到解存在的充分条件.第三章,我们讨论了一类含p-Laplacian算子型多点边值问题在共振条件下的可解性,用Leray-Schauder度与Brouwer度得出解存在的两个充分条件,我们的结果推广改进了现有文献的一些结果.第四章,我们还用Leray-Schauder度与Brouwer度讨论了一类含p-Laplacian算子型多点边值问题具有两个临界条件的可解性,这个讨论方法与已有的结果的讨论方法不同.
今后,我们还可以结合其他的一些工具,讨论共振问题的正解与多解的存在性,得到更全面的结果.
The ordinary differential equation boundary value problem is one of the most important branches of ordinary differential equations. It is resulted from physics, chemistry, biology, medicine, economics, engineering, cybernetics and so on. Boundary value problem at resonance have been studied by several authors in recent ten years. But the existence of high-order with complex boundary value conditons and equation including p-Laplacian operator has rarely been studied. So in this paper, we consider the problem as following: In section I, we give an introduction of the origin of the problems, the status of the research on related issues, and the trend of the future development. In section 2, result for solvability of 4-point boundary value problems of fourth order differential problem at resonance is obtained by using coincidence degree theory due to Mawhin. An sufficient condition is given under non-linear growth restriction on f. In section 3, we study the existence of p-Laplace-like equtions subject to multi-point boundary value problems at resonance. By using degree theory, we obtained two sufficient conditions for solvability, some known results are improved. In section 4, we use Brouwer degree and Leray-Schauder degree to discuss solvability of multi-point boundary value problem with two critical conditions. the methods used in the paper are different from some known results.
Later, we can consider boundary value problem at resonance with different tools to obtain the existence of postive solutions and multi-solutions.
今后,我们还可以结合其他的一些工具,讨论共振问题的正解与多解的存在性,得到更全面的结果.
The ordinary differential equation boundary value problem is one of the most important branches of ordinary differential equations. It is resulted from physics, chemistry, biology, medicine, economics, engineering, cybernetics and so on. Boundary value problem at resonance have been studied by several authors in recent ten years. But the existence of high-order with complex boundary value conditons and equation including p-Laplacian operator has rarely been studied. So in this paper, we consider the problem as following: In section I, we give an introduction of the origin of the problems, the status of the research on related issues, and the trend of the future development. In section 2, result for solvability of 4-point boundary value problems of fourth order differential problem at resonance is obtained by using coincidence degree theory due to Mawhin. An sufficient condition is given under non-linear growth restriction on f. In section 3, we study the existence of p-Laplace-like equtions subject to multi-point boundary value problems at resonance. By using degree theory, we obtained two sufficient conditions for solvability, some known results are improved. In section 4, we use Brouwer degree and Leray-Schauder degree to discuss solvability of multi-point boundary value problem with two critical conditions. the methods used in the paper are different from some known results.
Later, we can consider boundary value problem at resonance with different tools to obtain the existence of postive solutions and multi-solutions.
引文
[1]Agarwal R.P..Boundary Value Problems fbr Higher Order Differential Equations.Singapore:World Scientific,1986.
[2]郑祖庥.泛函微分方程理论.合肥:安徽教育出版社,1994.
[3]王国灿,丁培桂,郑成德.量子力学中的TFD方程边值问题的存在性.应用数学与力学,2000,21(2):615-622
[4]郭大钧.非线性泛函分析.(第二版).济南:山东科学技术出版社,1985.
[5]郭大钧,孙经先.抽象空间常微分方程.(第二版).济南:山东科学技术出版社,2003.
[6]郭大钧,孙经先,刘兆理.非线性常微分方程泛函方法.济南:山东科学技术出版社,1995.
