若干非线性算子方程的解及其应用

Nonlinear operator theory is an important theoretical foundation and basic tool of nonlinear sciences, and it is a research field of modern mathematics which has profound theories and extensive applications.The main contents of this project are: (1) the existence and the uniqueness of positive solutions or solutions for several classes of nonlinear operator equations which include Ax+Bx=x, A(x,x)+Bx=x,(A(x,y),B(x,y))=(x,y),AxBx+Cx=x, where A,B,C are general nonlinear operators in ordered Banach spaces;(2) some applications of nonlinear operator theory to nonlinear integral equations and differential equation boundary value problems. In this project, we first intend to transform the sum of operators into one operator,clarify a partial ordering in product spaces and establish the definition of algebra cones in Banach albegras; second, we intend to study the addition between nonlinear operators, operators which include the multiplication between elements of spaces,and then we will obtain the existence and the uniqueness of positive solutions or solutions of nonlinear operator eqations which include operator addition and eletent multiplication; finally, we intend to apply the nonliear operator theory obtained to some nonlinear integral equations and differential equation boundary value problems. The theory research of this project has great challenge and the research methods have much creativity.

非线性算子理论是非线性科学的重要理论基础与基本工具，是现代数学中一个既有深刻理论意义，又有广泛应用价值的研究方向。本项目主要研究内容有：（1）几类非线性算子方程解与正解的存在性和唯一性：算子方程主要包括Ax+Bx=x, A(x,x)+Bx=x, (A(x,y),B(x,y))=(x,y), AxBx+Cx=x等，其中A,B,C为序Banach空间中具有广泛意义的非线性算子;（2）非线性算子理论在某些积分方程和微分方程边值问题中的应用。本项目拟把和算子转化为单个算子，明确叉积空间的半序，明确Banach代数中乘锥的定义；进而考察非线性算子加法和含有元素乘法的算子的性质，给出含有算子加法和元素乘法的非线性算子方程解与正解的存在性与唯一性；将所得到的非线性算子理论应用于讨论一些非线性积分方程和微分方程边值问题。本项目的理论研究具有很大的挑战性，研究方法也具有较大的创新性。

本项目研究了半序空间中的非线性算子理论以及在非线性微积分方程中的应用，主要研究了带有算子之和的算子方程、研究了半序积空间中的算子方程以及半序Banach代数中的算子方程，获得了算子方程正解存在唯一的结论以及正解与参数的关系。这些算子方程的结论很好地应用到整数阶、分数阶微分方程边值问题，泛函微分方程、积分方程中，给出它们正解的存在唯一性结论以及正解与参数的关系。这些算子的研究很困难，它们的结论在文献中很少，因此所获结论很好补充完善了这方面的理论。所得到的微分方程边值问题的这些结论和所用方法在国内外同行中处于领先位置。本项目共发表论文22篇论文，出版1部专著，其中SCI论文16篇。项目主持人晋升为教授，成员杨晨晋升为副教授。培养研究生15名，毕业10名。期间获得山西省自然科学基金1项，获山西省自然科学奖二等奖1项。