Advances in Functional Equations, Second Edition—Editorial

It is our privilege to present the concluding editorial for this Special Issue, Advances in Functional Equations, Second Edition. This collection continues the mission of the first edition by showcasing contemporary developments in functional equations and inequalities, demonstrating both their theoretical depth and expanding applications in the broader mathematical sciences.

1. Context and Motivation

Functional equations remain one of the fundamental pillars of mathematical analysis. Their history is quite old, with the most important milestone going back to Cauchy’s Cours d’analyse, as shown in the work of Bradley and Sandifer [1], where functional equations, known today as Cauchy’s functional equations appeared explicitly. For a deeper historical discussion, see Aczél and Dhombres [2]. At present, the study of functional equations spans from classical problems such as Cauchy- and Jensen-type equations to modern generalizations involving stability, functional inequalities, and operator-based extensions. Their intrinsic connection to symmetry—in variables, domains, and solution structures—positions them naturally within the scope of Symmetry.

At the same time, the field has increasingly aligned itself with practical challenges, including applications in many fields of modern applied mathematics, such as economy (Eichhorn [3]), utility and game theory (Candeal et al. [4]), and social and behavioral sciences (Aczél [5]). This Special Issue was published to capture these evolving directions while remaining rooted in the core questions that have driven the subject for more than a century.

2. Highlights of This Special Issue

The works published here present advances along several key research axes:

Stability theory and generalizations. The story of stability theory started almost a century ago with a question posed by Ulam, which was answered in 1941 by Hyers in [6]. Since that time, several refinements of Ulam–Hyers stability and alternative stability concepts have been explored, particularly in nonlinear and generalized environments. Several contributions push beyond classical Euclidean settings and study stability phenomena under weaker assumptions or additional structural constraints.

Functional inequalities and symmetry-driven conditions. New inequalities of analytic relevance are established, and solution sets are characterized using symmetry arguments. These developments support deeper understanding of extremal problems and parameter-dependent solution behavior.

Multivariable extensions and domain generality. A number of articles treat functional equations in higher-dimensional contexts, expanding the understanding of the uniqueness, controllability, and regularity of solutions in multiple variables.

Connections to differential, stochastic, and dynamic systems. An increasing number of contributions show how functional-equation techniques guide the analysis of systems driven by randomness, impulses, or time evolution—reflecting a trend toward greater interdisciplinarity.

Together, these publications clearly demonstrate the vitality of the field. They also validate the inspiration behind this Special Issue—that advances in functional equations often arise at the intersection of symmetry, structure, and application.

3. Gaps and Ongoing Challenges

Despite the progress documented in this volume, several important gaps remain open for future exploration:

Unified methodologies for stability. Research is still fragmented across deterministic, stochastic, and dynamic frameworks. A coherent theory connecting classical Ulam–Hyers stability with probabilistic or evolutionary environments is greatly needed.

Complex geometries and functional spaces. While progress has been made, general theories for multivariable problems on irregular domains, manifolds, or infinite-dimensional spaces are still developing.

Stronger bridges to applications. The use of functional equation models in economics, engineering, physics, and data-driven sciences remains selective. Wider adoption requires additional numerical, computational, and empirical tools.

Computational approaches. As symbolic manipulation software and automated reasoning evolve, their potential for discovering, verifying, or approximating solutions to functional equations is far from fully utilized.

These challenges should not be viewed as limitations but rather as exciting invitations for research advancement.

4. Future Directions

Looking ahead, several promising avenues emerge from the insights collected in this Special Issue:

Integrated stability frameworks. Future work may focus on embedding stability concepts into stochastic, hybrid, or delay-driven models, enabling stronger theoretical consistency and practical impact.

Extended symmetry considerations. Symmetry properties in functional equations—involving invariance, periodicity, or group actions—can lead to more general solution classifications and versatile modeling tools.

Computational and algorithmic advancements. Development of systematic computational techniques, perhaps aided by symbolic AI tools, may accelerate the discovery of new solution forms and test conjectures.

Applications beyond traditional domains. Functional equations are increasingly relevant for modeling human behavior, optimization processes, network dynamics, and uncertainty quantification. Expanding these collaborations will strengthen both theory and practice.

Community building and interdisciplinary exchange. The diversity of works here shows that progress thrives where mathematical traditions meet. Continued collaboration among analysts, applied mathematicians, and computational researchers will drive the next breakthroughs.

5. Acknowledgments and Closing Remarks

We extend our heartfelt gratitude to all contributing authors for sharing their innovative work, and to the reviewers whose careful assessments ensured scientific quality. We also thank the editorial staff of Symmetry for their dedicated support throughout the process.

This Special Issue provides not only a snapshot of current advances but also a roadmap for the future. We hope it inspires continued exploration of functional equations and inequalities and motivates wide-ranging investigations into their symmetry-driven structures and growing applications.

We look forward to seeing how the next generation of researchers will build upon the foundations presented here—extending the boundaries of mathematical knowledge and enriching its contributions to science and society.