几类p-Laplacian多点边值问题及非线性微分方程组解的存在性的研究
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摘要
本论文主要应用Legget-Williams不动点定理、双锥不动点定理、Avery-Peterson不动点定理以及锥压缩与锥拉伸不动点定理等非线性分析的理论和方法,研究了一类具有反馈控制的非线性泛函微分系统正周期解的存在性,以及带有p-Laplace算子的多点边值问题正解的存在性和多重性问题.全文共分为三章:
第一章主要介绍了具有反馈控制的非线性泛函微分系统以及带有p-Laplace算子的多点边值问题的应用背景和国内外关于这些问题的研究现状与成果,并简述本文的主要研究成果.
第二章主要利用Avery-Peterson不动点定理,研究了具有反馈控制的非线性非自控泛函微分方程组的多重正周期解的存在性.我们证明了,对非线性项加以适当的增长性条件,所研究的方程组至少存在三个正周期解.
第三章研究带有p-Laplace算子的三阶三点边值问题。我们分别运用双锥不动点定理、Legget-Williams不动点定理、锥压缩与锥拉伸不动点定理讨论了这些问题正解的存在性,多重性以及不存在性.
This dissertation deals with the existence of positive periodic solutions to one kind of nonlinear functional differential system with feedback control, as well as the existence and multiplicity of positive solutions to p - Laplacian multi-point boundary value problems. The methods and techniques employed here are involved in nonlinear functional analysis, such as Avery-Peterson fixed-point theorem、cone tensile and compression fixed-point theorem、Leggett-Williams fixed-point theorem and double-cone fixed-point theorem. This dissertation consists of three chapters.
In Chapter 1, we introduce the background and research status on the nonlinear functional differential system with feedback control and p- Laplacian multi-point boundary value problems at home and abroad. We also present a brief survey of our results.
In Chapter 2, by using Avery-Peterson fixed-point theorem, we investigate the existence of multiple positive periodic solutions to the nonlinear non-autonomous functional differential system with feedback control, and prove that this system admits at least three positive periodic solutions under certain growth conditions imposed on the nonlinearity.
In Chapter 3 we investigate several kinds of p- Laplacian third-order three-point boundary value problems, we discuss the existence, multiplicity and no-existence of positive solution to these problems by using double-cone fixed-point theorem, Leggett-Williams fixed-point theorem and cone-tensile and cone-compression fixed-point theorem, separately.
第一章主要介绍了具有反馈控制的非线性泛函微分系统以及带有p-Laplace算子的多点边值问题的应用背景和国内外关于这些问题的研究现状与成果,并简述本文的主要研究成果.
第二章主要利用Avery-Peterson不动点定理,研究了具有反馈控制的非线性非自控泛函微分方程组的多重正周期解的存在性.我们证明了,对非线性项加以适当的增长性条件,所研究的方程组至少存在三个正周期解.
第三章研究带有p-Laplace算子的三阶三点边值问题。我们分别运用双锥不动点定理、Legget-Williams不动点定理、锥压缩与锥拉伸不动点定理讨论了这些问题正解的存在性,多重性以及不存在性.
This dissertation deals with the existence of positive periodic solutions to one kind of nonlinear functional differential system with feedback control, as well as the existence and multiplicity of positive solutions to p - Laplacian multi-point boundary value problems. The methods and techniques employed here are involved in nonlinear functional analysis, such as Avery-Peterson fixed-point theorem、cone tensile and compression fixed-point theorem、Leggett-Williams fixed-point theorem and double-cone fixed-point theorem. This dissertation consists of three chapters.
In Chapter 1, we introduce the background and research status on the nonlinear functional differential system with feedback control and p- Laplacian multi-point boundary value problems at home and abroad. We also present a brief survey of our results.
In Chapter 2, by using Avery-Peterson fixed-point theorem, we investigate the existence of multiple positive periodic solutions to the nonlinear non-autonomous functional differential system with feedback control, and prove that this system admits at least three positive periodic solutions under certain growth conditions imposed on the nonlinearity.
In Chapter 3 we investigate several kinds of p- Laplacian third-order three-point boundary value problems, we discuss the existence, multiplicity and no-existence of positive solution to these problems by using double-cone fixed-point theorem, Leggett-Williams fixed-point theorem and cone-tensile and cone-compression fixed-point theorem, separately.
