一维p-Laplacian方程多点边值问题的正解
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摘要
非线性泛函分析是分析数学中既有深刻理论又有广泛应用的研究学科,它以数学和自然科学中出现的非线性问题为背景,建立处理非线性问题的若干一般性理论和方法.因其能很好地解释自然界中诸多现象,近年来受到了国内外数学及自然科学界工作者的高度重视,逐渐形成了一门重要的学科.在处理实际问题所对应的各种非线性积分方程,微分方程和偏微分方程中发挥着不可替代的作用.
本文首先借助于泛函型的锥拉伸锥压缩不动点定理,给出了一类非线性项含有低阶导数的一维p-Laplacian微分方程四点边值问题至少存在一个,两个正解的充分条件.其次借助于Avery-Peterson不动点定理及一些分析技巧,对上述问题三个正解的存在性进行了讨论,建立了其至少有三个正解的存在性准则.所用方法可以推广到研究2m点边值问题.最后,我们分别利用单调迭代方法研究了当α=β,η=1-ξ的情形时,在没有假定所讨论边值问题上下解存在的条件下,讨论了边值问题(1),(2)对称正解的存在性,并且建立了相应的逼近正解的迭代格式.
Nonlinear functional analysis are research discipline in analysis mathematics both to have the profound theory and to have the widespread application. It takes the nonlinear problems appearing in mathematics and the natural science as background to establish some general theories and methods to handle nonlinear problem. Since it can commendably explain many kinds of natural phenomena, in recent years, which has received widely attention in domestic and foreign mathematics and natural science field, and formed an important discipline gradually. To handle all kinds of nonlinear integral equations, the differential equations and the partial differential equations in actual problems, it plays role which can not be substituted.
The dissertation is divided into three chapters:The first chapter introduces the background, methods of the related problems and the major work for this paper.
The second chapter, by using the fixed point index theory and the fixed point theorem of cone expansion-compression of functional type, we firstly obtain some sufficient conditions of the existence of at least one, two positive solution for boundary value problems with the one-dimensional p-LaplacianBy means of Avery-Peterson fixed point theorem on cone and some skill and techniques of analysis, we consider the positive solutions for boundary value problems with the one-dimensional p-Laplacian. Some sufficient conditions of the existence of at least three positive solutions are obtained. the method can be extended to the study of 2m-point boundary value problem. Finally, our approach is based on the monotone iterative technique, studying the caseα=β,η= 1-ξ. Without the assumption of the existence of lower and upper solution, we obtain not only the existence of positive solutions for the problems (1),(2), but also establish iterative schemes for approximating the solutions.
本文首先借助于泛函型的锥拉伸锥压缩不动点定理,给出了一类非线性项含有低阶导数的一维p-Laplacian微分方程四点边值问题至少存在一个,两个正解的充分条件.其次借助于Avery-Peterson不动点定理及一些分析技巧,对上述问题三个正解的存在性进行了讨论,建立了其至少有三个正解的存在性准则.所用方法可以推广到研究2m点边值问题.最后,我们分别利用单调迭代方法研究了当α=β,η=1-ξ的情形时,在没有假定所讨论边值问题上下解存在的条件下,讨论了边值问题(1),(2)对称正解的存在性,并且建立了相应的逼近正解的迭代格式.
Nonlinear functional analysis are research discipline in analysis mathematics both to have the profound theory and to have the widespread application. It takes the nonlinear problems appearing in mathematics and the natural science as background to establish some general theories and methods to handle nonlinear problem. Since it can commendably explain many kinds of natural phenomena, in recent years, which has received widely attention in domestic and foreign mathematics and natural science field, and formed an important discipline gradually. To handle all kinds of nonlinear integral equations, the differential equations and the partial differential equations in actual problems, it plays role which can not be substituted.
The dissertation is divided into three chapters:The first chapter introduces the background, methods of the related problems and the major work for this paper.