[7]Ma R.Y..Existence theorems for a second order m-point boundary value problem.J.Math.Anal.Appl.,1997,211:545-555
[8]Feng W.,Webb J.R.L..Solvability of m-point boundary value problems with nonlinear growth.J.Math.Appl.,1997,212:467-480
[9]Feng W.,Webb J.R.L..Solvability of three point boundary value problems at resonance.Nonlinear Analysis,1997,30:3227-3238
[10]Liu Bin,Yu Jianshe.Solvability of multi-point boundary value problem for second order ordinary differential equations.Appl.Math.Comput.,1998,89:133-146
[11]Liu Bin,Yu Jianshe.Solvability of multi-point boundary value problem at resonance(Ⅰ).Indian J.Pure and Appl.Math,2002,33:475-494
[12]Liu Bin.Solvability of multi-point boundary value problem at resonance(Ⅱ).Appl.Math.Comput.,2003,136:353-377
[13]Liu Bin,Yu Jianshe.Solvability of multi-point boundary value problem at resonance (Ⅲ).Appl.Math.Comput.,2002,129:119-143
[14]Liu Bin.Solvability of multi-point boundary value problem at resonance(Ⅳ).Appl.Math.Comput.,2003,143:275-299
[15]Du Z.J.,Lin X.J.,Ge W.G..On a third-order multi-point boundary value problem at resonance.J.Math.Anal.,2005,302:217-299
[16]林晓洁,杜增吉,葛渭高.具共振条件下的高阶多点边值问题解的存在性.应用数学和力学,2006,27(5):624-630
[17]杜增吉,林晓洁,葛渭高.具共振条件下一类三阶非局部边值问题的可解性.数学学报,2006,49(1):87-94
[18]Lin X.J.,Du Z.J.,Ge W.G..Solvability of multipoint boundary value problems at resonance for higher-order ordinary differential equations.Computers and Mathematics with Applications,2005,49:1-11
[19]Du Z.J.,Lin X.J.,Ge W.G..Some higher-order multi-point boundary value problem at resonance.Journal of Computational and Applied Mathematics,2005,177:55-65
[20]Nickolai Kosmatov.A multi-point boundary value problem with two critical conditions.Nonlinear Analysis,2006,65:622-633
[21]薛春艳,葛渭高.共振条件下多点边值问题解的存在性.数学学报,2005,48(2):281-290
[22]M.García-Huidobro,Chaitan P.Gupta,R.Manásevich.A Dirichelet-Neumann m-point BVP with a p-Laplacian-like operator.Nonlinear Anal,2005,62:1067-1089
[23]M.García-Huidobro,C.P.Gupta,R.Manásevich.An m-point boundary value problem of Neumann type for a p-Laplacian like operator.Nonlinear Anal,2004,56:1071-1089
[24]M.García-Huidobro,C.P.Gupta,R.Manásevich.Solvability for a non-linear three-point boundary value problem with p-Laplace-like operator at resonance.Abstr.Appl.Anal,2001,164:191-213
[25]Chen Shihua,Wei Ni,Wang Changping.Positive solution of fourth ordinary differential equation with four-point boundary conditions.Appl.Math.Lett.,2006,19:161-168
[26]廉立芳,葛渭高.共振情况下m点p-Laplacian算子边值问题解的存在性.应用数学学报,2005,28(2):288-295
[27]Mawhin J..Topological degree methods in nonlinear boundary value problems.IN:NSFCBMS Regional Conference Series in Mathematics,American Mathematical Society,Providence:Rhode Island,USA,1979.
[28]Ge Weigao,Ren Jingli.An extension of Mawhin's continuation theorem and its application to boundary value problems with a p-Laplacian.Nonlinear Anal.,2004,58:477-488
[29]Feng W.,Webb J.R.L.,Solvability of m-point boundary value problems with nonlinear growth.J.M.Anal.Appl.,1997,212:467-480
[30]Li Cuizhe,Li Zhiyan.Existence of solutions of third-order p-Laplacian resonant multipoint boundary value problems.Transactions of Beijing Institute of Technology,2004, 24(11):1024-1029(in Chinese)
[31]白定勇,马如云.p-Laplacian算子型奇异边值问题的正解.数学物理学报,2005,25(2):166-170
[32]Zhang Jianjun,Liu Wenbin,Chen Taiyong,Zhang Huixing.Solvability of p-Laplace equations subject to three-point boundary value problems.Appl.Math.Comput.,2006,179:688-695
[33]He Zhimin,Yu Jianshe.Periodic boundary value problem for first-order impulsive functional differential equations.J.Comp.Appl.Math.,2002,138:205-217
[34]夏道行等.实变函数论与泛函分析(下册).北京:人民教育出版社,1979.
