非线性算子正解与数值解及其应用

The nonlinear operator theory play a very important role to nonlinear science and has been widely used by biochemistry and engineering technology and so on. In this project, a class of nonlinear operator equations with a parameter and a perturbation which can be wide applied to various kinds of differential equations and integral equations will be investigated. On the one hand, fixed-point theory, topological degree theory, partial ordering method, variational method will be used to discuss the existence, uniqueness and multiplicity of positive solutions of the operator equations with (without) a perturbation and (or) a parameter. The relationship between the parameter and the solution, the perturbation and the main operator, the perturbation and the solution generated by itself, conditions for existence of positive solutions of two different operator equations with different properties will be discovered, respectively. On the other hand, based on classical methods such as Galerkin method, Adomain Decomposition methods, some methods for finding numerical solutions of the operator equations under consideration will be developed. Applications to some differential equations, integral equations and p-Laplacian problems on time scales will be provided to show the significance of the research works. It will be a good reference to study nonlinear science further.

非线性算子理论是非线性分析的重要工具，具有重要的理论意义与广泛的应用价值。本项目利用不动点理论、拓扑度、半序方法、变分方法等方法对一类带有参数以及扰动的和算子及其特殊形式的正解存在、唯一性、扰动生成的解、参数"分歧点"及位置等问题进行研究，揭示扰动算子以及参数对解的影响、扰动与扰动所生成解的关系、主算子与扰动算子为同类和不同类情形时正解存在性条件之间的关系；另一方面结合经典的Galerkin,Adomain Decomposition 等方法得到所关注算子的数值解方法；同时将研究成果应用于微分方程与积分方程，特别是时标上p-Laplacian算子问题的次调和解、同宿轨、异宿轨等问题。为进一步探讨非线性分析理论与应用提供有益的参考.

非线性算子理论是非线性分析的重要工具，具有重要的理论意义与广泛的应用价值。本项目利用不动点理论、Morse理论、拓扑度、半序方法、变分方法等方法对一类带有参数以及扰动的和算子及其特殊形式的正解存在、唯一性、扰动生成的解、参数“分歧点”及位置等问题进行研究，揭示扰动算子以及参数对解的影响、扰动与扰动所生成解的关系、主算子与扰动算子为同类和不同类情形时正解存在性条件之间的关系；另一方面结合经典的Galerkin,Adomain Decomposition 等方法得到所关注算子的数值解方法；同时将研究成果应用于微分方程与积分方程，特别是时标上p-Laplacian算子问题的次调和解、同宿轨、异宿轨等问题。为进一步探讨非线性分析理论与应用提供有益的参考，也为工程、生化等实际问题研究提供重要的理论支撑。