高阶非线性常微分方程边值问题的正解
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摘要
本文主要利用非线性泛函分析的方法研究高阶常微分方程正解的存在性.主要包括以下三个方面的内容:
第一章,研究了一类含参量非线性三阶三点微分方程边值问题的正解,并应用非线性泛函中的锥拉伸锥压缩不动点定理和vector field方法得到正解的存在性.最后讨论参量λ的取值范围得到多个正解的存在性.
第二章,研究了四阶常微分方程边值问题其中g(t)∈C((0,1),[0,+∞)),f(u,v)∈C([0,+∞)×(-∞,0),[0,+∞)),a≥0,b≥0,c∈0,d≥0且Δ=ac+ad+bc>0,并应用非线性泛函中的锥拉伸锥压缩不动点定理得到一个解和多个解的存在性结果.
第三章,研究了一类2n阶常微分方程边值问题正解的存在性,其中α_i,β_i,δ_i,γ_i≥0是常数,p_i=β_iδ_i+α_iδ_i+α_iγ_i>0,0≤i≤n-1
本文中,我们假设a(t):(0,1)→[0,∞)是连续函数,f:[0,1]×(0,+∞)×(-∞,0)×…×(-1)~((n-1))(0,+∞)→[0,+∞)是连续的,其中这里a(t)可以在t=0,1处奇异,f(t,v_1,v_2,…,v_n)可以在v_i=0,(i=1,2,…,n)处奇异.在条件M∫_0~1k_(n-1)(s,s)a(s)ds<(?)(定义见后)下应用非线性泛函中的锥拉伸锥压缩不动点定理得到两个解的结果.
In this paper,we study the higher order boundary value problem by using the knowledge of nonlinear functional analysis. This thesis includes the following three contents.
Chapter 1 We study the positive solution of a singular third-order three-piont boundary value problems with parameters .By means of the fixed point theorems on cones and vector field analysis,we obtain one positive solution and multiple positive solutions.
Chapter 2 We study the four order boundary value problem,including positive solution for a class of singular fourth-order nonlinear eigenvalue problems .g(t)∈C((0,1),[0,+∞)),f(u,v)∈C([0,+∞)×(-∞,0),[0,+∞)), a≥0, b≥0, c≥0, d≥0 andΔ= ac + ad + bc> 0,here g(t) is allowed to be singular at t = 0, t = 1.By means of the fixed point theorems on cones,we obtain one positive solution and multiple positive solutions.
Chapter 3 We study the hight order boundary value problem,the order is 2n. is considered,here a(t) is allowed to be singular at t=0,t=1,f(t,v_1,v_2,…,v_n) is allowed to be singular at v_i = 0(i = 1,2,…n).We can obtain existence of two positive solutions by Guo-KrasnoseVskii's fixed point theorem ,at the condition M∫_0~1 k_(n-1)(s,s)a(s)ds < (?).
第一章,研究了一类含参量非线性三阶三点微分方程边值问题的正解,并应用非线性泛函中的锥拉伸锥压缩不动点定理和vector field方法得到正解的存在性.最后讨论参量λ的取值范围得到多个正解的存在性.
第二章,研究了四阶常微分方程边值问题其中g(t)∈C((0,1),[0,+∞)),f(u,v)∈C([0,+∞)×(-∞,0),[0,+∞)),a≥0,b≥0,c∈0,d≥0且Δ=ac+ad+bc>0,并应用非线性泛函中的锥拉伸锥压缩不动点定理得到一个解和多个解的存在性结果.
第三章,研究了一类2n阶常微分方程边值问题正解的存在性,其中α_i,β_i,δ_i,γ_i≥0是常数,p_i=β_iδ_i+α_iδ_i+α_iγ_i>0,0≤i≤n-1
本文中,我们假设a(t):(0,1)→[0,∞)是连续函数,f:[0,1]×(0,+∞)×(-∞,0)×…×(-1)~((n-1))(0,+∞)→[0,+∞)是连续的,其中这里a(t)可以在t=0,1处奇异,f(t,v_1,v_2,…,v_n)可以在v_i=0,(i=1,2,…,n)处奇异.在条件M∫_0~1k_(n-1)(s,s)a(s)ds<(?)(定义见后)下应用非线性泛函中的锥拉伸锥压缩不动点定理得到两个解的结果.
In this paper,we study the higher order boundary value problem by using the knowledge of nonlinear functional analysis. This thesis includes the following three contents.
Chapter 1 We study the positive solution of a singular third-order three-piont boundary value problems with parameters .By means of the fixed point theorems on cones and vector field analysis,we obtain one positive solution and multiple positive solutions.
