Spectral properties and solutions for parameter dependent differential equations

拟议的研究是在非线性分析和微分方程领域。重点是应用谱理论研究含参数的非线性边值问题。结果的存在性，唯一性，多重性，积极性，节点属性，分歧和稳定性的解决方案是直接在相对于频谱的点（参数）的位置。 具有非线性或非局部边界条件的问题不太经典，因此处理它们的方法还没有完全发展。除了现有的非线性分析方法，新的方法和工具预计将被开发。特别是通过Matlab或其他编程语言进行数值分析，丰富了理论结果。 参数相关差分或微分方程在模拟真实的世界现象中有着广泛的应用。作为一个跨学科的项目，研究成果可应用于种群生物学、化学、经济学、通信网络、最优化、工程和力学等领域的一些数学模型的研究。

The proposed research is in the areas of Nonlinear Analysis and Differential Equations. The emphasis is on the investigation of parameter dependent nonlinear boundary value problems by applying spectral theories. Results on existence, uniqueness, multiplicity, positivity, nodal properties, bifurcation and stability of solutions are to be stated directly in terms of the location of the point (the parameters) relative to the spectra. Problems with nonlinear or nonlocal boundary conditions are less classical and therefore the methods to approach them are not completely developed. In addition to the existing methodologies in Nonlinear Analysis, new methods and tools are expected to be developed. In particular, the theoretical results will be enriched by numerical analysis through Matlab or other programming languages. Parameter dependent difference or differential equations have wide applications in modeling real world phenomena. As an interdisciplinary project, results from the research can be applied to the study of some mathematical models from population biology, chemistry, economics, communication networks, optimization, engineering and mechanics.