几类微分方程边值问题正解的存在性研究
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摘要
近年来，在数学、物理学、化学、生物学、医学、经济学、工程学、控制理论等许多科学领域中出现了各种各样的非线性问题，在解决这些非线性问题的过程中，逐渐形成了现代分析学中一个非常重要的分支-非线性泛函分析．它主要包括半序方法、拓扑方法和变分方法等内容，为当今科技领域中层出不穷的非线性问题提供了富有成效的理论工具，尤其是在处理应用学科中提出的各种非线性微分方程问题中发挥着重要的作用．1912年L．E．J．Brouwer对有限维空间建立了拓扑度的概念，1934年J．Leray和J．Schauder将这一概念推广到Banach空间的全连续场，后来E．Rothe，M．A．Krasnosel'skii，P．H．Rabinowitz，H．Amann，K．Deimling等对拓扑度理论、锥理论及其应用进行了深入的研究，国内张恭庆教授、郭大钧教授、陈文源教授、定光桂教授、孙经先教授等在非线性泛函分析的许多领域都取得了非常出色的成就(这方面的内容参见[1-12])．
奇异常微分方程是微分方程领域中一个重要的研究课题，由于它不断出现在各种应用科学中，例如：核物理、气体动力学、流体力学、边界层理论、非线性光学等，所以得到了广泛而深入的研究(见[16,46-49]及其中的参考文献)．
本文主要利用非线性泛函分析的拓扑度理论、锥理论和上下解方法等研究了几类非线性(奇异)常微分方程边值问题正解的存在性、多解等．主要内容如下：
第一章给出了后面几章要用到的关于不动点存在及不动点指数计算的几个引理，这些引理在本文主要结果的证明中是至关重要的．
第二章考虑了下述带有两个变参数的四阶微分方程边值问题其中A(t)，B(t)∈C[0，1],ξ_i∈(0，1)，a_i，b_i∈[0，+∞)，i=1，2，…，m-2为给定的常数．我们利用不动点指数理论得到了至少有一个和两个正解存在的充分条件．
第三章研究了奇异半正(n，p)特征值问题的正解．
第一节应用锥上的不动点指数定理，研究了下述(n，p)奇异边值问题正解的存在性其中n≥2，1≤p≤n-1固定，λ>0为一个常数，q∈L~1(0，1)，q≥0，a．e．f：[0，1]×R~+→R连续．
第二节去掉了第一节对非线性项中f连续的假设，用相对较弱的Caratheodory条件代替，同时去掉了f下方有界的严格限制，依然得到了上述问题正解及多解的存在性结果．
第四章研究奇异半正(k，n-k)共轭m-点边值问题其中，a_i∈[0，∞)，i=1，2，…，m-2，∑_(i=1)~(m-2)a_i>0，0<ξ_1<ξ_2<…<ξ_(m-2)<1为常数，m≥3，f：(0，1)×[0，∞)→[0，+∞)连续，p：(0，1)→(-∞，+∞)Lebesgue可积．我们利用不动点指数理论得到了上述问题正解存在的充分条件．
第五章研究非线性项变号的奇异高阶m-点边值问题其中a_i∈[0，∞)，i=1，2，…，m-2，∑_(i=1)~(m-2) a_i>0，0<ξ_1<ξ_2<…<ξ_(m-2)<1为常数，m≥3，f：(0，1)×(0，∞)→(-∞，+∞)连续．本章给出了上述问题的Green函数的一种简单表达式，并讨论了其性质，进一步利用不动点指数理论得到了其正解的存在性．
第六章研究了其中ψ_p(t)=|t|~(p-2)t，p>1，0<ξ<η<1为常数，α和β在[ξ，η)上右连续，在t=η左连续，在[ξ，η]上非减，α(ξ)=β(ξ)=0；∫_ξ~ηx(τ)dα(τ)与∫_ξ~ηψ_p(x"(τ))dβ(τ)分别表示x和ψ_p(x")关于α和β的Riemann-Stieltjes积分，且0<∫_ξ~ηdα(τ)<1，0<∫_ξ~ηdβ(τ)<1，利用上下解方法得到了上述问题伪C~3[0，1]正解和C~2[0，1]正解存在的充分必要条件．
In later years, all sorts of nonlinear problems have resulted from mathematics,physics, chemistry, biology, medicine, economics, engineering, cybernetics and so on. During the development of solving such problems, nonlinear functionalanalysis has been bing one of the most important research fields in modern mathematics. It mainly includes partial ordering method, topological degree method and the variational method. Also it provides a much effect theoretical tool for solving many nonlinear problems in the fields of the science and technology.And what is more, it is an important approach for studying nonlinear differential equations arising from many applied mathematics. L. E. J. Brouwer had established the conception of topological degree for finite dimensional space in 1912. J. Leray and J. Schauder had extend the conception to completely continuousfield of Banach space in 1934, afterward E. Rothe, M. A. Krasnosel'skii, P. H. Rabinowitz, H. Amann, K. Deimling had carried on embedded research on topological degree and cone theory. Many well known mathematicians in China, say Zhang Gongqing, Guo Dajun, Chen Wenyuan, Ding Guanggui, Sun Jingxianetc., had proud works in various fields of nonlinear functional analysis. (See [1-12]).
