### An axiomatic approach to a coincidence index for noncompact function pairs

#### Abstract

We prove that there is a coincidence index for the inclusion

$F(x)\in\Phi(x)$ when $\Phi$ is convex-valued and satisfies certain

compactness assumptions on countable sets. For $F$ we assume only that it

provides a coincidence index for single-valued finite-dimensional maps

(e.g. $F$ is a Vietoris map). For the special case $F={\rm id}$, the obtained

fixed point index is defined if $\Phi$ is countably condensing;

the assumptions in this case are even weaker than in

[M. Väth, < i> Fixed point theorems and fixed point index for countably condensing maps< /i> ,

Topol. Methods Nonlinear Anal. < b> 13< /b> (1999), 341–363].

$F(x)\in\Phi(x)$ when $\Phi$ is convex-valued and satisfies certain

compactness assumptions on countable sets. For $F$ we assume only that it

provides a coincidence index for single-valued finite-dimensional maps

(e.g. $F$ is a Vietoris map). For the special case $F={\rm id}$, the obtained

fixed point index is defined if $\Phi$ is countably condensing;

the assumptions in this case are even weaker than in

[M. Väth, < i> Fixed point theorems and fixed point index for countably condensing maps< /i> ,

Topol. Methods Nonlinear Anal. < b> 13< /b> (1999), 341–363].

#### Keywords

Degree theory; fixed point index; coincidence index; condensing operator; measure of noncompactness

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