The second chapter, by using the fixed point index theory and the fixed point theorem of cone expansion-compression of functional type, we firstly obtain some sufficient conditions of the existence of at least one, two positive solution for boundary value problems with the one-dimensional p-LaplacianBy means of Avery-Peterson fixed point theorem on cone and some skill and techniques of analysis, we consider the positive solutions for boundary value problems with the one-dimensional p-Laplacian. Some sufficient conditions of the existence of at least three positive solutions are obtained. the method can be extended to the study of 2m-point boundary value problem. Finally, our approach is based on the monotone iterative technique, studying the caseα=β,η= 1-ξ. Without the assumption of the existence of lower and upper solution, we obtain not only the existence of positive solutions for the problems (1),(2), but also establish iterative schemes for approximating the solutions.
引文
[1] R. P. Agarwal, D. O'Regan, P. J. Y. Wong, Positive solutions of differential, difference and integral equations. Kluwer Academic, Dordrecht. 1999.
[2] R.I. Avery, A.C. Peterson, Three positive ?xed points of nonlinear operators on ordered Banach spaces, Comput. Math. Appl. 42(2001)313 - 322.
[3] C. Bai, J.Fang, Existence of multiple positive solutions for nonlinear m-point boundary-value problems. Appl. Math. Comput. 140(2003), 297-305.
[4] C. Bai, J. Fang, Existence of multiple positive solutions for nonlinear multi-point boundary-value problems. J. Math. Anal. Appl. 281(2003), 76-85.
[5] J.Yang, Z.Wei, Existence of symmetric positive solutions for a class of Sturm-Liouville-like boundary value problems. Appl. Math. Comput.In Press.
[6] D.O,Regan, Some general existence principles and results for φ(y')'=qf(t, y, y'). SIAM J.Math.Anal. 1993,24(4), 648-668.
[7] I.Ly,D.Seck, Isoperimetric inequality for aninterior free boundary problem with p-Lapacian operator, Electronic J. Diff. Eqns. 2004,109:1-12.
[8] D.Jiang, X.Xu, Multiple positive solutions to a class of singular boundary value problems for the a one-dimensional p-Laplacian. Comput. Math. Appl. 47(2004),667-681.
[9] H. Feng, W. Ge, M. Jiang, Multiple positive solutions for m-point boundary-value problems with a one-dimensional p-Laplacian. Nonlinear Anal. 68(2008), 2269-2279.
[10] H. Feng, W. Ge, Triple symmetric positive solutions for multipoint boundary-value problem with one-dimensional p-Laplacian, Math. Comput. Modelling.47(2008), 186-195.
[11] F.Wong, Existence of positive solutions for m-point boundary-value problems.Appl. Math. lett. 12(1999), 11-17.
[12] 葛渭高,非线性常微分方程边值问题,北京:科学出版社,2007
[13] 郭大均,非线性泛函分析,山东科技出版社,2002.
[14] M. Garcia-Huidobro, C.P.Gupta, R. Manasevich, An multi-point boundary-value problem of Neumann type for p-Laplacian like operator. Nonlinear. Anal. 56(2004), 1071-1089.
[15] M. R. Grossinho, F. M. Minhos, A. I. Santos, Solvability of some third-order boundary value problems with asymmetric unbounded nonlinearities. Nonlinear. Anal. 62(2005), 1235-1250.
[16] M. D. R. Grossinho, F. M. Minhos, A. I. Santos, A third-order boundary value problem with one-sided Nagumo condition. Nonlinear. Anal. 63(2005), 247-256.
[17] Y. Guo, W. Ge, Three positive solutions for the one dimension p-Laplacian. J. Math. Anal. Appl. 286(2003), 491-508.
[18] C. P. Gupta, A generalized multi-point boundary value problem for second order ordinary differential equations. Appl. Math. Comput. 89(1998), 133-146.
[19] R.Glowinskl and J.Rappaz, , Approximation of a nonliear elliptic problem arising in a non Newtonian fluid flow model in glaciology, Math.Model.Numer.Anal. 2003,37(1),175-186.