引文
[1]R.I.Avery, J.Henderson, Two positive fixed points of nonlinear operators on ordered Banach spaces, Comm.Appl.Nonlinear Anal.8(2001),27-36.
[2]R.I.Avery, A.C.Peterson, Three positive fixed points of nonlinear operators on ordered Banach spaces, Computers Math.Appl.42(2001),313-322.
[3]S.Cheng, G.Zhang, Existence of positive periodic solutions for non-autonomous functional differential equations, Electronic J.Diff.Eqns.59(2001), 1-8.
[4]R.E.Gaines, J.L.Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer, Berlin, 1977.
[5]D.J.Guo, V.Lakshmikantham, Nonlinear Problems in Abstract Cones,Academic Press, Orlando, FL, 1988.
[6]W.S.C.Gurney, S.P.Blathe, R.M.Nishet, Nicholson' s blowflies revisited, Nature 287(1980), 17-21.
[7]H.F.Huo, W.T.Li, Positive solutions of a class of delay differential system with feedback control, Appl.Math.Comput.148(2004), 35-46.
[8]Y.Kuang, Delay Differential Equations with Application in Population Dynamics, Academic Press, New York, 1993.
[9]P.Liu, Y.K.Li, Multiple positive periodic solutions of nonlinear functional differential system with feedback control, J.Math.Anal.Appl.288(2003), 819-832.
[10]S.H.Saker, S.Agarwal, Oscillation and global attractivity in a periodic Nicholson' s blowflies model, Math.Comput.Modelling 35(2002),719-731.
[11]A.Wan, D.Q.Jiang, X.J.Xu, A new existence theory for positive periodic solutions to functional differential equations, Compuers Math. Appl.47(2004),1257-1262.
[12]D.Ye,M.Fan,W.P.Zhang,Periodic solutions of density dependent predator prey systems with Holling Type 2 functional response and infinite delays,ZAMM 85(2005),213-221.
[13]Y.Sun,Optimal existence criteria for symmetric positive solutions a three-point boundary value problem.Nonlinear Anal.66(2007) 1051-1063.
[14]葛渭高,任景莉,双锥不动点定理及其在非线性边值问题中的应用.数学年刊,27A(2006)155-168.
[15]Z.Bai,W.Ge,Existence of three positive solutions for some second-order boundary value problems.Comput.Math.Appl.48(2004)699-707.
[16]D.Ma,J.Han,X.Chen,Positive solution of boundary value problem for one-dimensinal p-Laplacian with singularities.J.Math.Anal.Appl.324(2006) 118-133.
[17]J.Wang,The existence of positive solutions for the one dimensional p-Laplacian,Proc.Amer.Math.Soc.125(1997) 2275-2283.
[18]钟承奎,范先令,陈文源,非线性泛函分析[M].兰州:兰州大学出版社,1998.
[19]Seshadev Padhi and Shilpee Srivastava,Multiple periodic solutions for a nonlinear first order functional differential equations with applications to population dynamics.Appl.Math.Comput.203(2008),no.1,1-6
[20]John R.Graef,Chuanxi Qian and Bo Yang,A three point boundary value problem for nonlinear fourth order differential equations.J.Math.Anal.Appl.287(2003)217-233.
[21]X.He,W.Ge and M.Peng,Multiple positive solutions for one-dimensional p- Laplacian boundary value problems.Appl.Math.Lett.15(2002) 937-943.
[22]X.He and W.Ge,A remark on some three-point boundary value problems for the one-dimensional p-Laplacian.ZAMM.82(2002) 728-731.
[23]X.He and W.Ge,Twin positive solutions for the one-dimensional p- Laplacian boundary value problems.Nonlinear Anal.56(2004)975-984.
[24]田元生,刘春根,三阶p-Laplacian方程三点奇异边值问题三个正解的存在性,应用数学学报(已接收发表).
[25]H.Su,Z.Wei and B.Wang,The existence of positive solutions for a nonlinear four-point singular boundary value problem with a p-Laplace operator.Nonlinear Anal.33(2007) 2204-2217.