[35]张恭庆,林源渠.泛函分析讲义(下册).北京:北京大学出版社,1987.
[36]Krasnoselskii M..Positive Solutions of Operator Equations.Groningen,Noorhoff,1964.
[37]Deimling Klaus.Nonlinear Functional Analysis.New York:Springer-Verlag,1985.
[38]马如云.非线性常微分方程非局部问题.北京:科学出版社,2004.
[39]郭志明,庾建设.二阶超线性差分方程周期解与次调和解的存在性.中国科学(A),2003,33(3):226-235
[40]Ma Shiwang,Wang Zhicheng,Yu Jianshe.An abstract existence theorem at resonance and its applications.Journal of Differential Equations,1998,145(2):274-293
[41]Yang Xiaojing.Multiple periodic solutions of a class of p-Laplacian.J.M.Anal.Appl.,2006,314:17-29
[42]鲁世平,葛渭高.具偏差变元的二阶p-Laplacian方程周期解存在性问题.数学学报,2005,48(5):841-850
[43]Peng Shiguo,Xu Zhiting.On the existence of periodic solutions fbr a class of p-Laplcacian system.J.Math.Anal.Appl.,2007,325:166-174
[2]郑祖庥.泛函微分方程理论.合肥:安徽教育出版社,1994.
[3]王国灿,丁培桂,郑成德.量子力学中的TFD方程边值问题的存在性.应用数学与力学,2000,21(2):615-622
[4]郭大钧.非线性泛函分析.(第二版).济南:山东科学技术出版社,1985.
[5]郭大钧,孙经先.抽象空间常微分方程.(第二版).济南:山东科学技术出版社,2003.
[6]郭大钧,孙经先,刘兆理.非线性常微分方程泛函方法.济南:山东科学技术出版社,1995.
[7]Ma R.Y..Existence theorems for a second order m-point boundary value problem.J.Math.Anal.Appl.,1997,211:545-555
[8]Feng W.,Webb J.R.L..Solvability of m-point boundary value problems with nonlinear growth.J.Math.Appl.,1997,212:467-480
[9]Feng W.,Webb J.R.L..Solvability of three point boundary value problems at resonance.Nonlinear Analysis,1997,30:3227-3238
[10]Liu Bin,Yu Jianshe.Solvability of multi-point boundary value problem for second order ordinary differential equations.Appl.Math.Comput.,1998,89:133-146
[11]Liu Bin,Yu Jianshe.Solvability of multi-point boundary value problem at resonance(Ⅰ).Indian J.Pure and Appl.Math,2002,33:475-494
[12]Liu Bin.Solvability of multi-point boundary value problem at resonance(Ⅱ).Appl.Math.Comput.,2003,136:353-377
[13]Liu Bin,Yu Jianshe.Solvability of multi-point boundary value problem at resonance (Ⅲ).Appl.Math.Comput.,2002,129:119-143
[14]Liu Bin.Solvability of multi-point boundary value problem at resonance(Ⅳ).Appl.Math.Comput.,2003,143:275-299
[15]Du Z.J.,Lin X.J.,Ge W.G..On a third-order multi-point boundary value problem at resonance.J.Math.Anal.,2005,302:217-299
[16]林晓洁,杜增吉,葛渭高.具共振条件下的高阶多点边值问题解的存在性.应用数学和力学,2006,27(5):624-630
[17]杜增吉,林晓洁,葛渭高.具共振条件下一类三阶非局部边值问题的可解性.数学学报,2006,49(1):87-94
[18]Lin X.J.,Du Z.J.,Ge W.G..Solvability of multipoint boundary value problems at resonance for higher-order ordinary differential equations.Computers and Mathematics with Applications,2005,49:1-11
[19]Du Z.J.,Lin X.J.,Ge W.G..Some higher-order multi-point boundary value problem at resonance.Journal of Computational and Applied Mathematics,2005,177:55-65
[20]Nickolai Kosmatov.A multi-point boundary value problem with two critical conditions.Nonlinear Analysis,2006,65:622-633
