三阶m-点非齐次边值问题的正解
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摘要
微分方程多点边值问题是非线性分析理论的一个重要分支,它起源于各种不同的应用数学和物理学领域,尤其是在弹性和稳定性理论中有着广泛的应用.特别地,常微分方程非齐次多点边值问题受到了人们的广泛关注,它是目前分析数学中研究最活跃的领域之一.因此,对微分方程非齐次多点边值问题的研究具有重要意义.
本硕士论文开展了以下三个方面的研究工作:首先,简述了课题的研究背景及本文的主要工作,并给出了本文用到的预备知识.其次,通过给予非线性项一些限制条件,建立了一类非线性三阶m?点非齐次边值问题解与正解的存在性准则,所用主要工具是Schauder不动点定理.最后,针对一类三阶m?点非齐次边值问题,通过运用Guo-Krasnoselskii不动点定理,在f满足超(次)线性时,给参数适当的取值,得到边值问题单调正解的存在性与不存在性.
Multi-point boundary value problems (BVPs for short) of di?erential equationsare an important branch in the theory of nonlinear analysis, which arise in di?erentfields of applied mathematics and physics and have wide applications on elasticityand stability theory. Especially, nonhomogeneous multi-point BVPs of di?erentialequations have received much attention from many authors. At present, it is oneof the most active fields that is studied in analytic mathematics. Therefore, it issignificant to study the nonhomogeneous multi-point BVPs of di?erential equations.
This thesis carries out research work in the following three aspects: first, we in-troduce the studying background of this thesis and our main works, and we list somepreliminary knowledge needed in this paper. Second, we discuss a class of nonlinearthird-order m? point BVP. By imposing some conditions on the nonlinear term, weconstruct some existence criteria of solution and positive solution for the third-orderm? point nonhomogeneous BVP. The mail tool used is the Schauder fixed pointtheorem. Finally, aiming at a class of third-order m? point nonhomogeneous BVP,The existence and nonexistence of monotone positive solution are discussed for suit-able parameter when f is superlinear or sublinear by using Guo-Krasnoselskii fixedpoint theorem.
本硕士论文开展了以下三个方面的研究工作:首先,简述了课题的研究背景及本文的主要工作,并给出了本文用到的预备知识.其次,通过给予非线性项一些限制条件,建立了一类非线性三阶m?点非齐次边值问题解与正解的存在性准则,所用主要工具是Schauder不动点定理.最后,针对一类三阶m?点非齐次边值问题,通过运用Guo-Krasnoselskii不动点定理,在f满足超(次)线性时,给参数适当的取值,得到边值问题单调正解的存在性与不存在性.
Multi-point boundary value problems (BVPs for short) of di?erential equationsare an important branch in the theory of nonlinear analysis, which arise in di?erentfields of applied mathematics and physics and have wide applications on elasticityand stability theory. Especially, nonhomogeneous multi-point BVPs of di?erentialequations have received much attention from many authors. At present, it is oneof the most active fields that is studied in analytic mathematics. Therefore, it issignificant to study the nonhomogeneous multi-point BVPs of di?erential equations.
This thesis carries out research work in the following three aspects: first, we in-troduce the studying background of this thesis and our main works, and we list somepreliminary knowledge needed in this paper. Second, we discuss a class of nonlinearthird-order m? point BVP. By imposing some conditions on the nonlinear term, weconstruct some existence criteria of solution and positive solution for the third-orderm? point nonhomogeneous BVP. The mail tool used is the Schauder fixed pointtheorem. Finally, aiming at a class of third-order m? point nonhomogeneous BVP,The existence and nonexistence of monotone positive solution are discussed for suit-able parameter when f is superlinear or sublinear by using Guo-Krasnoselskii fixedpoint theorem.