[26]X.He,Double positive solutions of a three-point boundary value problem for the one-dimensional p-Laplacian,Appl.Math.Lett.17(2004)867-873.
[27]X.He and W.Ge,Triple solutions for second-order three-point boundary value problems.J.Math.Anal.Appl.268(2002) 256-265.
[28]D.R.Anderson and J.M.Davis,Multiple solutions and eigenvalues for third-order right focal boundary value problem,J.Math.Anal.Appl.267(2002)135-157.
[29]R.I.Avery,J.M.Davis and J.Henderson,Three symmetric positive solutions for Lidstone problems by a generalization of the Leggett-Williams theorem,Electron.J.Differential Equations 2000(2000),No.40,pp.1-15.
[30]D.Cao and R.Ma,Positive solutions to a second order muliti-point boundary value problem,Electron.J.Differential Equations 2000(2000),No.65,pp.1-8.
[31]郭大钧,非线性泛函分析[M].山东:山东科技出版社,2001.
[32]X.He and W.Ge,A theorem about triple positive solutions for the one-dimensional p-Laplacian equations.(Chinese) Acta.Math.Appl.Sin. 26(2003),no.3,504-510.
[33]X.He and W.Ge,Existence of positive solutions for the one-dimensional p-Laplacian equations.(Chinese) Acta.Math.Sinica.46(2003),no.4,805-810.
[34]葛渭高,非线性常微分方程边值问题,科学出版社,2007年.
[35]D.Ji,M.Feng and W.Ge,Multiple positive solutions for multipoint boundary value problems with sign changing nonlinearity,Appl.Math.Comput.196(2008) 511-520.
[36]D.Ji,T.Yi,B.Yang and W.Ge,Triple positive pseudo-symmetric solutions to a four-point boundary value problem with p-Laplacian,Applied Mathematics Letters 21(2008) 268-274.
[37]H.Lian and W.Ge,Positive solutions for a four-point boundary value problem with the p-Laplacian,Nonlinear Analysis 68(2008) 3493-3503.
[38]H.Feng and W.Ge,Existence of three positive solutions for m-point boundary-value problems with one-dimensional p-Laplacian,Nonlinear Analysis 68(2008) 2017-2026.
[39]X.Zhang,M.Feng and W.Ge,Existence and nonexistence of positive solutions for a class of n-th-order three-point boundary value problems in Banach spaces.Nonlinear Anal.70(2009),no.2,584-597.
[40]B.Sun,J.Zhao,P.Yang,W.Ge,Successive iteration and positive solutions for a third-order multipoint generalized righ-focal boundary value problem with p-Laplacian.Nonlinear Anal.70(2009),no.1,220-230.
[41]W.Ge and C.Xue,Some fixed point theorems and existence of positive solutions of two-point boundary-value problems.Nonlinear Anal.70(2009),no.1,16-31.
[42]L.Wang,Y.Fan and W.Ge,Periodic solutions in a delayed predator-prey model with nonmonotonic functional response.Rocky Mountain J.Math.38 (2008), no.5, 1705—1719.
[43]Z.Gui, X.Yang and W.Ge, Existence of global exponential stability of periodic solutions of recurrent cellular neural networks with impulses and delays.Math.Comput.Simulation 79 (2008), no.1, 14—29.
[44]D.Ma and W.Ge, Existence and iteration of solutions for a multi-point boundary value problem with a />-Laplacian operator.(Chinese) Acta Math.Sinica (Chin.Ser.) 51 (2008), no.3, 447—456.
[45]D.Ma and W.Ge, Positive solution of multi-point boundary value problem for the one-dimensional /7-Laplacian with singularities.Rocky Mountain J.Math.37 (2007), no.4, 1229—1249.
[46]Y.Li, Positive periodic solutions for neutral functional differential equations with distributed delays and feedback control.Nonlinear Anal.Real World Appl.9 (2008), no.5, 2214—2221.
[47]Y.Li, Positive periodic solutions for a periodic neutral differential equation with feedback control.Nonlinear Anal.Real World Appl.6 (2005), no.1, 145—154.
[48]Y.Li, L.Zhu and P.Liu, Positive periodic solutions of nonlinear functional difference equations depending on a parameter.Comput.Math.Appl.48 (2004), no.10-11, 1453—1459.
