This paper proposes three new approaches for additive functional regression models with a functional response. The first is based on a reformulation of the linear regression model, while the last two are on the yet scarce case of additive nonlinear functional regression models. All proposals are based on extensions of similar models for scalar responses. One of the proposed nonlinear models is based on constructing a Spectral Additive Model, which is restricted to Hilbertian spaces. The other one extends the kernel estimator for functional response and more than one functional covariate. The latter can be applied to general metric spaces since it is only based on distances. The new approaches as well as real data sets are included in the developer version of R package \texttt{fda.usc} available on GitHub. The performances of the new proposals are compared with previous ones, which we review theoretically and practically in this paper. The simulation results show the advantages of the nonlinear proposals and the small loss of efficiency when the simulation scenario is truly linear. The supplementary material also provides a visualization tool for checking the linearity of the relationship between a single covariate and the response, as well as more simulation and data analysis results.

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