引文
[1] II’in V A, Moiseev E I. Nonlocal boundary value problem of the first kind for aSturm-Liouville operator in its di?erential and finite di?erence aspects. Di?erentialEquations, 1987, 23: 803-810
[2] II’in V A, Moiseev E I. Nonlocal boundary value problem of the second kind for aSturm-Liouville operator. Di?erential Equations, 1987, 23: 979-987
[3] Gupta C P. Solvability of a three-point nonlinear boundary value problem for a secondorder ordinary di?erential equation. J. Math. Anal. Appl., 1992, 168: 540-551
[4] Sun J P, Li W T, Zhao Y H. Three positive solutions for a nonlinear three-pointboundary value problems. J. Math. Anal. Appl., 2003, 288: 708-716
[5] Feng W, Webb J R L. Solvability of m-point boundary value problems with nonlineargrowth. J. Math. Anal. Appl., 1997, 212: 467-480
[6] Ma R. Solvability of singular second order m-point boundary value peoblems results.J. Math. Anal. Appl., 2005, 301: 124-134
[7] Ma R. Positive solutions for second order three-point boundary value problems. Appl.Math. Lett., 2001, 14: 1-5
[8] Thompson H B, Tisdell C. Three-point boundary value problems for second orderordinary di?erential equations. Math. Comput. Model., 2001, 34: 311-318
[9] Gregus M. Third Order Linear Di?erential Equations. in: Math. Appl., Reidel, Dor-drecht, 1987
[10] Du Z J, Ge W G, Lin X J. Existence of solutions for a class of third-order nonlinearboundary value problems. J. Math. Anal. Appl., 2004, 294: 104-112
[11] Feng Y, Liu S. Solvability of a third-order two-point boundary value problem. Appl.Math. Lett., 2005, 18: 1034-1040
[12] Grossinho M R, Minhos F M, Santos A I. Solvability of some third-order boundaryvalue problems with asymmetric unbounded nonlinearities. Nonlinear Anal., 2005, 62:1235-1250
[13] Feng Y. Existence and uniqueness results for a third-order implicit di?erential equa-tion. Computers and Mathematics with Applications, 2008, 56: 2507-2514
[14] Yao Q, Feng Y. The existence of solution for a third-order two-point boundary valueproblem. Appl. Math. Lett., 2002, 15: 227-232
[15] Li S. Positive solutions of nonlinear singular third-order two-point boundary valueproblem. J. Math. Anal. Appl., 2006, 323: 413-425
[16] Liu Z, Debnath L, Kang S M. Existence of monotone positive solutions to a third-ordertwo-point generalized right focal boundary value problem. Comput. Math. Appl.,2008, 55: 356-367
[17] Feng Y. Solution and positive solution of a semilinear third-order equation. J. Appl.Math. Comput., 2009, 29: 153-161
[18] Hopkins B, Kosmatov N. Third-order boundary value problems with sign-changingsolutions. Nonlinear Anal., 2007, 67: 126-137
[19] Bai Z B. Existence of solution for some third-order boundary-value problems. Elec-tronic J. Di?erential Equations, 2008, 25: 1-6
[20] Yao Q. Solutions and positive solution for a semilinear third-order two-point boundaryvalue problems. Appl. Math. Lett., 2004, 17: 1171-1175
[21] Anderson D R. Green s function for a third-order generalized right focal problem. J.Math. Anal. Appl., 2003, 288: 1-14
[22] Guo L J, Sun J P, Zhao Y H. Existence of positive solution for nonlinear third-orderthree-point boundary value problem. Nonlinear Anal., 2008, 68: 3151-3158
[23] Sun J P, Guo L J, Peng J G. Multiple nondecreasing positive solutions for a singularthird-order three-point BVP. Communications in Applied Analysis, 2008, 12: 91-100
[24] Sun J P, Ren Q Y, Zhao Y H. The upper and lower solution method for nonlinearthird-order three-point boundary value problem. Electronic Journal of QualitativeTheory of Di?erential Equations, 2010, 26: 1-8
[25] Ma R. Multiplicity results for a third order boundary value problem at resonance.Nonlinear Anal., 1998, 32: 493-499
[26] Sun Y. Positive solutions for third-order three-point nonhomogeneous boundary valueproblems. Appl. Math. Lett., 2009, 22: 45-51
[27] Yang B. Positive solutions of a third-order three-point boundary-value problem. Elec-tronic J. Di?erential Equations, 2008, 99: 1-10
[28] Yao Q. Positive solutions of singular third-order three-point boundary value problems.J. Math. Anal. Appl., 2009, 354: 207-212
[29] Du Z J, Lin X J, Ge W G. On a third-order multi-point boundary value problem atresonance. J. Math. Anal. Appl., 2005, 302: 217-229
