带p-Laplacian算子的微分方程多点边值问题
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摘要
p-laplacian算子边值问题在应用力学、天体物理和非线性偏微分方程中有着广泛的应用背景和非常重要的研究价值.本论文主要应用不动点定理对带p-laplacian算子的微分方程多点边值问题进行了深入的研究.首先,我们得到了非线性项含未知函数一阶导数的带p-laplacian算子的微分方程在射线上的多点边值问题在一定边界条件下多个正解的存在性.接着,又研究了非线性项含未知函数一阶导数的带p-laplacian算子的微分方程在(0,1)上的多点边值问题和一个奇异边值条件下的多个正解的存在性.
本文共分为四章,具体框架如下:
第一章,介绍带p-laplacian算子的微分方程的边值问题的背景及进展,并概述了本文的主要工作.
第二章,讨论了带p-laplacian算子多点边值问题在射线上的多个正解的存在性.主要利用Avery-Peterson不动点定理研究方程在边界条件下多个正解的存在性.
第三章,主要利用Ge-Bai不动点定理研究方程在边值条件下至少存在三个正解。
第四章,主要利用Avery-Peterson不动点定理研究方程在奇异边值条件下或三个正解的存在性。
The BVPs with p-laplacian arise in a variety of applied areas such as applied mechanics,astrophysics and nonlinear partial di?erential equations. The paper studies the boundaryvalue problems for ordinary di?erential equation with p-laplacian operator by using ?xedpoint theorems. Firstly, we get the su?cient condition of the existence of at least three posi-tive solutions for the BVPs with p-laplacian of nonlinear term with the ?rst order derivativeof unknown function on in?nite intervals. In addition, we study multi-point boundary valueproblem with p-Laplacian of nonlinear term with the ?rst order derivative of unknown func-tion on (0,1)
The paper consists of four chapters:
In chapter 1, we mainly introduce the background and the progress of the multi-pointBVPs with p-laplacian. In addition, we also present the organized of this paper.
In chapter 2, we get at least three positive solutions ofby using the Avery-Peterson’s ?xed point theorem.
In chapter 3, we get at least three positive solution ofby using Bai and Ge ?xed point theorem.
In chapter 4, we get the su?cient condition of the existence of at least three positivesolutions for withby using the Avery-Peterson’s ?xed point theorem.
本文共分为四章,具体框架如下:
第一章,介绍带p-laplacian算子的微分方程的边值问题的背景及进展,并概述了本文的主要工作.
第二章,讨论了带p-laplacian算子多点边值问题在射线上的多个正解的存在性.主要利用Avery-Peterson不动点定理研究方程在边界条件下多个正解的存在性.
第三章,主要利用Ge-Bai不动点定理研究方程在边值条件下至少存在三个正解。
第四章,主要利用Avery-Peterson不动点定理研究方程在奇异边值条件下或三个正解的存在性。
The BVPs with p-laplacian arise in a variety of applied areas such as applied mechanics,astrophysics and nonlinear partial di?erential equations. The paper studies the boundaryvalue problems for ordinary di?erential equation with p-laplacian operator by using ?xedpoint theorems. Firstly, we get the su?cient condition of the existence of at least three posi-tive solutions for the BVPs with p-laplacian of nonlinear term with the ?rst order derivativeof unknown function on in?nite intervals. In addition, we study multi-point boundary valueproblem with p-Laplacian of nonlinear term with the ?rst order derivative of unknown func-tion on (0,1)
The paper consists of four chapters:
In chapter 1, we mainly introduce the background and the progress of the multi-pointBVPs with p-laplacian. In addition, we also present the organized of this paper.
In chapter 2, we get at least three positive solutions ofby using the Avery-Peterson’s ?xed point theorem.
In chapter 3, we get at least three positive solution ofby using Bai and Ge ?xed point theorem.
In chapter 4, we get the su?cient condition of the existence of at least three positivesolutions for withby using the Avery-Peterson’s ?xed point theorem.
引文
[1] E. T. Bell. Men of Mathematics: The lives and the Achievements of the Great Math ematiciansfrom Zero to Poincaee, 1965.
[2] P. Harman. Mitton S.Cambridge Scienti?c Minds.Cambridge: Cambridge Univ. Press. 2002.
[3] J.C.F. Sturm. Memoire our les Equations Di?erentielles Lineaures du Second ord erJMPA,1836(1):106-186.