[20] V. A. Il'in, E. I. Moiseev, Nonlocal boundary value problem of the first kind for Sturm-Liouville operator in its differential and finite differece aspects.J. Diff. Eqs. 7(1987), 803-810.
[21] V. A. Il'in, E. I. Moiseev, Nonlocal boundary value problem of the second kind for Sturm-Liouville operator. Diff. Eqs. 23(1987), 979-987.
[22] D. Ji, H. Feng, W. Ge, The existence of symmetric positive solutions for some nonlinear equation systems. Appl. Math. Comput. 197(2008), 51-59.
[23] J.Diaz, F.de Thelin, On anonlinear parabolie problem arising in some medels related to turbulent flows. SIAM J.Math.Anal. 1994,24(4): 1085-1111.
[24] J.Zhao, W. Ge, Multiple symmetric positive solutions to a new kind of four point boundary value problem. Nonlinear Anal. 71(2009), 9-18.
[25] L. Kong, J. Wang, Multiple positive solutions for one-dimension p-Laplacian. Nonlinear Anal. 42(2000), 1327-1333.
[26] H.li, W. Ge, Positive solutions for a four-point boundary value problem with the p-Laplacian. Nonlinear. Anal. 68(2008),3493-3503.
[27] X.Li, Multiple positive solutions for some four-point boundary value problems with p-Laplacian Applied Mathematics and Computation. Appl. Math. Comput. 22(2008), 413-426.
[28] B. Liu, Positive solutions of a nonlinear four-point boundary value problems, Appl. Math. Comput. 155(2004), 179-203.
[29] R.Liang, J.Peng, J.Shen, Double positive solutions for a nonlinear fourpoint boundary value problem with a p-Laplacian operator, Nonlinear Anal. 68(2008),1881-1889.
[30] Y. Liu, P. Yang, W. Ge, Periodic solutions of higher order delay differential equations, Nonlinear Anal. 63(2005), 136-152.
[31] H. Lu, D. O'Regan, C. Zhong, Multiple positive solutions for one dimensional singular p-Laplacian, Appl. Math. Comput. 133(2002), 407-422.
[32] H. Lian, W. Ge, Positive solutions for a four-point boundary value problem with the p-Laplacian. Nonlinear Anal. 68(1998), 3493-3503.
[33] 马如云,非线性常微分方程非局部问题,北京:科学出版社,2001.
[34] Manasevich R.,Zanolin F, Time mappings and multiplicity of solutions for the one dimensional p-Laplacian. Nonlinear. Anal. 21(1993), 269-291.
[35] Manasevich R., Zanolin F, Periodic solutions for nonlinear systems with p-Laplacian like operators.J.D.E. 21(1993), 269-291.
[36] D.Ma, Z.Du, Existence and iteration of monotone positive solutions for multipoint boundary value problem with p-Laplacian operator. Comput. Math. Appl. 50(2005), 729-739.
[37]M.Xiong,J.Liu,P.Zeng Positive solutions for singular two-point boundary value problem for p-Laplacian equation,Acta Mathematica Scientia,27A(2007),549-558.(In Chinese)
[38]M.Xiong,S.Wang,The existence of positive solution for the one-dimensional p- Laplacian,Acta Mathematica Sinica,(Chinese Series),49(2006),165-172.(In Chinese)
[39]R.K.Nagle,K.L.Pothoven,On a third order nonlinear boundary value problem at resonance.J.Math.Anal.Appl.195(1995),149-159.
[40]H.Pang,H.Lian,W.Ge,Multiple positive solutions for second-order four-point boundary value problem.Appl.Math.lett.21(2008),656-661 .
[41]H.Pang,H.Lian,W Ge,Existence and monotone iteration of positive solutions for a three-point boundary value problem,Appl.Math.Comput.188(2008),1234-1243.