[30] Du Z J, Ge W G, Zhou M R. Singular perturbations for third-order nonlinear multi-point boundary value problem. J. Di?erential Equations, 2005, 218: 69-90
[31] Sun J P, Zhang H E. Existence of solutoins to third-order m? point boundary-valueproblems. Electronic J. Di?erential Equations, 2008, 125: 1-9
[32] Jin S, Lu S. Existence of solutions for a third-order multipoint boundary value problemwith p-Laplacian. Journal of the Franklin Institute, 2010, 347: 599-606
[33] Sun J P, Ren Q Y. Existence of solution for third-order m? point boundary valueproblem. Applied Mathematics E-Notes, 2010, 10: 208-274
[34] Chen H. Positive solutions for the nonhomogeneous three-point boundary value prob-lem of second-order di?erential equations. Mathematical and Computer Modelling,2007, 45: 844-852
[35] Ma R. Positive solutions for second-order three-point boundary value problems. Appl.Math. Lett., 2001, 14: 1-5
[36] Kong L, Kong Q. Multi-point boundary value problems of second-order di?erentialequations (I). Nonl. Anal., 2004, 58: 909-931
[37] Kong L, Kong Q. Multi-point boundary value problems of second-order di?erentialequations (II). Comm. Appl. Nonl. Anal., 2007, 14: 93-111
[38] Kong L, Kong Q. Second-order boundary value problems with nonhomogeneousboundary conditions (I). Math. Nach., 2005, 278: 173-193
[39] Kong L, Kong Q. Second-order boundary value problems with nonhomogeneousboundary conditions (II). J. Math. Anal. Appl., 2007, 330: 1393-1411
[40] Sun Y. Positive solutions for third-order three-point nonhomogeneous boundary valueproblems. Appl. Math. Lett., 2009, 22: 45-51
[41] Guo D, Lakshikantham V. Nonlinear Problems in Abstract Cones. New York: Aca-demic Press, 1988
[42] Guo D. Nonlinear Functional Analysis and Applications. Ji Nan: Academic Press,2001
[43]龚怀云,寿纪麟,王锦森.应用泛函分析.西安交通大学出版社, 1985
[44] Zeidler E. Nonlinear Functional Analysis and Applications, I: Fixed Point Theorems.New York: Springer-Verlag Press, 1986
[45] Amann H. Fixed point equations and nonlinear eigenvalue problems in ordered Ba-nach spaces. SIAM Rev., 1976, 18: 620-709
[46] Erbe L H, Wang H. On the existence of positive solutions of ordinary di?erentialequations. Proceedings of the American Mathematical Society, 1994, 120: 743-748
[2] II’in V A, Moiseev E I. Nonlocal boundary value problem of the second kind for aSturm-Liouville operator. Di?erential Equations, 1987, 23: 979-987
[3] Gupta C P. Solvability of a three-point nonlinear boundary value problem for a secondorder ordinary di?erential equation. J. Math. Anal. Appl., 1992, 168: 540-551
[4] Sun J P, Li W T, Zhao Y H. Three positive solutions for a nonlinear three-pointboundary value problems. J. Math. Anal. Appl., 2003, 288: 708-716
[5] Feng W, Webb J R L. Solvability of m-point boundary value problems with nonlineargrowth. J. Math. Anal. Appl., 1997, 212: 467-480
[6] Ma R. Solvability of singular second order m-point boundary value peoblems results.J. Math. Anal. Appl., 2005, 301: 124-134
[7] Ma R. Positive solutions for second order three-point boundary value problems. Appl.Math. Lett., 2001, 14: 1-5
[8] Thompson H B, Tisdell C. Three-point boundary value problems for second orderordinary di?erential equations. Math. Comput. Model., 2001, 34: 311-318
[9] Gregus M. Third Order Linear Di?erential Equations. in: Math. Appl., Reidel, Dor-drecht, 1987
[10] Du Z J, Ge W G, Lin X J. Existence of solutions for a class of third-order nonlinearboundary value problems. J. Math. Anal. Appl., 2004, 294: 104-112
[11] Feng Y, Liu S. Solvability of a third-order two-point boundary value problem. Appl.Math. Lett., 2005, 18: 1034-1040
[12] Grossinho M R, Minhos F M, Santos A I. Solvability of some third-order boundaryvalue problems with asymmetric unbounded nonlinearities. Nonlinear Anal., 2005, 62:1235-1250
[13] Feng Y. Existence and uniqueness results for a third-order implicit di?erential equa-tion. Computers and Mathematics with Applications, 2008, 56: 2507-2514
[14] Yao Q, Feng Y. The existence of solution for a third-order two-point boundary valueproblem. Appl. Math. Lett., 2002, 15: 227-232
[15] Li S. Positive solutions of nonlinear singular third-order two-point boundary valueproblem. J. Math. Anal. Appl., 2006, 323: 413-425