[4] J.Liouvill.Sur le d′eveloppment des fonctions ou partieo de fouctions en s′erieo dont leo diverstermes sont assujetties `a satisfaire a une m?enle′equmion di?erentiel les du second order eon-tenent un param`etre variable,JMPA,l 836(1):253-265;837(2):16-35;41 8-436.
[5] J. Leray. Schauder J. Topologie et equations fonctionelles. Ann. Sci. Ecole Norm, Sup 51:1934(3): 45-78.
[6] J. Leray. Les Probl`emes non Lineaires.L’enseignement Math, 1936(35): 139-151.
[7] J. Mathin. Topological degree methods in nonlinear boundary value problems.CBMS 40,Amer. Math.Soc., Providence, R. I.,1979.
[8] J.Mathin.Topological degree and boundary value problems for nonlinear di?erential equa-tions.In:Topological Methods for Ordinary Di?erential Equations.Spinger-Verlag,BerlinHeidelberg,1993.
[9] L.H.Thomas.The calculation of atomic ?elds.Proc.Camb.Phil.Soc.1927(23):542.548.
[10] E. Fermi. Un methodo statistico perla determunazione di alcune proori`eta dell’atoma.Rend.Accard. Naz. del Linceri. CI. sci. ?s. mat. e nat.1927(6):602-607.
[11] R.E.Kider.Unsteady ?ow of gas through a semi-in?nite porous medium.J.Appl Mech.,1957(27):329-332.
[12] M.A.Herrero,J.L.Vazquez.On the propagation properties of a nonlinear degenerateparabolic equation.Comm.Partial Di?.Eqns,1982(7):1381-1402.
[13] J.R.Esteban, J. L. Vazquez. On the equation ofthe turbulent ?ltration in dimensionaI porousmedia, Nonlinear Analysis, 1986(10): 1305-1325.
[14] M.del Pino, M.Elgueta, R.Manasevich, Ahomotopic de?erential di?erence and integral equa-tions[M]. Singapore, Springer, 2000.
[15] Erbe L H, Wang Haiyan, On the existence of positive solutions of ordinary diferential equa-tions, Proc Amer Math Soc,1994, 120(3):743-748.
[16] Wong Fu Hsiang. Existence of positive solutions for m-Laplacian BVPs, Appl .Math.Lett.,1999,12:11-17.
[17] Wang J Y, Gao W. J., A singular boundary value problem for the one-dimensional p-Laplacian,J. Math. Anal. Appl.,1996,201:851-866.
[18]孙伟平,葛渭高,一类非线性边值问题正解的存在性,数学学报, 2001,44(4):577-580.
[19]姚庆六,吕海琛,一维奇异p-Laplace方程的正解,数学学报, 1998,41(6):1255-1264.
[20]贺小明,葛渭高,一维p-Laplacian方程正解的三解定理,应用数学学报, 2003,26(3):504-510.
[21]李必文,具p-Laplacian算子奇异边值问题正解的存在性,数学物理学报, 2003, 23(3):257-264.
[22]白定勇,马如云, p-Laptacian算子型奇异边值问题的正解,数学物理学报, 2005,25(2):166-170.
[23]倪小虹,葛渭高,半无穷区间二阶微分方程边值问题的正解的存在性[J],系统科学与数学, 2006, 2:11.3一120 .
[24]郭飞,房庆平,王刚,二阶奇异边值问题的多解[J],曲阜师范大学学报, 2001,(4):11-15.
[25]马如云,奇异二阶边值问题的正解[J],数学学报, 1998,41(6):l 225～l 230.
[26] E R Kaufmann, N Kosmatov, A multiplicity result for a boundary value problem with in?nitelymany singulari?es[J], J. Math .Anal. Appl.,2002.269:444～453.
[27] Wong Fuhsiang, The existence of positive solutions for m-Laplacian BVPs, [J].Appl. Math.Lett., 1999,12:11～17.
[28]贺小明,一维p-Laplacian方程正解的存在性[J],数学学报, 2003, 46(4):805～8lO.
[29] Junyu Wang and Wenjie Gao, A Singular Boundary Value Problem for the One-Dimensionalp-Laplacian, J.Math.Ana.App.201,851-866,1996.
[30] Junyu Wang, The existence of positive solutions for the one-dimension p-laplacian[J].Proc.Amer.Soc,1997,125:2272-2283.