[42]C.Pang,Z.Wei,The exact number of positive solutions for a class of differential equations invoviong p-laplacian operator,Acta Mathematica Sinica,48(2005),1203-1208.(In Chinese)
[43]H.Su,Z.Wei,B.Wang,The existence of positive solutions for a nonlinear four- point singular boundary value problem with a p-Laplacian operator.Nonlinear.Anal.66(2007),2204-2217.
[44]H.Su,Z.Wei,X.Zhang and J.Liu,Positive solutions of n-order and m-order multi-point singular boundary value system.Appl.Math.Comput.188(2007),1234-1243.
[45]B.Sun,Y.Qu,W Ge,Existence and iteration of positive solutions for a multipoint one-dimensional p-Laplacian boundary value problem.Appl.Math.Comput.197(2008),389-398.
[46]B.Sun,W.Ge,Existence and iteration of positive solutions to a class of Sturm - Liouville-like p-Laplacian boundary value problems.Nonlinear.Anal.69(2008),1454-1461 .
[47] B. Sun, W. Ge, Existence and iteration of positive solutions for some p-Laplacian boundary value problems, Nonlinear Anal. 67(2007), 1820-1830.
[48] U.Janfalk, On eertain problem concerning the p-Laplace opertor, Likping studies in sciences and technology, dissertations, 1993, vol.326.
[49] J. Wang, W. Gao, A singular boundary value problem for one-dimension p-Laplacian. J. Math. Anal. Appl. 201(1996), 851-866.
[50] Y. Wang, W. Ge, Positive solutions for multi-point boundary value problems with a one-dimensional p-Laplacian, Nonlinear Anal. 66(2007), 1246-1256.
[51] Y. Wang, W. Ge, Existence of triple positive solutions for multi-point boundary value problems with a one dimensional p-Laplacian. Comput. Math. Appl. 54(2007), 793-807.
[52] Y. Wang, W. Ge, Existence of multiple positive solutions for multipoint boundary value problems with a one-dimensional p-Laplacian, Nonlinear Anal. 67(2007),476-485.
[53] Y. Wang, W. Ge, Multiple positive solutions for multipoint boundary value problems with one-dimensional p-Laplacian. J. Math. Anal. Appl. 327(2007). 1381-1395.
[54] X. Yang, Positive solutions for the one-dimensional p-Lpalacian. Math. Comput. Modelling. 35(2002), 129-135.
[55] 钟承奎,范先令,陈文源,非线性泛函分析引论,兰州大学出版社,2002.
[2] R.I. Avery, A.C. Peterson, Three positive ?xed points of nonlinear operators on ordered Banach spaces, Comput. Math. Appl. 42(2001)313 - 322.
[3] C. Bai, J.Fang, Existence of multiple positive solutions for nonlinear m-point boundary-value problems. Appl. Math. Comput. 140(2003), 297-305.
[4] C. Bai, J. Fang, Existence of multiple positive solutions for nonlinear multi-point boundary-value problems. J. Math. Anal. Appl. 281(2003), 76-85.
[5] J.Yang, Z.Wei, Existence of symmetric positive solutions for a class of Sturm-Liouville-like boundary value problems. Appl. Math. Comput.In Press.
[6] D.O,Regan, Some general existence principles and results for φ(y')'=qf(t, y, y'). SIAM J.Math.Anal. 1993,24(4), 648-668.
[7] I.Ly,D.Seck, Isoperimetric inequality for aninterior free boundary problem with p-Lapacian operator, Electronic J. Diff. Eqns. 2004,109:1-12.
[8] D.Jiang, X.Xu, Multiple positive solutions to a class of singular boundary value problems for the a one-dimensional p-Laplacian. Comput. Math. Appl. 47(2004),667-681.
[9] H. Feng, W. Ge, M. Jiang, Multiple positive solutions for m-point boundary-value problems with a one-dimensional p-Laplacian. Nonlinear Anal. 68(2008), 2269-2279.
