偏序空间中非线性映射及对微分方程的应用

The project is devoted to the following five aspects: (1) In the partially ordered Banach spaces，in disordered upper and lower solutions，reverse upper and lower solutions,or parallel upper and lower solutions premise, we discuss the conditions for the existence of the solution to the nonlinear operator equations and the conditions for existence of the fixed point to the nonlinear mappings. (2) In partially ordered Banach spaces, we study the solvability and multiple solution of the equation for non-cone operator, study the existence and number of fixed points for the non-cone mapping. For some non-cone mapping we calculate their topological degree or fixed point index. (3) For some nonlinear operator which does not have order monotonicity, we through the decomposition and transformation, so that it has a certain order monotone property(this is known as a "quasi monotone change" method), and then use order method to research. (4) We define several kind of nonlinear operator which rely on the partial order in Riesz space, the definition be similar to in partially ordered Banach space. We study the fixed points of the mappings or solution of the operator equations; If some problem is more difficult in partially ordered Banach spaces, we will be the first in Banach lattice to study them, and then extend to partially ordered Banach space. (5) Above abstract results are applied to integer order or fractional order integral equation, integer order or fractional order differential equation with impulses,matrix equation,elliptic partial differential equation, we will get impulsive solution, unbounded solution, sign-changing solution, multiple solution, structure of the solution set for the equations.

研究下述几类问题：(1)在偏序Banach空间中无序上下解、反向上下解或平行上下解前提下，探讨非线性算子方程解的存在与非线性映射不动点存在的条件.(2)研究偏序Banach空间中非锥算子方程的可解性和多解性、非锥映射不动点的存在性与个数，并对某些非锥映射进行拓扑度计算或不动点指数计算.(3)把若干不具有序单调性的非线性算子通过分解、转化使之具有一定的序单调性质（这称为“拟单调化”方法），然后用序方法加以研究.(4)将由序定义的几类非线性映射扩充定义在Riesz空间上，研究这些映射的不动点、算子方程的解；对偏序Banach空间中较难研究的问题首先在Banach格上解决，然后推广到一般偏序Banach空间上.(5)把上述各类研究结果应用到整数阶或分数阶积分方程、整数阶或分数阶含有脉冲项的常微分方程、矩阵方程、椭圆型偏微分方程类，得到脉冲解、无界解、变号解、多解性、解集的结构等方面的新结果

本项目是非线性泛函分析领域内的基础理论及其应用研究。虽然具有拓扑结构、代数结构、半序结构的抽象空间中的非线性算子已有很多成熟的成果，但远不能满足当前非线性问题研究的需要。于是半序空间中非线性映射性质、非线性算子方程解的研究及其对各类非线性方程的应用，是非线性泛函分析研究的重要方面。. 本课题对半序空间中的非线性映射、非线性算子方程的某些结果有不同程度的推广或者改进。把锥理论与临界点理论、拓扑度理论、不动点理论结合研究，把半序方法与变分方法、拓扑方法、解析方法相结合，对于若干类整数阶或分数阶微分方程、带有脉冲项的整数阶或分数阶微分方程、椭圆型偏微分方程等的已知结果，有不同程度的推广或者改进。对于不同类型的问题，得到所述方程正解、变号解、脉冲解、无界解的存在性、解的唯一性，多解性;解集的全局结构等方面的新结果。. 本项目在研期间，在提高研究能力、增加研究成果的基础上，项目组成员有1人主持申请到国家自然科学基金(青年)项目，有2人申请到山东省自然科学基金(面上)项目，有1人由副教授晋升为教授，有2人由讲师晋升为副教授。项目组成员在国内外学术刊物发表论文51篇，其中SCI检索42篇，其中ESI高被引论文9篇，成果获得山东省自然科学三等奖。人才培养方面，1名合作博士后合格出站，指导2名博士研究生与50名硕士研究生获得学位，其中有3名硕士生获得山东省优秀硕士论文奖。