Asymptotic linearity and Hadamard differentiability

For a mapping between Banach spaces, two weaker variants of the usual notion of asymptotic linearity are defined and explored. It is shown that, under inversion through the unit sphere, they correspond to Hadamard and weak Hadamard differentiability at the origin of the inversion. Nemytskii operators from Sobolev spaces to Lebesgue spaces over RNRN share these weaker properties but they are not asymptotically linear in the usual sense.