  We develop a strong-coupling theory of Bose-Einstein condensate-mediated
superconductivity in a hybrid system, which consists of a two-dimensional
electron gas with either (i) parabolic spectrum or (ii) relativistic Dirac
spectrum in the vicinity of a two-dimensional solid-state condensate of
indirect excitons. The Eliashberg equations are derived and the expressions for
the electron pairing self-energy due to the exchange interaction between
electrons mediated by a single Bogoliubov excitation (a bogolon) and the
bogolon pairs are found. Furthermore, we find the superconducting order
parameter and estimate the critical temperature of the superconducting
transition. The critical temperature reveals its linear dependence on the
dimensionless coupling constant. It is shown, that the bogolon-pair-mediated
interaction is the dominant mechanism of electron pairing in hybrid systems in
both the weak and strong coupling regimes. We calculate the effective
bogolon-electron interaction constant for both parabolic and linear electron
dispersions and examine the dependence of the critical temperature of electron
gas superconducting transition on exciton condensate density.
