In this paper, 2-microlocal Herz type Besov and Triebel-Lizorkin spaces with
variable exponents are introduced for the first time. Then, we give characterizations of
these spaces by so-called Peetre's maximal functions. Further, the atomic and molecular decompositions of these spaces are obtained. Finally, using the characterizations of
the spaces by local means and molecular decomposition we obtain the wavelet characterizations.
