Research Projects

Fixed point theory has long been a cornerstone of nonlinear analysis and mathematical modeling, offering powerful tools for solving equations that arise in diverse scientific and engineering contexts. This research project aims to explore and extend fixed point results in generalized metric spaces including partial metric [24], b−metric [9,14], cone metric spaces [20] and some others (see [3{7,10,11,13,16,22,23,25,26,28,30,34]) with a strong focus on their theoretical development and practical applications in modern computational frameworks. The first objective of this study is to establish new fixed point theorems in generalized metric spaces under relaxed conditions that go beyond classical contraction mappings. These results will address structures relevant in computer science, such as domain theory, recursive function theory, and program semantics, where convergence behavior of functions and operators is central to algorithm design and verification (see [1,12,15, 17,19,27,31,33]). The second major focus is the application of fixed point theory to the existence and uniqueness of solutions to classical and fractional differential equations. By constructing appropriate function spaces and employing advanced fixed point techniques, the project will contribute to analytical solutions of nonlinear models frequently found in physics, biology, and engineering, including population dynamics, viscoelasticity, and control systems with memory effects. Thirdly, the project will incorporate fixed point logic within uncertainty frameworks such as fuzzy [35], intuitionistic fuzzy [8], and neutrosophic environments [32]. This extension will allow for the development of robust decision-making models that handle imprecise, incomplete, or contradictory data. Applications include knowledge-based systems, recommendation engines, and risk assessment models, particularly in contexts where classical binary logic fails to provide actionable insights. Finally, the project will bridge fixed point theory with machine learning and optimization algorithms, particularly in areas such as neural network stability, iterative learning processes, and convergence analysis of gradient-based methods. Fixed point formulations will be used to design more mathematically grounded algorithms that ensure reliability and convergence under generalized assumptions. This integrative approach spanning pure theory and applied computation will yield a suite of analytical tools, models, and algorithms with cross-disciplinary impact. The outcomes of the project are expected to contribute to academic research in nonlinear analysis while supporting innovations in emerging areas such as intelligent systems, data-driven modeling, and computational intelligence.