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1
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Epsilon Nielsen coincidence theory

100%
Fenille M.
Open Mathematics
|
2014
|
tom 12
|
nr 9
1337-1348
EN
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.
2
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Strong surjectivity of maps from 2-complexes into the 2-sphere

100%
Fenille M., Neto O.
Open Mathematics
|
2010
|
tom 8
|
nr 3
421-429
EN
Given a model 2-complex K P of a group presentation P, we associate to it an integer matrix ΔP and we prove that a cellular map f: K P → S 2 is root free (is not strongly surjective) if and only if the diophantine linear system ΔP Y = $$ \overrightarrow {deg} $$(f) has an integer solution, here $$ \overrightarrow {deg} $$(f)is the so-called vector-degree of f
3
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Strong surjectivity of mappings of some 3-complexes into $$ M_{Q_8 } $$

76%
Aniz C.
Open Mathematics
|
2008
|
tom 6
|
nr 4
497-503
EN
Let K be a CW-complex of dimension 3 such that H 3(K;ℤ) = 0 and $$ M_{Q_8 } $$ the orbit space of the 3-sphere $$ \mathbb{S}^3 $$ with respect to the action of the quaternion group Q 8 determined by the inclusion Q 8 ⊆ $$ \mathbb{S}^3 $$. Given a point a ∈ $$ M_{Q_8 } $$, we show that there is no map f:K → $$ M_{Q_8 } $$ which is strongly surjective, i.e., such that MR[f,a]=min{#(g −1(a))|g ∈ [f]} ≠ 0.
4
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Minimization of the number of periodic points for smooth self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers

76%
Graff G., Kaczkowska A.
Open Mathematics
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2012
|
tom 10
|
nr 6
2160-2172
EN
Let f be a smooth self-map of m-dimensional, m ≥ 4, smooth closed connected and simply-connected manifold, r a fixed natural number. For the class of maps with periodic sequence of Lefschetz numbers of iterations the authors introduced in [Graff G., Kaczkowska A., Reducing the number of periodic points in smooth homotopy class of self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers, Ann. Polon. Math. (in press)] the topological invariant J[f] which is equal to the minimal number of periodic points with the periods less or equal to r in the smooth homotopy class of f. In this paper the invariant J[f] is computed for self-maps of 4-manifold M with dimH 2(M; ℚ) ≤ 4 and estimated for other types of manifolds. We also use J[f] to compare minimization of the number of periodic points in smooth and in continuous categories.
5
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An Ulam stability result on quasi-b-metric-like spaces

64%
Alsulami H. H., Gülyaz S., Karapınar E., Erhan İ. M.
Open Mathematics
|
2016
|
tom 14
|
nr 1
1087-1103
EN
In this paper a class of general type α-admissible contraction mappings on quasi-b-metric-like spaces are defined. Existence and uniqueness of fixed points for this class of mappings is discussed and the results are applied to Ulam stability problems. Various consequences of the main results are obtained and illustrative examples are presented.
6
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A cohomological index of Fuller type for parameterized set-valued maps in normed spaces

52%
Skiba R.
Open Mathematics
|
2014
|
tom 12
|
nr 8
1164-1197
EN
We construct a cohomological index of the Fuller type for set-valued flows in normed linear spaces satisfying the properties of existence, excision, additivity, homotopy and topological invariance. In particular, the constructed index detects periodic orbits and stationary points of set-valued dynamical systems, i.e., those generated by differential inclusions. The basic methods to calculate the index are also presented.
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