The singular ordinary differential equation is an important aspect of differentialequation, it arise in the fields of gas dynamics, newtonian fluid mechanics, nuclear physics, the theory of boundary layer, nonlinear optics and so on. Theorefore,it has been considered extensively(See [16,46-49] and reference therein).
The present paper mainly investigates existence of positive solutions, multiplicityfor several classes of ordinary differential equations as well as singular differential equation boundary value problem by using topological degree, cone theory and lower and upper solution method. The main contents are as follows:
Chapter 1 gives serval lemmas on existence of fixed point and computation of fixed point index, which play an important role in next chapters.
Chapter 2 considers the positive solution of fourth-order differential equation boundary value problem withe two varible parameterwhere A(t),B(t)∈C[0,1],ξ_i∈(0,1),a_i,b_i∈[0,+∞),i=1,2,…,m-2are given constants. Using the fixed point index theory, we obtain the sufficient condition for the above BVP has at least one and two positive solution.
Chapter 3 considers the positive solution of singular semipositone (n,p) eigenvalue problem.
In section 1, using the fixed point index theorem on cones, we investigates the existence of positive solution for the following singular (n, p) boundary value problem.where n≥2,1≤p≤n-1 are fixed.λ>0 is a constant, q∈L~1(0，1)，q≥0，a．e．f：[0，1]×R~+→R is continuous.
In section 2, we replace the suppose of continuity on f by relative weaker Caratheodory condition, and removed the strict restriction of lower bounded on f. we still obtained the existence of positive solution and multiplicity for the above BVP.
Chapter 4 investigates singular semipositone (k, n - k) conjugate m-point boundary value problemwhere a_i∈[0,∞),i=1,2,…,m-2,∑_(i=1)~(m-2)a_i>0,0<ξ_1<ξ_2<…<ξ_(m-2)<1 are constants,m≥3,f：(0,1)×[0,∞)→[0，+∞)is continuous,p:(0,1)→(-∞,+∞) Lebesgue integrable. By using the fixed point indextheory, we obtain the sufficient condition of positive solution for the above BVP.
Chapter 5 investigates the following sigular higher-order m-point boundary value problemwhere a_i∈[0，∞)，i=1，2，…，m-2，∑_(i=1)~(m-2) a_i>0，0<ξ_1<ξ_2<…<ξ_(m-2)<1 are constants,m≥3，f：(0，1)×[0，∞)→[0，+∞)is continuous. we givea simpler expression of Green's function for the above problem, and discuss it's properties. then by using the fixed point index theory, we obtained the existence of positive solution for the above BVP.
