三类非线性分数阶微分方程边值问题正解的存在性研究
发布时间：2018-08-30 09:50
【摘要】：近半个世纪以来由于分数阶微积分理论的发展和广泛应用,分数阶微分方程的理论研究得到迅猛发展,分数阶微分方程的一些模型也被广泛的应用在具体的科学领域中,例如于经济、化学、生物、物理、医学等学科.因此,在具体的学术研究中一个可解的分数阶微分方程模型能在现代社会中产生巨大的影响.本文对三类非线性分数阶微分方程边值问题正解的存在性进行了研究.第一类研究的是分数阶微分方程多点边值问题迭代正解的存在性.本文运用迭代技巧和0u正算子研究了下列多点边值问题正解的存在唯一性:得到的结论是如果函数f(t,u(t))满足Lipschitz条件,并且在Lipschitz常数满足一定条件下,就可以得到正解的存在性和唯一性.第二类是对以下带有p-Laplacian算子的分数阶微分方程多点边值问题正解的存在性和不存在性进行了研究,本文讨论了参数l的取值范围,通过运用Guo-Krasnoselskii不动点定理得到正解存在性和不存在性的充分条件.第三类研究的是以下带有p-Laplacian算子的分数阶奇异微分方程积分边值问题正解的存在性:文中通过运用上下解的方法和Schauder不动点定理,通过证明修正后微分方程边值问题存在正解,规避了方程的奇异性,得到了当参数l在特定范围时正解的存在性.
[Abstract]:In the last half century, with the development and wide application of fractional calculus theory, the theoretical research of fractional differential equations has been developed rapidly, and some models of fractional differential equations have also been widely used in specific scientific fields. Such as economics, chemistry, biology, physics, medicine and so on. Therefore, a solvable fractional differential equation model can have a great influence in modern society. In this paper, the existence of positive solutions for boundary value problems of three nonlinear fractional differential equations is studied. The first is the existence of iterative positive solutions for multipoint boundary value problems of fractional differential equations. In this paper, the existence and uniqueness of the positive solution of the following multipoint boundary value problems are studied by means of iterative technique and 0u positive operator: the conclusion is obtained that if the function f (t u (t) satisfies the Lipschitz condition and the Lipschitz constant satisfies some conditions), The existence and uniqueness of positive solution can be obtained. The second is to study the existence and nonexistence of positive solutions for multipoint boundary value problems of fractional differential equations with p-Laplacian operator. In this paper, the range of parameter l is discussed. By using Guo-Krasnoselskii fixed point theorem, sufficient conditions for the existence and non-existence of positive solutions are obtained. In the third category, the existence of positive solutions for integral boundary value problems of fractional singular differential equations with p-Laplacian operator is studied. By using the method of upper and lower solutions and Schauder fixed point theorem, the existence of positive solutions to the boundary value problems of modified differential equations is proved. The singularity of the equation is avoided and the existence of positive solution is obtained when the parameter l is in a specific range.
【学位授予单位】：华北电力大学(北京)
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.8
本文编号：2212717
[Abstract]:In the last half century, with the development and wide application of fractional calculus theory, the theoretical research of fractional differential equations has been developed rapidly, and some models of fractional differential equations have also been widely used in specific scientific fields. Such as economics, chemistry, biology, physics, medicine and so on. Therefore, a solvable fractional differential equation model can have a great influence in modern society. In this paper, the existence of positive solutions for boundary value problems of three nonlinear fractional differential equations is studied. The first is the existence of iterative positive solutions for multipoint boundary value problems of fractional differential equations. In this paper, the existence and uniqueness of the positive solution of the following multipoint boundary value problems are studied by means of iterative technique and 0u positive operator: the conclusion is obtained that if the function f (t u (t) satisfies the Lipschitz condition and the Lipschitz constant satisfies some conditions), The existence and uniqueness of positive solution can be obtained. The second is to study the existence and nonexistence of positive solutions for multipoint boundary value problems of fractional differential equations with p-Laplacian operator. In this paper, the range of parameter l is discussed. By using Guo-Krasnoselskii fixed point theorem, sufficient conditions for the existence and non-existence of positive solutions are obtained. In the third category, the existence of positive solutions for integral boundary value problems of fractional singular differential equations with p-Laplacian operator is studied. By using the method of upper and lower solutions and Schauder fixed point theorem, the existence of positive solutions to the boundary value problems of modified differential equations is proved. The singularity of the equation is avoided and the existence of positive solution is obtained when the parameter l is in a specific range.
【学位授予单位】：华北电力大学(北京)
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.8
【参考文献】
相关期刊论文 前3条
1 张金陵;张福珍;;一类分数阶奇异微分方程边值问题正解的存在性[J];常熟理工学院学报;2011年08期
2 张淑琴;;EXISTENCE OF SOLUTION FOR A BOUNDARY VALUE PROBLEM OF FRACTIONAL ORDER[J];Acta Mathematica Scientia;2006年02期
3 李翠哲,葛渭高;一维p-Laplacian奇异Sturm-Liouville边值问题的正解[J];应用数学;2002年03期
相关硕士学位论文 前1条
1 于瑶;非线性分数阶微分方程边值问题的多重正解[D];东北师范大学;2009年
，本文编号：2212717
本文链接：https://www.wllwen.com/kejilunwen/yysx/2212717.html
