Fixed Point Theorems for e2-Contractions Maps
DOI:
https://doi.org/10.66147/lnaa.20264280Keywords:
Fixed point theorems, $e_2$-contraction, metric spaceAbstract
We introduce and study a new class of contractive maps on metric spaces defined by a contraction condition on the second elementary symmetric polynomial of the three pairwise distances of any triple of distinct points. A map satisfying this condition is called an e2-contraction; the corresponding strict version is a strict e2-contraction. For complete metric spaces containing at least three points, every e2-contraction is continuous and has a fixed point if and only if it admits no periodic point of prime period 2, with at most two fixed points in total. For arbitrary metric spaces containing at least three points, a strict e2-contraction with no period-2 point for which some orbit admits a convergent subsequence necessarily has a fixed point. As consequences, we recover the classical Banach and Edelstein fixed point theorems, and show that on spaces where every point is an accumulation point every e2-contraction is a Banach contraction. Explicit finite examples show that e2-contractions (resp. strict e2-contractions) strictly extend the Banach (resp. Edelstein) and perimeter-contracting classes of [11] (resp. [2]).
References
S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math., 3 (1922), 133-181. DOI: https://doi.org/10.4064/fm-3-1-133-181
C. Bey, E. Petrov, E. and R. Salimov, On three-point generalisations of Banach and Edelstein fixed point theorems. Filomat, 39 (1) (2025), 185-195. DOI: https://doi.org/10.2298/FIL2501185B
D.W. Boyd and J.S.W. Wong, On nonlinear contractions. Proc. Am. Math. Soc., 20 (1969), 458--464. DOI: https://doi.org/10.1090/S0002-9939-1969-0239559-9
A. Branciari, A fixed point theorem of Banach--Caccioppoli type on a class of generalised metric spaces. Publ. Math. Debrecen 57 (1--2) (2000), 31-37. DOI: https://doi.org/10.5486/PMD.2000.2133
Lj.B. Ciric, A generalisation of Banach's contraction principle. Proc. Amer. Math. Soc. 45(2) (1974), 267-273. DOI: https://doi.org/10.1090/S0002-9939-1974-0356011-2
M. Edelstein, On fixed and periodic points under contractive mappings. J. Lond. Math. Soc., 37 (1962), 74--79. DOI: https://doi.org/10.1112/jlms/s1-37.1.74
E. Karapinar, On the novelty of "Contracting perimeters of triangles in metric space". Results in Nonlinear Anal., 8 (2025), 115-123. DOI: https://doi.org/10.31838/rna/2025.08.01.010
W.A. Kirk, Fixed points of asymptotic contractions. J. Math. Anal. Appl., 277 (2) (2003), 645-650. DOI: https://doi.org/10.1016/S0022-247X(02)00612-1
A. Meir and E. Keeler, A theorem on contraction mappings. J. Math. Anal. Appl., 28 (1969), 326-329. DOI: https://doi.org/10.1016/0022-247X(69)90031-6
S.B. Jr. Nadler, Multi-valued contraction mappings. Pacific J. Math., 30 (1969), 475-488. DOI: https://doi.org/10.2140/pjm.1969.30.475
E. Petrov, Fixed point theorem for mappings contracting perimeters of triangles. J. Fixed Point Theory Appl., 25 (2023), 1-11. DOI: https://doi.org/10.1007/s11784-023-01078-4
E. Petrov, Periodic points of mappings contracting total pairwise distances. J. Fixed Point Theory Appl., 27 (2) (2025), 1-11. DOI: https://doi.org/10.1007/s11784-025-01190-7
D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl., 94 (2012), 1-6. DOI: https://doi.org/10.1186/1687-1812-2012-94
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Copyright (c) 2026 Maher Berzig
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