Triple Fixed-Point Theorems: Generalized Analysis with AI/Crypto Applications

This study uses important findings from Hilbert Space (HS) theory to suggest new fixed-point theorems for Three Self-Mappings (3-SMs) that apply to Generalized Inner Product Spaces (GIPS). We made it apparent when Common Fixed Points (CFPs) can exist and be unique, even when the contraction criteria aren’t as strict. This improvement makes the theory better for systems with more than one operator. Our results are important in two main areas: (1) Artificial Intelligence (AI), where we use fixed-point analysis to make sure that Deep Equilibrium Models (DEMs) will work correctly, and (2) Cryptography (Crypto), where we use the mathematical properties of inner product spaces to make new lattice-based designs for Post-Quantum Protocols (PQPs). Real-world examples from nonlinear analysis and computational proof of the presented theorems support the theoretical contributions. This work combines advanced functional analysis with modern problems in Machine Learning (ML) and cybersecurity, giving it both mathematical depth and real-world usefulness.