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

Elena V. Bogacheva, Yuri E. Gliklikh

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.

Keywords


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

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