[31] DAQING JIANG, Upper and Lower Solutions Method and a Singular Superlinear BoundaryValue Problem for the One-Dimensional p-laplacian, Compu.and Math.App.42(2001)927-940.
[32] Yongkun Li, Linghong Lu .Existence of positive solutions of p-Laplacian di?erence equations,Appl. Math. Lett. 19 (2006) 1019-1023.,Aug.2006.
[33] Meiqiang Feng, Xuemei Zhang, Weigao Ge ,Exact number of solutions for a class of two-pointboundary value problems with one-dimensional p-Laplacian, J. Math. Anal. Appl. 338 (2008)784-792.
[34] He Xiaoming, Ge Weigao, Multiple Positive Solutions for one-Dimensional p- Laplacian Bound-ary Value Problems [J].Appl. Math.Lett., 2002, 15:937-943.
[35] Jiang Daqing, Xuxiaojie, Multiple Positive Solution to a Class of Singular Boundary ValueProblems For the One-Dimensional P-Laplacian [J], Comput. Math.Appl. 2004,47:667-681.
[36] LU Hai-shen, BAI Zhan-bing, A Necessary and Su?cient Condition for the Existence of Posi-tive Solutions to the Singular p-Laplacian,应用泛函分析学报. Vol. 6, No. 4, Decem. , 2004.
[37] Xue Chunyan, Ge Weigao, New ?xed point theorems and applications to boundary valueproblems with p-Laplacian, Comput.Math. Appl. 53 (2007) 706-716.
[38] Youyu Wang, Weigao Ge,Nonlinear Analysis 67 (2007) 476– 485
[39] Youyu Wang, Weigao Ge,Nonlinear Analysis 66 (2007) 1246– 1256
[40] Yanping Guo,Yude Ji, Xiujun Liu,Journal of Computational and Applied Mathematics 216(2008) 144– 156
[41] S. Liang, J. Zhang, The existence of countably many positive solutions for one-dimensional p-Laplacian with in?nitely many singularities on the half-line, Appl. Math. Comput 201 (2008)210-220.
[42] Hanying Fenga, Weigao Ge,Ming Jiang,Nonlinear Analysis 68 (2008) 2269– 2279
[43] D.Ma, Z.Du, W. Ge, Existence and iteration of monotone positive solutions for multi pointboundary value problem with p-Laplacian operator, Comput. Math. Appl. 50 (2005) 729-739.
[44] B.Q. Yan, Multiple unbounded solutions of boundary value problems for second-order di?er-ential equations on the half-line, Nonlinear Anal. 51 (2002) 1031-1044.
[45] M. Zima, On positive solution of boundary value problems on the half-line, J. Math. Anal.Appl. 259 (2001) 127-136.
[46] H. Lian et al., Triple positive solutions for boundary value problems on in?nite intervals,Nonlinear Anal. 67 (2007) 2199-2207.
[47] Y.S. Liu, Existence and unboundedness of positive solutions for singular boundary value prob-lems on half-line, Appl. Math. Comput. 1404 (2003) 543-556.
[48] R.I. Avery, A.C. Peterson, Three ?xed points of nonlinear operators on ordered Banach spaces,Comput. Math. Appl. 42 (2001) 313-322.
[49] X.Zhou, W.Ge, Existence of at least three positive solutions for multi-point boundary valueproblem on in?nite intervals with p-Laplacian operator, J. Appl. Math. Comput. 28 (2008)391-403.
[50] Z. Bai, W. Ge, Existence of three positive solutions for some second-order boundary valueproblems, Comput. Math. Appl. 48 (2004) 699-707.
[51] Y.Guo, Y.Ji, X.Liu, Multiple positive solutions for some multi-point boundary value problemswith p-Laplacian, J. Comput. Appl. Math. 216 (2008) 144-156.
[52] D. Ma, Z. Du, W. Ge, Existence and iteration of monotone positive solutions for multi pointboundary value problem with p-Laplacian operator, Comput. Math. Appl. 50 (2005) 729-739.
[53] H.L u¨, D.O. Regan, C.Zhong, Multiple positive solutions for the one-dimensional singularp-Laplacian, Appl. Math. Comput. 133 (2002) 407-422.
[2] P. Harman. Mitton S.Cambridge Scienti?c Minds.Cambridge: Cambridge Univ. Press. 2002.