Chapter 2 We study the four order boundary value problem,including positive solution for a class of singular fourth-order nonlinear eigenvalue problems .g(t)∈C((0,1),[0,+∞)),f(u,v)∈C([0,+∞)×(-∞,0),[0,+∞)), a≥0, b≥0, c≥0, d≥0 andΔ= ac + ad + bc> 0,here g(t) is allowed to be singular at t = 0, t = 1.By means of the fixed point theorems on cones,we obtain one positive solution and multiple positive solutions.
Chapter 3 We study the hight order boundary value problem,the order is 2n. is considered,here a(t) is allowed to be singular at t=0,t=1,f(t,v_1,v_2,…,v_n) is allowed to be singular at v_i = 0(i = 1,2,…n).We can obtain existence of two positive solutions by Guo-KrasnoseVskii's fixed point theorem ,at the condition M∫_0~1 k_(n-1)(s,s)a(s)ds < (?).
引文
[1]. A.Palamides, G.Smyrlis, Positive solution to a singular third-point boundary value problem with an indefinitely signed Green's function, Nonlinear Anal[J],2008,68:2104-2118.
[2]. Y.sun, Positive Solution of singular third-order three-point boundary value problem[J],Math.Anal.Appl.2005,306:589-603.
[3].郭大钧,非线性泛函分析[M],济南:山东科技出版社,2003.
[4].郭大钧,孙经先,抽象空间常微分方程[M],济南:山东科技出版社,2003.
[5].周友明,四阶非线性特征值问题的正解[J],系统科学与数学,2004,24:433-442.
[6]. Yansheng Liu, Multiple Positive Solutions of Nonlinear Singular Boundary Value Problem for Fourth-Order Equations[J], Applied Mathematics Letters , 2004,17:747-757.
[7].康平,刘立山,二阶奇异微分方程边值问题正解的存在性[J]:数学物理学报,2008,28:073-080.
[8]. Nussbaum R D,Eigenvectors of Nonlinear Positive Operators and the Linear Krein-Rutman Theorem In Fixed Point Theory[M] ,New York: 1980,886:309-330.
[9]. Guo Dajun, Lakshmikantham V.Nonlinear Problems in Abstract Cones[M],NewYork:Academic Press. 1988.
[10]. Lishan Liu, Xingguang Zhang,Yonghong Wu.Existence of positive solution for singular higher-order differential equations[J].Nonlinear Anal.2008,68:3948-3961.
[11].刘建,张克梅,四阶奇异半正边值问题的正解[J].工程数学学报,2006,23(3):430-434.
[12].Guoqing Chai, Existence of positive solutions for fourth-order boundary value problem with variable parameters[J], Nonlinear Anal.66(2007)870-880.
[13].孙经先,刘衍胜,Multiple positive solution of singular third-order periodic boundary value problem[J],数学物理学报,2005,25B(1):81-88.
[14].马如云,吴红萍,一类四阶两点边值问题多个正解的存在性[J],数学物理学报,2002,22A(2):244-249.
[15]. Ma R, Positive solutions of a nonlinear three-point boundary value problem[J],Electron J Diff Eqns, 1999,34:1-8.
[16].韦忠礼,李秀珍,一类超线性四阶奇异边值问题的正解[J],工程数学学报,2005,22(1):84-88.
[17].姚庆六,一类非线性三阶三点边值问题的解和正解[J],数学杂志,2007,27(6):704-708.
[18].姚庆六,一类二阶三点非线性边值问题的正解存在性与多解性[J],数学学报,2002,45(6):1057-1064.
[19].孙永平,张新光,三阶三点边值问题无穷多个正解的存在性[J],工程数学学报,2004:21(4):661-664
[20].蒋达清,三阶非线性常微分方程正解的存在性[J],东北师范大学学报,1996(4):6-10.
[21].Cabada A, The method of lower and upper solution for second ,third,fourth and higher order boundary value problems[J], Math Anal Appl.1994,185:302-320.
[22].Li Shuhong, Positive solution of nonlinear singular third-order two-piont boundary value problem [J], Math Appl,2006,323:413-425.
[23].J.Henderson and H.wang, Positive solutions for nonlinear eigenvalue problem[J],Math Anal Appl,1997,208:252-259.
[24].冯育强,刘三阳,关于三阶边值问题解的存在性[J],应用数学,2003,16(3):108-111.
[25].姚庆六,三阶常微分方程的某些特征值问题的正解[J],数学物理学报,2003,20A(5):513-519.
[26].马巧珍,—类四阶半正边值问题的正解的存在性[J],工程数学学报,2002.19(3):133-136.
