Symmetry in Fixed-Point Theory and Optimization: Computations and Applications

Fixed-point theory and optimization constitute two of the most dynamic and interconnected branches of modern mathematical analysis. The convergence, stability, and efficiency of optimization algorithms often depend on the geometric and structural properties of the spaces in which they operate. In this context, symmetry serves as a fundamental principle for developing robust mathematical models.

This Special Issue highlights recent advancements that bridge abstract mathematical results and functional computational tools. The nine articles published reflect a wide spectrum of research, categorized into three primary thematic pillars.

1. Overview of Contributions

1.1. Theoretical Foundations in Abstract and Non-Euclidean Spaces

• Gabeleh et al. [1] explore best proximity theory in metrically convex Menger Probabilistic Metric (PM) spaces, providing existence results validated in CAT(0) spaces.
• Khan et al. [2] introduce an efficient iterative algorithm for mappings in CAT(0) spaces with applications to polynomiography.
• Moore [3] presents a novel approach to relational modeling through object-oriented geometric figures and N-Euclidean geometry.
• Yu et al. [4] examine geometric symmetry in uniformly convex Banach spaces to establish weak convergence for generalized Ishikawa iterative algorithms.

1.2. Iterative Algorithms for Optimization and Signal Processing

• Belhenniche et al. [5] investigate reinforcement learning algorithms, employing Zamfirescu’s fixed-point theorem to establish convergence in infinite-dimensional policy spaces.
• Thammasiri et al. [6] propose an inertial forward–backward–forward algorithm with a moving point projection technique for monotone inclusions and image restoration.

1.3. Fractional Modeling and Multi-Disciplinary Applications

• Alharbi [7] develops a Caputo fractional-order model for regulating blood oxygen saturation, utilizing fixed-point theory to prove the existence of optimal control strategies.
• Boonyopakorn and Ketcham [8] introduce an intelligent monitoring system for elderly care using CNN-LSTM architectures and symmetry-aware keypoint analysis.
• Sompong et al. [9] establish well-posedness and Ulam–Hyers stability for nonlinear implicit Hilfer fractional differential equations in weighted spaces.

2. Conclusions and Future Directions

The research presented in this Special Issue underscores the continuing vitality of fixed-point theory as a foundational tool for modern optimization. Based on the findings published here, several future research directions emerge:

• Physics-Informed Reinforcement Learning: Building on the convergence analysis of weak contraction mappings [5], future work could integrate fixed-point constraints directly into neural network loss functions. This ensures that AI-driven control systems remain within mathematically stable bounds while optimizing complex feedback loops.
• Personalized Physiological Modeling: The fractional-order models for oxygen regulation [7] and implicit differential equations [9] provide a roadmap for “Digital Twin” technologies. Future studies should focus on identifying patient-specific fractional orders to tailor optimal control strategies in real-time biomedical applications.
• Optimization on Manifolds and Graphs: The success of algorithms in CAT(0) and symmetric Banach spaces [2,4] suggests a logical expansion toward optimization on non-linear manifolds and large-scale graphs, which is particularly relevant for high-dimensional data clustering and the study of complex network symmetries.
• Symmetry-Aware Edge Computing: Integrating symmetry-aware pose estimation [8] into low-power edge devices presents a challenge. Developing “lightweight” fixed-point algorithms that maintain high accuracy with minimal computational overhead will be crucial for the next generation of non-intrusive monitoring systems.
• Adaptive Inertial Schemes: The moving point projection technique [6] could be further generalized to include adaptive step sizes and stochastic perturbations, enhancing its robustness in solving large-scale, ill-posed inverse problems in dynamic environments.