[3] J.C.F. Sturm. Memoire our les Equations Di?erentielles Lineaures du Second ord erJMPA,1836(1):106-186.
[4] J.Liouvill.Sur le d′eveloppment des fonctions ou partieo de fouctions en s′erieo dont leo diverstermes sont assujetties `a satisfaire a une m?enle′equmion di?erentiel les du second order eon-tenent un param`etre variable,JMPA,l 836(1):253-265;837(2):16-35;41 8-436.
[5] J. Leray. Schauder J. Topologie et equations fonctionelles. Ann. Sci. Ecole Norm, Sup 51:1934(3): 45-78.
[6] J. Leray. Les Probl`emes non Lineaires.L’enseignement Math, 1936(35): 139-151.
[7] J. Mathin. Topological degree methods in nonlinear boundary value problems.CBMS 40,Amer. Math.Soc., Providence, R. I.,1979.
[8] J.Mathin.Topological degree and boundary value problems for nonlinear di?erential equa-tions.In:Topological Methods for Ordinary Di?erential Equations.Spinger-Verlag,BerlinHeidelberg,1993.
[9] L.H.Thomas.The calculation of atomic ?elds.Proc.Camb.Phil.Soc.1927(23):542.548.
[10] E. Fermi. Un methodo statistico perla determunazione di alcune proori`eta dell’atoma.Rend.Accard. Naz. del Linceri. CI. sci. ?s. mat. e nat.1927(6):602-607.
[11] R.E.Kider.Unsteady ?ow of gas through a semi-in?nite porous medium.J.Appl Mech.,1957(27):329-332.
[12] M.A.Herrero,J.L.Vazquez.On the propagation properties of a nonlinear degenerateparabolic equation.Comm.Partial Di?.Eqns,1982(7):1381-1402.
[13] J.R.Esteban, J. L. Vazquez. On the equation ofthe turbulent ?ltration in dimensionaI porousmedia, Nonlinear Analysis, 1986(10): 1305-1325.
[14] M.del Pino, M.Elgueta, R.Manasevich, Ahomotopic de?erential di?erence and integral equa-tions[M]. Singapore, Springer, 2000.
[15] Erbe L H, Wang Haiyan, On the existence of positive solutions of ordinary diferential equa-tions, Proc Amer Math Soc,1994, 120(3):743-748.
[16] Wong Fu Hsiang. Existence of positive solutions for m-Laplacian BVPs, Appl .Math.Lett.,1999,12:11-17.
[17] Wang J Y, Gao W. J., A singular boundary value problem for the one-dimensional p-Laplacian,J. Math. Anal. Appl.,1996,201:851-866.
[18]孙伟平,葛渭高,一类非线性边值问题正解的存在性,数学学报, 2001,44(4):577-580.
[19]姚庆六,吕海琛,一维奇异p-Laplace方程的正解,数学学报, 1998,41(6):1255-1264.
[20]贺小明,葛渭高,一维p-Laplacian方程正解的三解定理,应用数学学报, 2003,26(3):504-510.
[21]李必文,具p-Laplacian算子奇异边值问题正解的存在性,数学物理学报, 2003, 23(3):257-264.
[22]白定勇,马如云, p-Laptacian算子型奇异边值问题的正解,数学物理学报, 2005,25(2):166-170.
[23]倪小虹,葛渭高,半无穷区间二阶微分方程边值问题的正解的存在性[J],系统科学与数学, 2006, 2:11.3一120 .
[24]郭飞,房庆平,王刚,二阶奇异边值问题的多解[J],曲阜师范大学学报, 2001,(4):11-15.
[25]马如云,奇异二阶边值问题的正解[J],数学学报, 1998,41(6):l 225～l 230.
[26] E R Kaufmann, N Kosmatov, A multiplicity result for a boundary value problem with in?nitelymany singulari?es[J], J. Math .Anal. Appl.,2002.269:444～453.
[27] Wong Fuhsiang, The existence of positive solutions for m-Laplacian BVPs, [J].Appl. Math.Lett., 1999,12:11～17.
[28]贺小明,一维p-Laplacian方程正解的存在性[J],数学学报, 2003, 46(4):805～8lO.
[29] Junyu Wang and Wenjie Gao, A Singular Boundary Value Problem for the One-Dimensionalp-Laplacian, J.Math.Ana.App.201,851-866,1996.
