Fixed Point Theory, Nonlinear Differential Equations and Computational Algorithms on Data Analytics

拟议的研究与应用数学和计算中的三个密切相关的领域重叠：不动点理论、非线性微分方程和数据分析的计算算法。每个主题本身都是重要和实质性的。将这三个领域(不动点理论、非线性微分方程和神经网络)沿着从理论到实际应用的连续体连接在一起是创新的，也是我研究的中心重点。 不动点理论作为研究微分方程的一种非常强大和重要的工具，在生物学、化学、经济学、工程学、博弈论、物理学和计算机科学等领域有着广泛的应用。这项研究的重点在于发展和应用不动点理论中的新的数学方法来研究来自种群动力学、反应扩散系统、图像分类、神经网络和社会活动的非线性微分方程解。特别是，涉及具有非局部边界或初始条件的多参数的微分方程组将被考虑。在应用中，这类微分系统代表气体扩散、热传导、弹性光束和一般的反馈控制，通过反馈控制，过程沿其演化的状态之和等于来自物理、化学或生物的真实过程中的初始状态。 除了泛函分析、非线性分析和线性算子理论中的拓扑方法外，还将应用迭代、单调映射、上下解等构造技术。在计算方面，将使用真实世界的数据集开发、实施和测试新的算法。将对系统性能进行评估和记录。将采用包括数值方法和计算机模拟在内的计算方法。还将开发新的工具和方法。 这些结果将在非线性分析、动力系统、数学建模和数据分析的计算算法等多个领域做出贡献。特别是最后一个领域，最近受到了相当大的关注。我们的工作将继续为我们加拿大研究界的福祉做出贡献，无论是在基础方面还是在应用方面。

The proposed research overlaps with three closely related areas in applied mathematics and computing: fixed point theory, nonlinear differential equations and computational algorithms for data analytics. Each topic on its own is important and substantial. Connecting the three areas together (fixed-point theory, nonlinear differential equations, and neural networks) along a continuum from theory to real-world applications is innovative and the central focus of my research. As a very powerful and important tool in studying differential equations, fixed point theory has diverse applications in various fields such as Biology, Chemistry, Economics, Engineering, Game Theory, Physics, and Computer Sciences. The research lies in the development and application of new mathematical techniques in fixed point theory to study solutions for nonlinear differential equations arising from population dynamics, reaction-diffusion systems, image classifications, neural networks and social activities. In particular, systems of differential equations involving multiple parameters with non-local boundary or initial conditions will be considered. In applications, this class of differential systems represents gas diffusion, heat conduction, elastic beam and in general feedback controls by which the “sum” of the states of the process along its evolution equals the initial state in real processes from Physics, Chemistry or Biology. In addition to topological methods from functional analysis, nonlinear analysis and linear operator theory, constructional techniques such as iteration, monotone mappings, upper and lower solutions will also be applied. On the computational side, new algorithms will be developed, implemented and tested using real-world data sets. System performance will be evaluated and documented. Computational approaches including numerical methods and computer simulations will be applied. New tools and methods will also be developed. The results will contribute across several fields in nonlinear analysis, dynamical systems, mathematical modeling and computational algorithms for data analytics. The last field, in particular, has received considerable attention of late. Our work will continue to contribute to the well-being of our Canadian research community, both in fundamental and applied terms.