Neuro-quantum iterative algorithms: A hybrid convergence framework for fixed point problems in functional spaces
*Vishali KansalCorresponding authorvishalikansal88@gmail.comDepartment of MathematicsUniversity Institute of Sciences (UIS)Chandigarh UniversityGharuan, Mohali, Punjab, 140413, IndiaView full profile → , Naveen Kumarimnaveenphd@gmail.comDepartment of MathematicsUniversity Institute of Sciences (UIS)Chandigarh UniversityGharuan, Mohali, Punjab, 140413, IndiaView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 Feb 2026
- Published Online:
- 25 Aug 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JIM-2637
- Pages:
- 2509–2518
Abstract
This paper presents a ground-breaking framework—Neuro-Quantum Iterative Algorithm (NQIA)—that unifies neural meta-learning with quantum-inspired operator dynamics to solve fixed point problems in abstract functional spaces. Unlike conventional schemes limited by strict contractiveness or static update rules, NQIA introduces an adaptive, layered convergence mechanism governed by novel neuro-symbolic residual control and unitary evolution protocols. We establish original theorems that rigorously demonstrate strong convergence in Hilbert spaces, weak convergence in Banach spaces, and residual minimization stability under hybrid topological conditions. Specifically, the Layered Convergence Theorem ensures robust progression of iterates under firmly non-expansive mappings; the Optimal Ω-Selection Theorem derives the spectral-radius minimizing learning rate from eigenvalue analysis; the Residual Stability Theorem introduces a neurosymbolic energy-based correction scheme that guarantees contracting and convergence; and the Topological Residual-Contraction Theorem extends the approach to reflexive Banach spaces equipped with weak topology. This research opens novel directions in the theory and practice of fixed-point algorithms, transcending classical assumptions through hybrid dynamical learning.
Keywords
Subject Classifications
References
[1] R. P. Agarwal, M. Meehan, and D. O’Regan, Fixed Point Theory and Applications. Cambridge, U.K.: Cambridge Univ. Press (2001).
[2] É. Picard, “Sur les équations différentielles contenant une fonction non linéaire de l’inconnue,” J. Math. Pures Appl., vol. 6, pp. 217–272 (1890).
[3] W. R. Mann, “Mean value methods in iteration,” Proc. Amer. Math. Soc., vol. 4, no. 3, pp. 506–510 (1953).
[4] V. Berinde, Iterative Approximation of Fixed Points. Berlin, Germany: Springer, (2007).
[5] B. E. Rhoades, “A comparison of various definitions of contractive mappings,” Trans. Amer. Math. Soc., vol. 226, pp. 257–290 (1977).
[6] F. E. Browder, “Nonlinear operators and nonlinear equations of evolution in Banach spaces,” in Proc. Sympos. Pure Math., vol. 18. Providence, RI, USA: Amer. Math. Soc., pp. 1–308 (1967).
[7] Y. Bengio, A. Courville, and P. Vincent, “Representation learning: A review and new perspectives,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 35, no. 8, pp. 1798–1828 (Aug. 2013).
[8] M. Schuld, I. Sinayskiy, and F. Petruccione, “An introduction to quantum machine learning,” Contemp. Phys., vol. 56, no. 2, pp. 172–185 (2015).
[9] N. Mani, A. Somal, S. Beniwal, and R. Shukla, “Solving systems of linear integral equations via fixed point theory in extended parametric Sb-metric spaces,” Int. J. Differ. Equ., vol. 2025, Art. no. 1226661 (2025).
[10] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory. Cambridge, U.K.: Cambridge Univ. Press, (1990).
[11] C. Finn, P. Abbeel, and S. Levine, “Model-agnostic meta-learning for fast adaptation of deep networks,” in Proc. 34th Int. Conf. Mach. Learn., pp. 1126–1135 (2017).
[12] N. Kumar and S. S. Chauhan (Gonder), “A study of convergence behavior of fixed-point iterative processes via computer simulation,” Adv. Appl. Math. Sci., vol. 19, no. 9, pp. 943–953 (2020).
[13] Y. Chen and H. Hassani, “Learning to accelerate fixed point iterations,” in Adv. Neural Inf. Process. Syst., vol. 34 (2021).
[14] M. M. Bronstein, J. Bruna, Y. LeCun, A. Szlam, and P. Vandergheynst, “Geometric deep learning: Going beyond Euclidean data,” IEEE Signal Process. Mag., vol. 34, no. 4, pp. 18–42 (Jul. 2017).
[15] N. Kumar and S. S. Chauhan (Gonder), “Rapid convergence of new FP iterative algorithm,” Commun. Comput. Inf. Sci., vol. 2081, pp. 89–106 (2024).
[16] Y. Liu, S. Arunachalam, and K. Temme, “A rigorous and robust quantum speed-up in supervised learning,” Nat. Phys., vol. 19, no. 2, pp. 190–197 (Feb. 2023).
[17] N. Kumar and S. S. Chauhan, “Rapid convergence of new fixed point iterative algorithm,” J. Adv. Res. Dyn. Control Syst., vol. 10, no. 8, pp. 2303–2310 (2018).
[18] P. Patil, S. Kulkarni, R. Deshmukh, and A. Joshi, “Fixed-point theorems in training deep neural networks,” J. Interdiscip. Math., vol. 29, no. 3, pp. 701–708 (2026).
[19] S. S. Chauhan, P. Tyagi, and N. Mani, “Rational contraction mapping in F-modular b-metric spaces,” J. Interdiscip. Math., vol. 27, no. 8, pp. 1781–1786 (2024).
[20] M. Arjovsky, L. Bottou, I. Gulrajani, and D. Lopez-Paz, “Invariant risk minimization,” J. Mach. Learn. Res., vol. 22, no. 1, pp. 1–74 (2021).
