Fixed Point Theory from Early Foundations to Contemporary Challenges

This book demonstrates the significance, applicability, and widespread nature of fixed point theorems in contexts outside of mathematics, including engineering, computer science, economics, and biological sciences. In the real world, fixed point theory is used to solve problems where stability, balance, or repeated processes are involved, such as predicting economic equilibrium, optimizing traffic flow, ensuring robots move precisely, or stabilizing medical devices like pacemakers. It is also used in artificial intelligence, engineering systems, and biological modeling where stable solutions in complex systems are needed. The authors not only highlight modern hurdles, but also explore how the field has accommodated and grown in response to them. The book provides comprehensive coverage of both the classical underpinnings and the cutting-edge advancements in fixed point theory. Each concept is illustrated with well-crafted, original examples that are carefully chosen to demonstrate both theoretical depth and practical significance. The chapters conclude with open problems and future research directions, encouraging further exploration. This book is designed to serve as both a guide for those entering the field as well as a resource for seasoned researchers looking to deepen their understanding of its modern applications.

Introductory Perspectives on Fixed Points.- Generalized Spaces: A Metric Evolution.- Renowned Theorems in Fixed Point Theory: Topological and Discrete Perspectives.- Foundational Extensions of the Contraction Mapping.- Control Process.- Equilibrium Points, Dynamics and Synchronization of Neural Networks.- Modeling and Simulation.- Medical Image Processing.

Sumati Kumari Panda, Ph.D., is a Professor of Mathematics at the GMR Institute of Technology, India.  Her research areas include fractional calculus, fixed point theory, neural networks, and their applications. She has published more than 100 research papers in reputed international journals and presented her work at several national and international conferences. She is currently serving as an Academic Editor for Scientific Reports (Springer, Scopus & SCIE-indexed). Dr. Panda received her Ph.D. in Mathematics from K.L. University in 2015.

Velusamy Vijayakumar, Ph.D., is an Assistant Professor at the Vellore Institute of Technology (VIT), Vellore, India. His research interests include fractional calculus, dynamical systems, mathematical control theory, and neural networks. Dr. Vijayakumar has authored over 220 research articles in reputed scientific journals.  Dr. Vijayakumar received his B.Sc, M.Sc, M.Phil, and Ph.D. degrees in Mathematics from Bharathiar University, Coimbatore, Tamil Nadu, India, in 2002, 2004, 2006, and 2016 respectively.

Ravi P. Agarwal, Ph.D., is an Emeritus Research Professor in the Department of Mathematics and Systems Engineering at the Florida Institute of Technology (USA). He has authored or co-authored more than 50 books and more than 2,000 research articles. He has received numerus honors and awards from several universities of the world. His research interests include nonlinear analysis, differential and difference equations, fixed point theory, and general inequalities. Dr. Agarwal received his Ph.D. at the Indian Institute of Technology, Madras, India, in 1973.