[49]Y.Li and L.Zhu, Positive periodic solutions of nonlinear functional differential equations.Appl.Math.Comput.156 (2004), no.2, 329—339.
[2]R.I.Avery, A.C.Peterson, Three positive fixed points of nonlinear operators on ordered Banach spaces, Computers Math.Appl.42(2001),313-322.
[3]S.Cheng, G.Zhang, Existence of positive periodic solutions for non-autonomous functional differential equations, Electronic J.Diff.Eqns.59(2001), 1-8.
[4]R.E.Gaines, J.L.Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer, Berlin, 1977.
[5]D.J.Guo, V.Lakshmikantham, Nonlinear Problems in Abstract Cones,Academic Press, Orlando, FL, 1988.
[6]W.S.C.Gurney, S.P.Blathe, R.M.Nishet, Nicholson' s blowflies revisited, Nature 287(1980), 17-21.
[7]H.F.Huo, W.T.Li, Positive solutions of a class of delay differential system with feedback control, Appl.Math.Comput.148(2004), 35-46.
[8]Y.Kuang, Delay Differential Equations with Application in Population Dynamics, Academic Press, New York, 1993.
[9]P.Liu, Y.K.Li, Multiple positive periodic solutions of nonlinear functional differential system with feedback control, J.Math.Anal.Appl.288(2003), 819-832.
[10]S.H.Saker, S.Agarwal, Oscillation and global attractivity in a periodic Nicholson' s blowflies model, Math.Comput.Modelling 35(2002),719-731.
[11]A.Wan, D.Q.Jiang, X.J.Xu, A new existence theory for positive periodic solutions to functional differential equations, Compuers Math. Appl.47(2004),1257-1262.
[12]D.Ye,M.Fan,W.P.Zhang,Periodic solutions of density dependent predator prey systems with Holling Type 2 functional response and infinite delays,ZAMM 85(2005),213-221.
[13]Y.Sun,Optimal existence criteria for symmetric positive solutions a three-point boundary value problem.Nonlinear Anal.66(2007) 1051-1063.
[14]葛渭高,任景莉,双锥不动点定理及其在非线性边值问题中的应用.数学年刊,27A(2006)155-168.
[15]Z.Bai,W.Ge,Existence of three positive solutions for some second-order boundary value problems.Comput.Math.Appl.48(2004)699-707.
[16]D.Ma,J.Han,X.Chen,Positive solution of boundary value problem for one-dimensinal p-Laplacian with singularities.J.Math.Anal.Appl.324(2006) 118-133.
[17]J.Wang,The existence of positive solutions for the one dimensional p-Laplacian,Proc.Amer.Math.Soc.125(1997) 2275-2283.
[18]钟承奎,范先令,陈文源,非线性泛函分析[M].兰州:兰州大学出版社,1998.
[19]Seshadev Padhi and Shilpee Srivastava,Multiple periodic solutions for a nonlinear first order functional differential equations with applications to population dynamics.Appl.Math.Comput.203(2008),no.1,1-6
[20]John R.Graef,Chuanxi Qian and Bo Yang,A three point boundary value problem for nonlinear fourth order differential equations.J.Math.Anal.Appl.287(2003)217-233.
[21]X.He,W.Ge and M.Peng,Multiple positive solutions for one-dimensional p- Laplacian boundary value problems.Appl.Math.Lett.15(2002) 937-943.
[22]X.He and W.Ge,A remark on some three-point boundary value problems for the one-dimensional p-Laplacian.ZAMM.82(2002) 728-731.
[23]X.He and W.Ge,Twin positive solutions for the one-dimensional p- Laplacian boundary value problems.Nonlinear Anal.56(2004)975-984.
[24]田元生,刘春根,三阶p-Laplacian方程三点奇异边值问题三个正解的存在性,应用数学学报(已接收发表).
[25]H.Su,Z.Wei and B.Wang,The existence of positive solutions for a nonlinear four-point singular boundary value problem with a p-Laplace operator.Nonlinear Anal.33(2007) 2204-2217.
[26]X.He,Double positive solutions of a three-point boundary value problem for the one-dimensional p-Laplacian,Appl.Math.Lett.17(2004)867-873.
