On the acyclicity of the solution sets of operator equations

A parameter-dependent completely continuous map is considered. The acyclicity of the set of fixed points of this map is proved for some fixed value of the parameter under the assumption that for close values of the parameter the map has a unique fixed point. The results obtained are used to prove the acyclicity of the set of fixed points of a ‘nonscattering’ map, as well as to study the topological structure of the set of fixed points of an abstract Volterra map.
Bibliography: 13 titles.