[10] H. Feng, W. Ge, Triple symmetric positive solutions for multipoint boundary-value problem with one-dimensional p-Laplacian, Math. Comput. Modelling.47(2008), 186-195.
[11] F.Wong, Existence of positive solutions for m-point boundary-value problems.Appl. Math. lett. 12(1999), 11-17.
[12] 葛渭高,非线性常微分方程边值问题,北京:科学出版社,2007
[13] 郭大均,非线性泛函分析,山东科技出版社,2002.
[14] M. Garcia-Huidobro, C.P.Gupta, R. Manasevich, An multi-point boundary-value problem of Neumann type for p-Laplacian like operator. Nonlinear. Anal. 56(2004), 1071-1089.
[15] M. R. Grossinho, F. M. Minhos, A. I. Santos, Solvability of some third-order boundary value problems with asymmetric unbounded nonlinearities. Nonlinear. Anal. 62(2005), 1235-1250.
[16] M. D. R. Grossinho, F. M. Minhos, A. I. Santos, A third-order boundary value problem with one-sided Nagumo condition. Nonlinear. Anal. 63(2005), 247-256.
[17] Y. Guo, W. Ge, Three positive solutions for the one dimension p-Laplacian. J. Math. Anal. Appl. 286(2003), 491-508.
[18] C. P. Gupta, A generalized multi-point boundary value problem for second order ordinary differential equations. Appl. Math. Comput. 89(1998), 133-146.
[19] R.Glowinskl and J.Rappaz, , Approximation of a nonliear elliptic problem arising in a non Newtonian fluid flow model in glaciology, Math.Model.Numer.Anal. 2003,37(1),175-186.
[20] V. A. Il'in, E. I. Moiseev, Nonlocal boundary value problem of the first kind for Sturm-Liouville operator in its differential and finite differece aspects.J. Diff. Eqs. 7(1987), 803-810.
[21] V. A. Il'in, E. I. Moiseev, Nonlocal boundary value problem of the second kind for Sturm-Liouville operator. Diff. Eqs. 23(1987), 979-987.
[22] D. Ji, H. Feng, W. Ge, The existence of symmetric positive solutions for some nonlinear equation systems. Appl. Math. Comput. 197(2008), 51-59.
[23] J.Diaz, F.de Thelin, On anonlinear parabolie problem arising in some medels related to turbulent flows. SIAM J.Math.Anal. 1994,24(4): 1085-1111.
[24] J.Zhao, W. Ge, Multiple symmetric positive solutions to a new kind of four point boundary value problem. Nonlinear Anal. 71(2009), 9-18.
[25] L. Kong, J. Wang, Multiple positive solutions for one-dimension p-Laplacian. Nonlinear Anal. 42(2000), 1327-1333.
[26] H.li, W. Ge, Positive solutions for a four-point boundary value problem with the p-Laplacian. Nonlinear. Anal. 68(2008),3493-3503.
[27] X.Li, Multiple positive solutions for some four-point boundary value problems with p-Laplacian Applied Mathematics and Computation. Appl. Math. Comput. 22(2008), 413-426.
[28] B. Liu, Positive solutions of a nonlinear four-point boundary value problems, Appl. Math. Comput. 155(2004), 179-203.
[29] R.Liang, J.Peng, J.Shen, Double positive solutions for a nonlinear fourpoint boundary value problem with a p-Laplacian operator, Nonlinear Anal. 68(2008),1881-1889.
[30] Y. Liu, P. Yang, W. Ge, Periodic solutions of higher order delay differential equations, Nonlinear Anal. 63(2005), 136-152.
[31] H. Lu, D. O'Regan, C. Zhong, Multiple positive solutions for one dimensional singular p-Laplacian, Appl. Math. Comput. 133(2002), 407-422.
[32] H. Lian, W. Ge, Positive solutions for a four-point boundary value problem with the p-Laplacian. Nonlinear Anal. 68(1998), 3493-3503.
[33] 马如云,非线性常微分方程非局部问题,北京:科学出版社,2001.