[27]X.He and W.Ge,Triple solutions for second-order three-point boundary value problems.J.Math.Anal.Appl.268(2002) 256-265.
[28]D.R.Anderson and J.M.Davis,Multiple solutions and eigenvalues for third-order right focal boundary value problem,J.Math.Anal.Appl.267(2002)135-157.
[29]R.I.Avery,J.M.Davis and J.Henderson,Three symmetric positive solutions for Lidstone problems by a generalization of the Leggett-Williams theorem,Electron.J.Differential Equations 2000(2000),No.40,pp.1-15.
[30]D.Cao and R.Ma,Positive solutions to a second order muliti-point boundary value problem,Electron.J.Differential Equations 2000(2000),No.65,pp.1-8.
[31]郭大钧,非线性泛函分析[M].山东:山东科技出版社,2001.
[32]X.He and W.Ge,A theorem about triple positive solutions for the one-dimensional p-Laplacian equations.(Chinese) Acta.Math.Appl.Sin. 26(2003),no.3,504-510.
[33]X.He and W.Ge,Existence of positive solutions for the one-dimensional p-Laplacian equations.(Chinese) Acta.Math.Sinica.46(2003),no.4,805-810.
[34]葛渭高,非线性常微分方程边值问题,科学出版社,2007年.
[35]D.Ji,M.Feng and W.Ge,Multiple positive solutions for multipoint boundary value problems with sign changing nonlinearity,Appl.Math.Comput.196(2008) 511-520.
[36]D.Ji,T.Yi,B.Yang and W.Ge,Triple positive pseudo-symmetric solutions to a four-point boundary value problem with p-Laplacian,Applied Mathematics Letters 21(2008) 268-274.
[37]H.Lian and W.Ge,Positive solutions for a four-point boundary value problem with the p-Laplacian,Nonlinear Analysis 68(2008) 3493-3503.
[38]H.Feng and W.Ge,Existence of three positive solutions for m-point boundary-value problems with one-dimensional p-Laplacian,Nonlinear Analysis 68(2008) 2017-2026.
[39]X.Zhang,M.Feng and W.Ge,Existence and nonexistence of positive solutions for a class of n-th-order three-point boundary value problems in Banach spaces.Nonlinear Anal.70(2009),no.2,584-597.
[40]B.Sun,J.Zhao,P.Yang,W.Ge,Successive iteration and positive solutions for a third-order multipoint generalized righ-focal boundary value problem with p-Laplacian.Nonlinear Anal.70(2009),no.1,220-230.
[41]W.Ge and C.Xue,Some fixed point theorems and existence of positive solutions of two-point boundary-value problems.Nonlinear Anal.70(2009),no.1,16-31.
[42]L.Wang,Y.Fan and W.Ge,Periodic solutions in a delayed predator-prey model with nonmonotonic functional response.Rocky Mountain J.Math.38 (2008), no.5, 1705—1719.
[43]Z.Gui, X.Yang and W.Ge, Existence of global exponential stability of periodic solutions of recurrent cellular neural networks with impulses and delays.Math.Comput.Simulation 79 (2008), no.1, 14—29.
[44]D.Ma and W.Ge, Existence and iteration of solutions for a multi-point boundary value problem with a />-Laplacian operator.(Chinese) Acta Math.Sinica (Chin.Ser.) 51 (2008), no.3, 447—456.
[45]D.Ma and W.Ge, Positive solution of multi-point boundary value problem for the one-dimensional /7-Laplacian with singularities.Rocky Mountain J.Math.37 (2007), no.4, 1229—1249.
[46]Y.Li, Positive periodic solutions for neutral functional differential equations with distributed delays and feedback control.Nonlinear Anal.Real World Appl.9 (2008), no.5, 2214—2221.
[47]Y.Li, Positive periodic solutions for a periodic neutral differential equation with feedback control.Nonlinear Anal.Real World Appl.6 (2005), no.1, 145—154.
[48]Y.Li, L.Zhu and P.Liu, Positive periodic solutions of nonlinear functional difference equations depending on a parameter.Comput.Math.Appl.48 (2004), no.10-11, 1453—1459.
[49]Y.Li and L.Zhu, Positive periodic solutions of nonlinear functional differential equations.Appl.Math.Comput.156 (2004), no.2, 329—339.