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

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.