[16] Liu Z, Debnath L, Kang S M. Existence of monotone positive solutions to a third-ordertwo-point generalized right focal boundary value problem. Comput. Math. Appl.,2008, 55: 356-367
[17] Feng Y. Solution and positive solution of a semilinear third-order equation. J. Appl.Math. Comput., 2009, 29: 153-161
[18] Hopkins B, Kosmatov N. Third-order boundary value problems with sign-changingsolutions. Nonlinear Anal., 2007, 67: 126-137
[19] Bai Z B. Existence of solution for some third-order boundary-value problems. Elec-tronic J. Di?erential Equations, 2008, 25: 1-6
[20] Yao Q. Solutions and positive solution for a semilinear third-order two-point boundaryvalue problems. Appl. Math. Lett., 2004, 17: 1171-1175
[21] Anderson D R. Green s function for a third-order generalized right focal problem. J.Math. Anal. Appl., 2003, 288: 1-14
[22] Guo L J, Sun J P, Zhao Y H. Existence of positive solution for nonlinear third-orderthree-point boundary value problem. Nonlinear Anal., 2008, 68: 3151-3158
[23] Sun J P, Guo L J, Peng J G. Multiple nondecreasing positive solutions for a singularthird-order three-point BVP. Communications in Applied Analysis, 2008, 12: 91-100
[24] Sun J P, Ren Q Y, Zhao Y H. The upper and lower solution method for nonlinearthird-order three-point boundary value problem. Electronic Journal of QualitativeTheory of Di?erential Equations, 2010, 26: 1-8
[25] Ma R. Multiplicity results for a third order boundary value problem at resonance.Nonlinear Anal., 1998, 32: 493-499
[26] Sun Y. Positive solutions for third-order three-point nonhomogeneous boundary valueproblems. Appl. Math. Lett., 2009, 22: 45-51
[27] Yang B. Positive solutions of a third-order three-point boundary-value problem. Elec-tronic J. Di?erential Equations, 2008, 99: 1-10
[28] Yao Q. Positive solutions of singular third-order three-point boundary value problems.J. Math. Anal. Appl., 2009, 354: 207-212
[29] Du Z J, Lin X J, Ge W G. On a third-order multi-point boundary value problem atresonance. J. Math. Anal. Appl., 2005, 302: 217-229
[30] Du Z J, Ge W G, Zhou M R. Singular perturbations for third-order nonlinear multi-point boundary value problem. J. Di?erential Equations, 2005, 218: 69-90
[31] Sun J P, Zhang H E. Existence of solutoins to third-order m? point boundary-valueproblems. Electronic J. Di?erential Equations, 2008, 125: 1-9
[32] Jin S, Lu S. Existence of solutions for a third-order multipoint boundary value problemwith p-Laplacian. Journal of the Franklin Institute, 2010, 347: 599-606
[33] Sun J P, Ren Q Y. Existence of solution for third-order m? point boundary valueproblem. Applied Mathematics E-Notes, 2010, 10: 208-274
[34] Chen H. Positive solutions for the nonhomogeneous three-point boundary value prob-lem of second-order di?erential equations. Mathematical and Computer Modelling,2007, 45: 844-852
[35] Ma R. Positive solutions for second-order three-point boundary value problems. Appl.Math. Lett., 2001, 14: 1-5
[36] Kong L, Kong Q. Multi-point boundary value problems of second-order di?erentialequations (I). Nonl. Anal., 2004, 58: 909-931
[37] Kong L, Kong Q. Multi-point boundary value problems of second-order di?erentialequations (II). Comm. Appl. Nonl. Anal., 2007, 14: 93-111
[38] Kong L, Kong Q. Second-order boundary value problems with nonhomogeneousboundary conditions (I). Math. Nach., 2005, 278: 173-193
[39] Kong L, Kong Q. Second-order boundary value problems with nonhomogeneousboundary conditions (II). J. Math. Anal. Appl., 2007, 330: 1393-1411
[40] Sun Y. Positive solutions for third-order three-point nonhomogeneous boundary valueproblems. Appl. Math. Lett., 2009, 22: 45-51
[41] Guo D, Lakshikantham V. Nonlinear Problems in Abstract Cones. New York: Aca-demic Press, 1988
[42] Guo D. Nonlinear Functional Analysis and Applications. Ji Nan: Academic Press,2001
[43]龚怀云,寿纪麟,王锦森.应用泛函分析.西安交通大学出版社, 1985
[44] Zeidler E. Nonlinear Functional Analysis and Applications, I: Fixed Point Theorems.New York: Springer-Verlag Press, 1986
[45] Amann H. Fixed point equations and nonlinear eigenvalue problems in ordered Ba-nach spaces. SIAM Rev., 1976, 18: 620-709
[46] Erbe L H, Wang H. On the existence of positive solutions of ordinary di?erentialequations. Proceedings of the American Mathematical Society, 1994, 120: 743-748