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In this paper, we show a series of abstract results on fixed point regularitywith respect to a parameter. They are based on a Taylor development taking intoaccount a loss of regularity phenomenon, typically occurring for compositionoperators acting on spaces of functions with finite regularity. We generalizethis approach to higher order differentiability, through the notion of ann-graded family.  We then give applications to the fixed point of a non linear map, and tolinear response in the context of (uniformly) expanding dynamics (theorem 3 andcorollary 2), in the spirit of Gou\"ezel-Liverani weak spectral perturbationtheorem.

A regularity result for fixed points, with applications to linear response