### Topological index for condensing maps on Finsler manifolds with applications to functional-differential equations of neutral type

DOI: http://dx.doi.org/10.12775/TMNA.2005.033

#### Abstract

The topological index for maps of infinite-dimensional Finsler manifolds,

condensing with respect to internal Kuratowski's measure of non-compactness,

is constructed under the hypothesis that the manifold can be embedded into

a certain Banach linear space as a neighbourhood retract so that the Finsler

norm in tangent spaces and the restriction of the norm from enveloping space

on the tangent spaces are equivalent. It is shown that the index is an

internal topological characteristic, i.e. it does not depend on the choice

of enveloping space, embedding, etc. The total index (Lefschetz number) and

the Nielsen number are also introduced. The developed machinery is applied

to investigation of functional-differential equations of neutral type on

Riemannian manifolds. A certain existence and uniqueness theorem is proved.

It is shown that the shift operator, acting in the manifold of $C^1$-curves,

is condensing, its total index is calculated to be equal to the Euler

characteristic of (compact) finite-dimensional Riemannian manifold where

the equation is given. Some examples of calculating the Nielsen number are

also considered.

condensing with respect to internal Kuratowski's measure of non-compactness,

is constructed under the hypothesis that the manifold can be embedded into

a certain Banach linear space as a neighbourhood retract so that the Finsler

norm in tangent spaces and the restriction of the norm from enveloping space

on the tangent spaces are equivalent. It is shown that the index is an

internal topological characteristic, i.e. it does not depend on the choice

of enveloping space, embedding, etc. The total index (Lefschetz number) and

the Nielsen number are also introduced. The developed machinery is applied

to investigation of functional-differential equations of neutral type on

Riemannian manifolds. A certain existence and uniqueness theorem is proved.

It is shown that the shift operator, acting in the manifold of $C^1$-curves,

is condensing, its total index is calculated to be equal to the Euler

characteristic of (compact) finite-dimensional Riemannian manifold where

the equation is given. Some examples of calculating the Nielsen number are

also considered.

#### Keywords

Topological index; condensing maps; Finsler manifolds; functional-differential equations of neutral type on manifolds

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