[34] Manasevich R.,Zanolin F, Time mappings and multiplicity of solutions for the one dimensional p-Laplacian. Nonlinear. Anal. 21(1993), 269-291.
[35] Manasevich R., Zanolin F, Periodic solutions for nonlinear systems with p-Laplacian like operators.J.D.E. 21(1993), 269-291.
[36] D.Ma, Z.Du, Existence and iteration of monotone positive solutions for multipoint boundary value problem with p-Laplacian operator. Comput. Math. Appl. 50(2005), 729-739.
[37]M.Xiong,J.Liu,P.Zeng Positive solutions for singular two-point boundary value problem for p-Laplacian equation,Acta Mathematica Scientia,27A(2007),549-558.(In Chinese)
[38]M.Xiong,S.Wang,The existence of positive solution for the one-dimensional p- Laplacian,Acta Mathematica Sinica,(Chinese Series),49(2006),165-172.(In Chinese)
[39]R.K.Nagle,K.L.Pothoven,On a third order nonlinear boundary value problem at resonance.J.Math.Anal.Appl.195(1995),149-159.
[40]H.Pang,H.Lian,W.Ge,Multiple positive solutions for second-order four-point boundary value problem.Appl.Math.lett.21(2008),656-661 .
[41]H.Pang,H.Lian,W Ge,Existence and monotone iteration of positive solutions for a three-point boundary value problem,Appl.Math.Comput.188(2008),1234-1243.
[42]C.Pang,Z.Wei,The exact number of positive solutions for a class of differential equations invoviong p-laplacian operator,Acta Mathematica Sinica,48(2005),1203-1208.(In Chinese)
[43]H.Su,Z.Wei,B.Wang,The existence of positive solutions for a nonlinear four- point singular boundary value problem with a p-Laplacian operator.Nonlinear.Anal.66(2007),2204-2217.
[44]H.Su,Z.Wei,X.Zhang and J.Liu,Positive solutions of n-order and m-order multi-point singular boundary value system.Appl.Math.Comput.188(2007),1234-1243.
[45]B.Sun,Y.Qu,W Ge,Existence and iteration of positive solutions for a multipoint one-dimensional p-Laplacian boundary value problem.Appl.Math.Comput.197(2008),389-398.
[46]B.Sun,W.Ge,Existence and iteration of positive solutions to a class of Sturm - Liouville-like p-Laplacian boundary value problems.Nonlinear.Anal.69(2008),1454-1461 .
[47] B. Sun, W. Ge, Existence and iteration of positive solutions for some p-Laplacian boundary value problems, Nonlinear Anal. 67(2007), 1820-1830.
[48] U.Janfalk, On eertain problem concerning the p-Laplace opertor, Likping studies in sciences and technology, dissertations, 1993, vol.326.
[49] J. Wang, W. Gao, A singular boundary value problem for one-dimension p-Laplacian. J. Math. Anal. Appl. 201(1996), 851-866.
[50] Y. Wang, W. Ge, Positive solutions for multi-point boundary value problems with a one-dimensional p-Laplacian, Nonlinear Anal. 66(2007), 1246-1256.
[51] Y. Wang, W. Ge, Existence of triple positive solutions for multi-point boundary value problems with a one dimensional p-Laplacian. Comput. Math. Appl. 54(2007), 793-807.
[52] Y. Wang, W. Ge, Existence of multiple positive solutions for multipoint boundary value problems with a one-dimensional p-Laplacian, Nonlinear Anal. 67(2007),476-485.
[53] Y. Wang, W. Ge, Multiple positive solutions for multipoint boundary value problems with one-dimensional p-Laplacian. J. Math. Anal. Appl. 327(2007). 1381-1395.
[54] X. Yang, Positive solutions for the one-dimensional p-Lpalacian. Math. Comput. Modelling. 35(2002), 129-135.
[55] 钟承奎,范先令,陈文源,非线性泛函分析引论,兰州大学出版社,2002.