# Epsilon Nielsen coincidence theory

Open Mathematics (2014)

- Volume: 12, Issue: 9, page 1337-1348
- ISSN: 2391-5455

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topMarcio Fenille. "Epsilon Nielsen coincidence theory." Open Mathematics 12.9 (2014): 1337-1348. <http://eudml.org/doc/269804>.

@article{MarcioFenille2014,

abstract = {We construct an epsilon coincidence theory which generalizes, in some aspect, the epsilon fixed point theory proposed by Robert Brown in 2006. Given two maps f, g: X → Y from a well-behaved topological space into a metric space, we define µ ∈(f, g) to be the minimum number of coincidence points of any maps f 1 and g 1 such that f 1 is ∈ 1-homotopic to f, g 1 is ∈ 2-homotopic to g and ∈ 1 + ∈ 2 < ∈. We prove that if Y is a closed Riemannian manifold, then it is possible to attain µ ∈(f, g) moving only one rather than both of the maps. In particular, if X = Y is a closed Riemannian manifold and idY is its identity map, then µ ∈(f, idY) is equal to the ∈-minimum fixed point number of f defined by Brown. If X and Y are orientable closed Riemannian manifolds of the same dimension, we define an ∈-Nielsen coincidence number N ∈(f, g) as a lower bound for µ ∈(f, g). Our constructions and main results lead to an epsilon root theory and we prove a Minimum Theorem in this special approach.},

author = {Marcio Fenille},

journal = {Open Mathematics},

keywords = {Nielsen number; Riemannian manifold; Epsilon homotopy; Epsilon coincidence; Minimum Theorem; epsilon homotopy; epsilon coincidence; minimum theorem},

language = {eng},

number = {9},

pages = {1337-1348},

title = {Epsilon Nielsen coincidence theory},

url = {http://eudml.org/doc/269804},

volume = {12},

year = {2014},

}

TY - JOUR

AU - Marcio Fenille

TI - Epsilon Nielsen coincidence theory

JO - Open Mathematics

PY - 2014

VL - 12

IS - 9

SP - 1337

EP - 1348

AB - We construct an epsilon coincidence theory which generalizes, in some aspect, the epsilon fixed point theory proposed by Robert Brown in 2006. Given two maps f, g: X → Y from a well-behaved topological space into a metric space, we define µ ∈(f, g) to be the minimum number of coincidence points of any maps f 1 and g 1 such that f 1 is ∈ 1-homotopic to f, g 1 is ∈ 2-homotopic to g and ∈ 1 + ∈ 2 < ∈. We prove that if Y is a closed Riemannian manifold, then it is possible to attain µ ∈(f, g) moving only one rather than both of the maps. In particular, if X = Y is a closed Riemannian manifold and idY is its identity map, then µ ∈(f, idY) is equal to the ∈-minimum fixed point number of f defined by Brown. If X and Y are orientable closed Riemannian manifolds of the same dimension, we define an ∈-Nielsen coincidence number N ∈(f, g) as a lower bound for µ ∈(f, g). Our constructions and main results lead to an epsilon root theory and we prove a Minimum Theorem in this special approach.

LA - eng

KW - Nielsen number; Riemannian manifold; Epsilon homotopy; Epsilon coincidence; Minimum Theorem; epsilon homotopy; epsilon coincidence; minimum theorem

UR - http://eudml.org/doc/269804

ER -

## References

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- [6] Gonçalves D.L., Coincidence theory, In: Handbook of Topological Fixed Point Theory, Springer, Dordrecht, 2005, 3–42 http://dx.doi.org/10.1007/1-4020-3222-6_1 Zbl1081.55002
- [7] Milnor J., Morse Theory, Ann. of Math. Stud., 51, Princeton University Press, Princeton, 1963
- [8] Munkres J.R., Topology, 2nd ed., Prentice-Hall, Englewood Cliffs, 2000
- [9] Vick J.W., Homology Theory, 2nd ed., Grad. Texts in Math., 145, Springer, New York, 1994 http://dx.doi.org/10.1007/978-1-4612-0881-5

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