Chapter 6 investigateswhereψ_p(t)=|t|~(p-2)t，p>1，0<ξ<η<1,are constants,αandβare rightcontinuous on [ξ,η), left continuous at t =η, and nondecreasing on [ξ,η], withα(ξ)=β(ξ)=0;∫_ξ~ηx(τ)dα(τ)and∫_ξ~ηψ_p(x"(τ))dβ(τ)denote the Riemann-Stieltjes integrals of x andψ_p(x") with respect toαandβ, respectively, and0<∫_ξ~ηdα(τ)<1 and 0<∫_ξ~ηdβ(τ)<1.By using the lower and upper solutionmethod, we obtain the sufficient and necessary condition of pseudoC~3[0,1] positive solution and C~2[0,1]positive solution.
奇异常微分方程是微分方程领域中一个重要的研究课题，由于它不断出现在各种应用科学中，例如：核物理、气体动力学、流体力学、边界层理论、非线性光学等，所以得到了广泛而深入的研究(见[16,46-49]及其中的参考文献)．
本文主要利用非线性泛函分析的拓扑度理论、锥理论和上下解方法等研究了几类非线性(奇异)常微分方程边值问题正解的存在性、多解等．主要内容如下：
第一章给出了后面几章要用到的关于不动点存在及不动点指数计算的几个引理，这些引理在本文主要结果的证明中是至关重要的．
第二章考虑了下述带有两个变参数的四阶微分方程边值问题其中A(t)，B(t)∈C[0，1],ξ_i∈(0，1)，a_i，b_i∈[0，+∞)，i=1，2，…，m-2为给定的常数．我们利用不动点指数理论得到了至少有一个和两个正解存在的充分条件．
第三章研究了奇异半正(n，p)特征值问题的正解．
第一节应用锥上的不动点指数定理，研究了下述(n，p)奇异边值问题正解的存在性其中n≥2，1≤p≤n-1固定，λ>0为一个常数，q∈L~1(0，1)，q≥0，a．e．f：[0，1]×R~+→R连续．
第二节去掉了第一节对非线性项中f连续的假设，用相对较弱的Caratheodory条件代替，同时去掉了f下方有界的严格限制，依然得到了上述问题正解及多解的存在性结果．
第四章研究奇异半正(k，n-k)共轭m-点边值问题其中，a_i∈[0，∞)，i=1，2，…，m-2，∑_(i=1)~(m-2)a_i>0，0<ξ_1<ξ_2<…<ξ_(m-2)<1为常数，m≥3，f：(0，1)×[0，∞)→[0，+∞)连续，p：(0，1)→(-∞，+∞)Lebesgue可积．我们利用不动点指数理论得到了上述问题正解存在的充分条件．
第五章研究非线性项变号的奇异高阶m-点边值问题其中a_i∈[0，∞)，i=1，2，…，m-2，∑_(i=1)~(m-2) a_i>0，0<ξ_1<ξ_2<…<ξ_(m-2)<1为常数，m≥3，f：(0，1)×(0，∞)→(-∞，+∞)连续．本章给出了上述问题的Green函数的一种简单表达式，并讨论了其性质，进一步利用不动点指数理论得到了其正解的存在性．
第六章研究了其中ψ_p(t)=|t|~(p-2)t，p>1，0<ξ<η<1为常数，α和β在[ξ，η)上右连续，在t=η左连续，在[ξ，η]上非减，α(ξ)=β(ξ)=0；∫_ξ~ηx(τ)dα(τ)与∫_ξ~ηψ_p(x"(τ))dβ(τ)分别表示x和ψ_p(x")关于α和β的Riemann-Stieltjes积分，且0<∫_ξ~ηdα(τ)<1，0<∫_ξ~ηdβ(τ)<1，利用上下解方法得到了上述问题伪C~3[0，1]正解和C~2[0，1]正解存在的充分必要条件．
In later years, all sorts of nonlinear problems have resulted from mathematics,physics, chemistry, biology, medicine, economics, engineering, cybernetics and so on. During the development of solving such problems, nonlinear functionalanalysis has been bing one of the most important research fields in modern mathematics. It mainly includes partial ordering method, topological degree method and the variational method. Also it provides a much effect theoretical tool for solving many nonlinear problems in the fields of the science and technology.And what is more, it is an important approach for studying nonlinear differential equations arising from many applied mathematics. L. E. J. Brouwer had established the conception of topological degree for finite dimensional space in 1912. J. Leray and J. Schauder had extend the conception to completely continuousfield of Banach space in 1934, afterward E. Rothe, M. A. Krasnosel'skii, P. H. Rabinowitz, H. Amann, K. Deimling had carried on embedded research on topological degree and cone theory. Many well known mathematicians in China, say Zhang Gongqing, Guo Dajun, Chen Wenyuan, Ding Guanggui, Sun Jingxianetc., had proud works in various fields of nonlinear functional analysis. (See [1-12]).
