几类分数阶方程边值问题解的定性研究

Fractional calculus is the theory of arbitrary order differential and integral, compared with integer order differential equations, fractional differential equations can describe some natural physical processes and dynamic system processes better. Combined with the theory of fractional calculus, this project is systematically devoted to study the following fractional equations by using the thought and method of nonlinear functional analysis. The main contents include three points: (1) We study the boundary value problems of sequential fractional differential equation(equations); (2) We study the boundary value problems of equation with left, right fractional intergral; (3) We study the boundary value problems of fractional differential equation on the infinite intervals. By the application of the fixed point theory, the cone theory, topological degree, monotone iterative and critical point theory, we get the existence, uniqueness and multiplicity of solutions for equations which are discussed above, especially the nonlinear term is singular, sign changing. We also discuss the relationship between the equation parameters and the number of solutions, the influence of the existence of solutions, at the same time, get the specific example analysis, so as to improve and develop the research in boundary value problems of fractional equations.

分数阶微积分是关于任意阶微分和积分的理论，同整数阶微分方程相比，分数阶微分方程能够更好的描述某些自然物理过程和动态系统过程。结合分数阶微积分理论，本项目致力于应用非线性泛函分析的思想方法研究如下几类分数阶方程，主要包括：(1) 研究序列分数阶微分方程(组)边值问题；(2) 研究含有左，右分数阶积分的方程边值问题；(3) 研究无穷区间上的分数阶微分方程边值问题。利用不动点定理、锥理论、拓扑度理论、单调迭代、临界点理论等方法得到上述方程解的存在性、唯一性、多重性，特别是非线性项奇异，变号的情况下，得到方程正解的存在性，并将讨论方程中参数与解的个数之间的关系，以及对方程解的存在性的影响，同时得到具体的实例分析，从而完善和发展分数阶方程边值问题的研究。

分数阶微积分是关于任意阶微分和积分的理论，同整数阶微分方程相比，分数阶微分方程能够更好的描述某些自然物理过程和动态系统过程。结合分数阶微积分理论，本项目致力于应用非线性泛函分析的思想方法研究如下几类分数阶方程，主要包括：(1) 研究分数阶微分方程(组)边值问题；(2) 研究含有左，右分数阶积分的方程和椭圆方程(组)问题；(3) 研究无穷区间上的分数阶微分方程边值问题。利用不动点定理、锥理论、拓扑度理论、单调迭代、临界点理论等方法得到上述方程解的存在性、唯一性、多重性，特别是非线性项奇异，变号的情况下，得到方程正解的存在性，并将讨论方程中参数与解的个数之间的关系，以及对方程解的存在性的影响，同时得到具体的实例分析，从而完善和发展方程边值问题的研究。