Functional Equations and Stability Theory

Functional Equations and Stability Theory: Methods and Applications in Science and Engineering presents current advancements in the theory of functional equations and their stability, with a particular emphasis on their applications across science, engineering, and computational domains. Functional equations—long-standing pillars of mathematical analysis—have experienced a transformative evolution through the lens of Hyers–Ulam stability theory and its generalizations, which now intersect with key disciplines including artificial intelligence, cognitive science, and cybersecurity. This book provides readers with both principles of classical theory along with contemporary applications, thereby advancing the discourse on functional equations beyond its traditional confines. The book provides readers with both foundational studies and emerging methodologies, exploring how stability properties—such as Ulam–Hyers and generalized stability—are not only central to pure mathematics but also critically relevant to the modeling of real-world systems. By presenting methods that span theoretical results, fixed point methods, approximation techniques, and interdisciplinary case studies, this book offers a comprehensive perspective for readers interested in both mathematical rigor and real-world impact.

The book features chapters on advanced topics including classical and generalized stability of Cauchy, Jensen, Drygas, and Pexider-type equations; functional equations in restricted domains and abstract metric spaces; fixed point techniques for analyzing nonlinear and operatorial equations; stability in functional inequalities and mixed-type systems; applications to cognitive modeling, behavioral dynamics, and learning systems; functional frameworks in information theory, entropy, and communication systems; operator-valued and matrix functional equations; stability-informed modeling in cybersecurity, robotics, and intelligent environments; and, approximation and numerical approaches to stability problems. As such, the authors address both computational and mathematical dimensions, empowering readers to apply functional equations and stability theory to contemporary scientific problems.