The singular ordinary differential equation is an important aspect of differentialequation, it arise in the fields of gas dynamics, newtonian fluid mechanics, nuclear physics, the theory of boundary layer, nonlinear optics and so on. Theorefore,it has been considered extensively(See [16,46-49] and reference therein).
The present paper mainly investigates existence of positive solutions, multiplicityfor several classes of ordinary differential equations as well as singular differential equation boundary value problem by using topological degree, cone theory and lower and upper solution method. The main contents are as follows:
Chapter 1 gives serval lemmas on existence of fixed point and computation of fixed point index, which play an important role in next chapters.
Chapter 2 considers the positive solution of fourth-order differential equation boundary value problem withe two varible parameterwhere A(t),B(t)∈C[0,1],ξ_i∈(0,1),a_i,b_i∈[0,+∞),i=1,2,…,m-2are given constants. Using the fixed point index theory, we obtain the sufficient condition for the above BVP has at least one and two positive solution.
Chapter 3 considers the positive solution of singular semipositone (n,p) eigenvalue problem.
In section 1, using the fixed point index theorem on cones, we investigates the existence of positive solution for the following singular (n, p) boundary value problem.where n≥2,1≤p≤n-1 are fixed.λ>0 is a constant, q∈L~1(0，1)，q≥0，a．e．f：[0，1]×R~+→R is continuous.
In section 2, we replace the suppose of continuity on f by relative weaker Caratheodory condition, and removed the strict restriction of lower bounded on f. we still obtained the existence of positive solution and multiplicity for the above BVP.
Chapter 4 investigates singular semipositone (k, n - k) conjugate m-point boundary value problemwhere a_i∈[0,∞),i=1,2,…,m-2,∑_(i=1)~(m-2)a_i>0,0<ξ_1<ξ_2<…<ξ_(m-2)<1 are constants,m≥3,f：(0,1)×[0,∞)→[0，+∞)is continuous,p:(0,1)→(-∞,+∞) Lebesgue integrable. By using the fixed point indextheory, we obtain the sufficient condition of positive solution for the above BVP.
Chapter 5 investigates the following sigular higher-order m-point boundary value problemwhere a_i∈[0，∞)，i=1，2，…，m-2，∑_(i=1)~(m-2) a_i>0，0<ξ_1<ξ_2<…<ξ_(m-2)<1 are constants,m≥3，f：(0，1)×[0，∞)→[0，+∞)is continuous. we givea simpler expression of Green's function for the above problem, and discuss it's properties. then by using the fixed point index theory, we obtained the existence of positive solution for the above BVP.
Chapter 6 investigateswhereψ_p(t)=|t|~(p-2)t，p>1，0<ξ<η<1,are constants,αandβare rightcontinuous on [ξ,η), left continuous at t =η, and nondecreasing on [ξ,η], withα(ξ)=β(ξ)=0;∫_ξ~ηx(τ)dα(τ)and∫_ξ~ηψ_p(x"(τ))dβ(τ)denote the Riemann-Stieltjes integrals of x andψ_p(x") with respect toαandβ, respectively, and0<∫_ξ~ηdα(τ)<1 and 0<∫_ξ~ηdβ(τ)<1.By using the lower and upper solutionmethod, we obtain the sufficient and necessary condition of pseudoC~3[0,1] positive solution and C~2[0,1]positive solution.
引文
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