A History of Contraction Principles
Abstract views: 65
/ 
PDF downloads: 32
Keywords:
fixed point, quasi-metric, Rus-Hicks-Rhoades (RHR) map, T-orbitally complete.Abstract
Recently we obtained several extensions of the Banach contraction principle, which appears frequently in many literature. From our 2023 Metatheorem, we deduce Theorem H on the equivalent formulations of completeness of quasi-metric spaces. From Theorem H, we derive the Banach contraction principle, its extended form (Theorem Q), the Rus-Hicks-Rhoades contraction principle (Theorem P), and others. Consequently, our Theorem H contains wellknown theorems of Banach, Covitz-Nadler, Oettli-Thera, Rus-Hicks-Rhoades, and some others.
References
S. Park, Foundations of ordered fixed point theory, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 61(2) (2022) 1–51.
S. Park, History of the metatheorem in ordered fixed point theory, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 62 (2023) 373–410.
S. Park, Some metric fixed point theorems hold for quasi-metric spaces, Results in Nonlinear Analysis 6 (2023) 116–127.
I.A. Rus, Teoria punctului fix, II, Univ. Babes-Bolyai, Cluj, 1973.
T.L. Hicks, B.E. Rhoades, A Banach type fixed point theorem, Math. Japon. 24 (1979) 327–330.
S.B. Nadler, Jr. Multi-valued contraction mappings, Pacific J. Math. 30 (1969) 475–488.
H. Covitz, S.B. Nadler, Jr. Multi-valued contraction mappings in generalized metric spaces, Israel J. Math. 8 (1970) 5–11.
S. Park, The realm of the Rus-Hicks-Rhoades maps in the metric fixed point theory, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 63(1) (2024) 1–50.
S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales. Fund. Math. 3 (1922) 133–181.
V. Berinde, A. Petrusel, J.A. Rus, Remarks on the terminology of the mappings in fixed point iterative methods in metric spaces, Fixed Point Theory 24(2) (2023) 525–540. https://doi.org/10.24193/fpt-ro.2023.2.05
S. Park, Relatives of a theorem of Rus-Hicks-Rhoades, Letters Nonlinear Anal. Appl. 1 (2023) 57–63. ISSN 2985-874X
S. Park, Almost all about Rus-Hicks-Rhoades maps in quasi-metric spaces, Adv. Th. Nonlinear Anal. Appl. 7(2) (2023) 455–471. https://doi.org/10.31197/atnaa.1185449
S. Park, The use of quasi-metric in the metric fixed point theory, J. Nonlinear Convex Anal. 25(7) (2024) 1553–1564.
S. Park, Improving many metric fixed point theorems, Letters Nonlinear Anal. Appl. 2(2) (2024) 35–61.
H. Aydi, M. Jellali, E. Karapinar, On fixed point results for α-implicit contractions in quasi-metric spaces and consequences, Nonlinear Anal. Model. Control, 21(1) (2016) 40–56.
M. Jleli, B. Samet, Remarks on G-metric spaces and fixed point theorems, Fixed Point Theory Appl. (2012) 2012:210.
S. Park, Remarks on the metatheorem in ordered fixed point theory, Advanced Mathematical Analysis and Its Applications (Edited by P. Debnath, D.F.M. Torres, Y.J. Cho) CRC Press (2023), 11–27. https://doi.org/10.1201/9781003388678-2
W. Oettli, M. Th´era, Equivalents of Ekeland’s principle, Bull. Austral. Math. Soc. 48 (1993) 385–392.
V.I. Istr˘a¸tescu, Fixed Point Theory: An Introduction, D. Reidel Publ. Co., Dortrecht, Holland, 1981.
R. Cacciopoli, Una teorem generale sull’esistenza di elementi uniti in una transformazione funzionale, Rend. Accad. Nazion. Lincei 6(11) (1930) 794–799.
F.F. Bonsall, Lectures on some fixed point theorems of functional analysis, Tata Institute of Fundamental Research, Bombay, 1962.
W.A. Kirk, Contraction mappings and extensions. Chapter 1, Handbook of Metric Fixed Point Theory (W.A. Kirk and
B. Sims, eds.), Kluwer Academic Publ. (2001) 1–34.
J.T. Markin, A fixed point theorem for set-valued mappings. Bull. Amer. Math. Soc. 74 (1968) 639–640.
S. Park, Several recent episodes on the metric completeness, Mathematical Analysis: Theory and Applications (Editors:
P. Debnath, H. M. Srivastava, D. F. M. Torres, and Y. J. Cho), CRC Press. (2025) 15–34.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2025 Results in Nonlinear Analysis
This work is licensed under a Creative Commons Attribution 4.0 International License.