[27].吴红萍,马如云,一类四阶边值问题正解的存在性[J],应用泛函分析学报,2000.2(4):324-348.
[28].马如云,奇异二阶边值问题的正解[J],数学学报,1998,41(6):1225-1230.
[29].Liu Yansheng, multiple positive solutions of nonlinear singular boundary value problem for fourth order equations[J]. Applied Mathematics Letters, 2004,17(7).747-757.
[30].Guo Dajun, Lakshmikantham V, Liu Xinzhi.Nonlinear integral equations in abstract space[M], Kluwer Academic Publishers, 1987.
[2]. Y.sun, Positive Solution of singular third-order three-point boundary value problem[J],Math.Anal.Appl.2005,306:589-603.
[3].郭大钧,非线性泛函分析[M],济南:山东科技出版社,2003.
[4].郭大钧,孙经先,抽象空间常微分方程[M],济南:山东科技出版社,2003.
[5].周友明,四阶非线性特征值问题的正解[J],系统科学与数学,2004,24:433-442.
[6]. Yansheng Liu, Multiple Positive Solutions of Nonlinear Singular Boundary Value Problem for Fourth-Order Equations[J], Applied Mathematics Letters , 2004,17:747-757.
[7].康平,刘立山,二阶奇异微分方程边值问题正解的存在性[J]:数学物理学报,2008,28:073-080.
[8]. Nussbaum R D,Eigenvectors of Nonlinear Positive Operators and the Linear Krein-Rutman Theorem In Fixed Point Theory[M] ,New York: 1980,886:309-330.
[9]. Guo Dajun, Lakshmikantham V.Nonlinear Problems in Abstract Cones[M],NewYork:Academic Press. 1988.
[10]. Lishan Liu, Xingguang Zhang,Yonghong Wu.Existence of positive solution for singular higher-order differential equations[J].Nonlinear Anal.2008,68:3948-3961.
[11].刘建,张克梅,四阶奇异半正边值问题的正解[J].工程数学学报,2006,23(3):430-434.
[12].Guoqing Chai, Existence of positive solutions for fourth-order boundary value problem with variable parameters[J], Nonlinear Anal.66(2007)870-880.
[13].孙经先,刘衍胜,Multiple positive solution of singular third-order periodic boundary value problem[J],数学物理学报,2005,25B(1):81-88.
[14].马如云,吴红萍,一类四阶两点边值问题多个正解的存在性[J],数学物理学报,2002,22A(2):244-249.
[15]. Ma R, Positive solutions of a nonlinear three-point boundary value problem[J],Electron J Diff Eqns, 1999,34:1-8.
[16].韦忠礼,李秀珍,一类超线性四阶奇异边值问题的正解[J],工程数学学报,2005,22(1):84-88.
[17].姚庆六,一类非线性三阶三点边值问题的解和正解[J],数学杂志,2007,27(6):704-708.
[18].姚庆六,一类二阶三点非线性边值问题的正解存在性与多解性[J],数学学报,2002,45(6):1057-1064.
[19].孙永平,张新光,三阶三点边值问题无穷多个正解的存在性[J],工程数学学报,2004:21(4):661-664
[20].蒋达清,三阶非线性常微分方程正解的存在性[J],东北师范大学学报,1996(4):6-10.
[21].Cabada A, The method of lower and upper solution for second ,third,fourth and higher order boundary value problems[J], Math Anal Appl.1994,185:302-320.
[22].Li Shuhong, Positive solution of nonlinear singular third-order two-piont boundary value problem [J], Math Appl,2006,323:413-425.
[23].J.Henderson and H.wang, Positive solutions for nonlinear eigenvalue problem[J],Math Anal Appl,1997,208:252-259.
[24].冯育强,刘三阳,关于三阶边值问题解的存在性[J],应用数学,2003,16(3):108-111.
[25].姚庆六,三阶常微分方程的某些特征值问题的正解[J],数学物理学报,2003,20A(5):513-519.
[26].马巧珍,—类四阶半正边值问题的正解的存在性[J],工程数学学报,2002.19(3):133-136.
[27].吴红萍,马如云,一类四阶边值问题正解的存在性[J],应用泛函分析学报,2000.2(4):324-348.
[28].马如云,奇异二阶边值问题的正解[J],数学学报,1998,41(6):1225-1230.
[29].Liu Yansheng, multiple positive solutions of nonlinear singular boundary value problem for fourth order equations[J]. Applied Mathematics Letters, 2004,17(7).747-757.
[30].Guo Dajun, Lakshmikantham V, Liu Xinzhi.Nonlinear integral equations in abstract space[M], Kluwer Academic Publishers, 1987.