  This paper proposes a new nonlinear approach for additive functional
regression with functional response based on kernel methods along with some
slight reformulation and implementation of the linear regression and the
spectral additive model. The latter methods have in common that the covariates
and the response are represented in a basis and so, can only be applied when
the response and the covariates belong to a Hilbert space, while the proposed
method only uses the distances among data and thus can be applied to those
situations where any of the covariates or the response is not Hilbertian
typically normed or even metric spaces with a real vector structure. A
comparison of these methods with other procedures readily available in R is
performed in a simulation study and in real datasets showing the results the
advantages of the nonlinear proposals and the small loss of efficiency when the
simulation scenario is truly linear. The comparison is done in the Hilbertian
case as it is the only scenario where all the procedures can be compared.
Finally, the supplementary material provides a visualization tool for checking
the linearity of the relationship between a single covariate and the response
and a link to a GitHub repository where the code and data are available.} %and
an example considering that the response is not Hilbertian.
