IJLTEMAS International Journal of Latest Technology in Engineering, Management & Applied Science (IJLTEMAS) IJLTEMAS 2278-2540 IJLTEMAS 487 10.51583/IJLTEMAS.2026.150900136 Functional Analysis A Comparative Analysis of Fixed Point Results in Banach and Hilbert Spaces Raja Ram Singh Dr. Assistant Professor, Department of Mathematics, S. D. College Kaler, Arwal, Magadh University, Bodh-Gaya, Bihar, India India 15 9 2760 2768 23 09 2026 28 09 2026 21 10 2026 Normed space Banach space Hilbert space inner product unique fixed point contraction mapping non-expansive mapping Abstract

The main objective of this paper is to use contraction mapping and non-expansive operators for unique fixed point results in real Banach and Hilbert spaces. A fundamental role of the fixed point theory plays in optimization and nonlinear functional analysis. This work is a generalization of the unique fixed point and its discussion. We have collected different ideas for unique fixed points in real Banach and Hilbert spaces. Also, we formulate and compare fixed point conditions involving contractive inequalities, the role of completeness, nom structure,inner product geometry, lower semicontinuous auxiliary function and Hausdorff distance for set valuede maps. Finally, we have conducted a comparative study and analysis of unique fixed points in Banach and Hilbert spaces. New comparative study, observation, differences analysis and implementation of these structure features are also provided.

 References 1.Banach,S.,(1922).Sur les Operation Dansles Ensembles Abstracts Set leur. Applications Equation Integrals Fundamental Math.V. 3, PP.133-181. Kannan,R.,(1968). Some Results on Fixed-point,Bull. Calcutta.Math. Soc.V.10, PP.71-76. Meir,A.and Keeler,E.,(1969).Theorem on Contraction Mapping. Journal of Mathematical Analysis and Application.V. 28. PP. 326−329. Chatterjea S. K., (1972). Fixed point Theorem .C R. Acad. Bulg Sci.V.25, PP.727-730, Ekeland, I., (1979). Non Convex Minimization Problems, Bulletin of the American Math. Society.V.1. PP.443−474 Reich, S., (1983). Some Problems and Results in Fixed Point Theory Contemporary Mathematics. V. 21. PP.179−187. Mustafa, and Sims, B., (2009). Fixed-point Theorem for Contractive Mapping in Complete G-Metric space, Fixed-point Theory and Application, Article no. 917175 Farshid, K., Erdal, K., and Hassan, K., (2016). Some applications of Carist are Fixed points. Dhage, B.C., (19992). Generalized Metric Space and Mapping with Fixed-points. Bull. Cal. Math. Soc. 84, PP. 329−336. Sedge, S. Shobe, N. and Zhou, H. Y.,(2007). A Common Fixed-point Theorem In Metric Space. Journal of Fixed-point Theory and Application, PP.1−13. T. Suzuki, T.,(2008). Fixed-point Theorem and Convergence Theorem for Some Generalized Nonexpansive Mapping, j. Math. Ana. Appl.340, PP.1088-1085 Ciric L .B.(1993). On Some Non-Expansive Type Mapping and Fixed-points, Indian, J. Pure Appl. Math. V.3. , PP. 145-149 Bose, S. C. (1978). Common Fixed-point of Mapping in Uniformly Convex Spaces. J. Landan, Math. Soc.V.18. PP.151-156. Reach, S., and kr. P. k.,(1979). Weak Convergence Theorem for Non–Expansive Mapping in Banach Space, j. , Math. Anal. Application. V. 67.PP. 274-276. Mannan, A. M., and Hossain, M.,(2021). A Study of Caristi’s Fixed Point Theorem in Normed Space and its Applications. Advanced in Pure Mathematics,V.11, PP.169-179 16.Goebel, K., and Zostkiehlick, E.,(1971).Some Fixed-point Theorems in Banach Space. Coll. Math.V- 23, PP.103-106. Belluce, L.P., and Kirk, W.A., (1969).Nonexpansive Mapping and Fixed Point in Banach Spaces. Illinois. J.Math. V.11.PP. 474-479. 18.Zeidler,E.,(1986).Nonlinear Functional Anzlysis and its Application: Nonlinear Monotone Operator. Springer-verlag.