[30] Junyu Wang, The existence of positive solutions for the one-dimension p-laplacian[J].Proc.Amer.Soc,1997,125:2272-2283.
[31] DAQING JIANG, Upper and Lower Solutions Method and a Singular Superlinear BoundaryValue Problem for the One-Dimensional p-laplacian, Compu.and Math.App.42(2001)927-940.
[32] Yongkun Li, Linghong Lu .Existence of positive solutions of p-Laplacian di?erence equations,Appl. Math. Lett. 19 (2006) 1019-1023.,Aug.2006.
[33] Meiqiang Feng, Xuemei Zhang, Weigao Ge ,Exact number of solutions for a class of two-pointboundary value problems with one-dimensional p-Laplacian, J. Math. Anal. Appl. 338 (2008)784-792.
[34] He Xiaoming, Ge Weigao, Multiple Positive Solutions for one-Dimensional p- Laplacian Bound-ary Value Problems [J].Appl. Math.Lett., 2002, 15:937-943.
[35] Jiang Daqing, Xuxiaojie, Multiple Positive Solution to a Class of Singular Boundary ValueProblems For the One-Dimensional P-Laplacian [J], Comput. Math.Appl. 2004,47:667-681.
[36] LU Hai-shen, BAI Zhan-bing, A Necessary and Su?cient Condition for the Existence of Posi-tive Solutions to the Singular p-Laplacian,应用泛函分析学报. Vol. 6, No. 4, Decem. , 2004.
[37] Xue Chunyan, Ge Weigao, New ?xed point theorems and applications to boundary valueproblems with p-Laplacian, Comput.Math. Appl. 53 (2007) 706-716.
[38] Youyu Wang, Weigao Ge,Nonlinear Analysis 67 (2007) 476– 485
[39] Youyu Wang, Weigao Ge,Nonlinear Analysis 66 (2007) 1246– 1256
[40] Yanping Guo,Yude Ji, Xiujun Liu,Journal of Computational and Applied Mathematics 216(2008) 144– 156
[41] S. Liang, J. Zhang, The existence of countably many positive solutions for one-dimensional p-Laplacian with in?nitely many singularities on the half-line, Appl. Math. Comput 201 (2008)210-220.
[42] Hanying Fenga, Weigao Ge,Ming Jiang,Nonlinear Analysis 68 (2008) 2269– 2279
[43] D.Ma, Z.Du, W. Ge, Existence and iteration of monotone positive solutions for multi pointboundary value problem with p-Laplacian operator, Comput. Math. Appl. 50 (2005) 729-739.
[44] B.Q. Yan, Multiple unbounded solutions of boundary value problems for second-order di?er-ential equations on the half-line, Nonlinear Anal. 51 (2002) 1031-1044.
[45] M. Zima, On positive solution of boundary value problems on the half-line, J. Math. Anal.Appl. 259 (2001) 127-136.
[46] H. Lian et al., Triple positive solutions for boundary value problems on in?nite intervals,Nonlinear Anal. 67 (2007) 2199-2207.
[47] Y.S. Liu, Existence and unboundedness of positive solutions for singular boundary value prob-lems on half-line, Appl. Math. Comput. 1404 (2003) 543-556.
[48] R.I. Avery, A.C. Peterson, Three ?xed points of nonlinear operators on ordered Banach spaces,Comput. Math. Appl. 42 (2001) 313-322.
[49] X.Zhou, W.Ge, Existence of at least three positive solutions for multi-point boundary valueproblem on in?nite intervals with p-Laplacian operator, J. Appl. Math. Comput. 28 (2008)391-403.
[50] Z. Bai, W. Ge, Existence of three positive solutions for some second-order boundary valueproblems, Comput. Math. Appl. 48 (2004) 699-707.
[51] Y.Guo, Y.Ji, X.Liu, Multiple positive solutions for some multi-point boundary value problemswith p-Laplacian, J. Comput. Appl. Math. 216 (2008) 144-156.
[52] D. Ma, Z. Du, W. Ge, Existence and iteration of monotone positive solutions for multi pointboundary value problem with p-Laplacian operator, Comput. Math. Appl. 50 (2005) 729-739.
[53] H.L u¨, D.O. Regan, C.Zhong, Multiple positive solutions for the one-dimensional singularp-Laplacian, Appl. Math. Comput. 133 (2002) 407-422.