[21]薛春艳,葛渭高.共振条件下多点边值问题解的存在性.数学学报,2005,48(2):281-290
[22]M.García-Huidobro,Chaitan P.Gupta,R.Manásevich.A Dirichelet-Neumann m-point BVP with a p-Laplacian-like operator.Nonlinear Anal,2005,62:1067-1089
[23]M.García-Huidobro,C.P.Gupta,R.Manásevich.An m-point boundary value problem of Neumann type for a p-Laplacian like operator.Nonlinear Anal,2004,56:1071-1089
[24]M.García-Huidobro,C.P.Gupta,R.Manásevich.Solvability for a non-linear three-point boundary value problem with p-Laplace-like operator at resonance.Abstr.Appl.Anal,2001,164:191-213
[25]Chen Shihua,Wei Ni,Wang Changping.Positive solution of fourth ordinary differential equation with four-point boundary conditions.Appl.Math.Lett.,2006,19:161-168
[26]廉立芳,葛渭高.共振情况下m点p-Laplacian算子边值问题解的存在性.应用数学学报,2005,28(2):288-295
[27]Mawhin J..Topological degree methods in nonlinear boundary value problems.IN:NSFCBMS Regional Conference Series in Mathematics,American Mathematical Society,Providence:Rhode Island,USA,1979.
[28]Ge Weigao,Ren Jingli.An extension of Mawhin's continuation theorem and its application to boundary value problems with a p-Laplacian.Nonlinear Anal.,2004,58:477-488
[29]Feng W.,Webb J.R.L.,Solvability of m-point boundary value problems with nonlinear growth.J.M.Anal.Appl.,1997,212:467-480
[30]Li Cuizhe,Li Zhiyan.Existence of solutions of third-order p-Laplacian resonant multipoint boundary value problems.Transactions of Beijing Institute of Technology,2004, 24(11):1024-1029(in Chinese)
[31]白定勇,马如云.p-Laplacian算子型奇异边值问题的正解.数学物理学报,2005,25(2):166-170
[32]Zhang Jianjun,Liu Wenbin,Chen Taiyong,Zhang Huixing.Solvability of p-Laplace equations subject to three-point boundary value problems.Appl.Math.Comput.,2006,179:688-695
[33]He Zhimin,Yu Jianshe.Periodic boundary value problem for first-order impulsive functional differential equations.J.Comp.Appl.Math.,2002,138:205-217
[34]夏道行等.实变函数论与泛函分析(下册).北京:人民教育出版社,1979.
[35]张恭庆,林源渠.泛函分析讲义(下册).北京:北京大学出版社,1987.
[36]Krasnoselskii M..Positive Solutions of Operator Equations.Groningen,Noorhoff,1964.
[37]Deimling Klaus.Nonlinear Functional Analysis.New York:Springer-Verlag,1985.
[38]马如云.非线性常微分方程非局部问题.北京:科学出版社,2004.
[39]郭志明,庾建设.二阶超线性差分方程周期解与次调和解的存在性.中国科学(A),2003,33(3):226-235
[40]Ma Shiwang,Wang Zhicheng,Yu Jianshe.An abstract existence theorem at resonance and its applications.Journal of Differential Equations,1998,145(2):274-293
[41]Yang Xiaojing.Multiple periodic solutions of a class of p-Laplacian.J.M.Anal.Appl.,2006,314:17-29
[42]鲁世平,葛渭高.具偏差变元的二阶p-Laplacian方程周期解存在性问题.数学学报,2005,48(5):841-850
[43]Peng Shiguo,Xu Zhiting.On the existence of periodic solutions fbr a class of p-Laplcacian system.J.Math.Anal.Appl.,2007,325:166-174