Semi-Hyperbolic Mappings, Condensing Operators, and Neutral Delay Equations

Semi-hyperbolic mappings in Banach spaces are Lipschitz continuous and not necessarily invertible. Like hyperbolic mappings, they involve a splitting into stable and unstable spaces, but a slight leakage from the strict invariance of the spaces is possible and the unstable subspaces are assumed to be finite dimensional. It is shown that semi-hyperbolic mappings are locallyψ-contracting, whereψis the Hausdorff measure of noncompactness, and that a linear operator is semi-hyperbolic if and only if it isψ-contracting and has no spectral values on the unit circle. A bishadowing result, which combines both direct and indirect forms of shadowing, is extended to semi-hyperbolic mappings in Banach spaces with locally condensing continuous comparison mappings. The result is applied to linear neutral delay equations with nonsmooth perturbations.