Let $p>2$ be a prime and let $X$ be a compactified PEL Shimura variety of type (A) or (C) such that $p$ is an unramified prime for the PEL datum. Using the geometric approach of Andreatta, Iovita, Pilloni, and Stevens we define the notion of families of overconvergent locally analytic $p$-adic modular forms of Iwahoric level for $X$. We show that the system of eigenvalues of any finite slope cuspidal eigenform of Iwahoric level can be deformed to a family of systems of eigenvalues living over an open subset of the weight space. To prove these results, we actually construct eigenvarieties of the expected dimension that parametrize systems of eigenvalues appearing in the space of families of cuspidal forms.
There is a problem in the normalizing factor of the Hecke operators at p, that is different from the ones in AIP15b and BSP15. This does not change the density of the set of classical points in the eigenvariety, but the explicit bound of Theorem 6.7 should be changed. (Or one can of course modify the definition of the Hecke operators accordingly to AIP15b and BSP15.)