Repulsive fixed points of multivalued transformations and the fixed point index

1975 ◽  
Vol 218 (1) ◽  
pp. 9-18 ◽  
Author(s):  
Christian C. Fenske ◽  
Heinz-Otto Peitgen
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Fixed Point Index ◽  
Point Index

Equivariant Fixed Point Index and the Period-Doubling Cascades

1991 ◽  
Vol 43 (4) ◽  
pp. 738-747 ◽  
Author(s):  
L. H. Erbe ◽  
K. Gęba ◽  
W. Krawcewicz
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Fixed Point Index ◽  
Period Doubling ◽  
Nonlinear Problems ◽  
Point Index ◽  
Homotopy Classes ◽  
Equivariant Maps

Properties of fixed points of equivariant maps have been studied by several authors including A. Dold (cf. [2], 1982), H. Ulrich (cf. [9], 1988), A. Marzantowicz (cf. [7], 1975) and others. Closely related is the work of R. Rubinsztein (cf. [8], 1976) in which he investigated homotopy classes of equivariant maps between spheres. There have been many attempts to introduce and effectively apply these concepts to nonlinear problems. In particular we mention the work of E. Dancer (cf. [1], 1982) in which some applications to nonlinear problems are given.

scholarly journals Fixed points of local actions of nilpotent Lie groups on surfaces

2016 ◽  
Vol 37 (4) ◽  
pp. 1238-1252 ◽  
Author(s):  
MORRIS W. HIRSCH
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Lie Group ◽  
Fixed Point Index ◽  
Nilpotent Lie Groups ◽  
Compact Set ◽  
Real Surface ◽  
Nilpotent Lie Group ◽  
Point Index ◽  
Local Flow

Let$G$be a connected nilpotent Lie group with a continuous local action on a real surface$M$, which might be non-compact or have non-empty boundary$\unicode[STIX]{x2202}M$. The action need not be smooth. Let$\unicode[STIX]{x1D711}$be the local flow on$M$induced by the action of some one-parameter subgroup. Assume$K$is a compact set of fixed points of$\unicode[STIX]{x1D711}$and$U$is a neighborhood of$K$containing no other fixed points.Theorem.If the Dold fixed-point index of$\unicode[STIX]{x1D711}_{t}|U$is non-zero for sufficiently small$t>0$,then$\mathsf{Fix}(G)\cap K\neq \varnothing$.

A bound on the fixed-point index for surface mappings

1997 ◽  
Vol 17 (6) ◽  
pp. 1393-1408 ◽  
Author(s):  
MICHAEL R. KELLY
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Homotopy Class ◽  
Fixed Point Index ◽  
Type Inequality ◽  
Point Index

In this paper we consider the class of surface mappings consisting of those maps which have the least number of fixed points possible among all maps in their homotopy class. When the surface has non-empty boundary, we show that for mappings in this class the index of a fixed point is bounded above by $1$ and below by $2\chi -1$. This generalizes a well known result for pseudo-Anosov homeomorphisms. A proof of a Jiang–Guo type inequality is also given.

scholarly journals Nontrivial Solutions of Systems of Perturbed Hammerstein Integral Equations with Functional Terms

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 330
Author(s):  
Gennaro Infante
Keyword(s):  
Fixed Point ◽  
Boundary Conditions ◽  
Integral Equations ◽  
General Class ◽  
Fixed Point Index ◽  
Nontrivial Solutions ◽  
Point Index ◽  
Hammerstein Integral Equations ◽  
Functional Boundary ◽  
Functional Boundary Conditions

We discuss the solvability of a fairly general class of systems of perturbed Hammerstein integral equations with functional terms that depend on several parameters. The nonlinearities and the functionals are allowed to depend on the components of the system and their derivatives. The results are applicable to systems of nonlocal second order ordinary differential equations subject to functional boundary conditions, this is illustrated in an example. Our approach is based on the classical fixed point index.

scholarly journals Fixed point index and decompositions of isolated invariant compacta

2004 ◽  
Vol 141 (1-3) ◽  
pp. 207-223
Author(s):  
Francisco R. Ruiz del Portal ◽  
José M. Salazar
Keyword(s):  
Fixed Point ◽  
Fixed Point Index ◽  
Point Index
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