  Limits on $\beta^+$EC (here EC denotes electron capture) and ECEC processes
in $^{112}$Sn have been obtained using a 380 cm$^3$ HPGe detector and an
external source consisting of 53.355 g enriched tin (94.32% of $^{112}$Sn). A
limit with 90% C.L. on the $^{112}$Sn half-life of $4.7\times 10^{20}$ y for
the ECEC(0$\nu$) transition to the $0^+_3$ excited state in $^{112}$Cd (1871.0
keV) has been established. This transition is discussed in the context of a
possible enhancement of the decay rate by several orders of magnitude given
that the ECEC$(0\nu)$ process is nearly degenerate with an excited state in the
daughter nuclide. Prospects for investigating such a process in future
experiments are discussed. The limits on other $\beta^+$EC and ECEC processes
in $^{112}$Sn were obtained on the level of $(0.6-8.7)\times 10^{20}$ y at the
90% C.L.
